geosphere/0000755000176200001440000000000014335012510012225 5ustar liggesusersgeosphere/NAMESPACE0000644000176200001440000000014314161531052013446 0ustar liggesusersuseDynLib(geosphere, .registration = TRUE) import(Rcpp) import(sp) exportPattern("^[^\\.\\_]") geosphere/ChangeLog0000644000176200001440000000605013472155746014024 0ustar liggesusers-- to do inconsistency reported by Ryan Case: bearing is expressed as -180 to 180 whereas bearingRhum is from 0 to 360 -- Current version dist* functions threw an error when using a two row matrix for p1, and missing p2. Fixed by setting "drop=FALSE". Bug report and fix by Miles Wood. -- 2 November 2018, version 1.5.7 distHaversine bugged in some cases for antipodes: https://stackoverflow.com/questions/45889616/why-does-disthaversine-return-nan-for-some-pairs-of-coordinates# dist2gc always returned a positive value (the manual said it should be negative if it was at the left of the circle); in contrast with what the manual stated. Getting the sign is now an option. Reported by Charles Mohan, Hughan Ross and David Cooley dist2gc used the bearing for an ellipsoid, not for a spheroid. Reported by Greg Whittier -- 15 June 2016, version 1.5-5 Fixed error in gcMaxLat (reported by William Smith) Bug fix in antipodal (reported by Gareth Davies) Bug fix in gcIntermediate when crossing the date line -- 7 July 2015, version 1.4-3 Included C-version of GeographicLib by C.F.F. Karney. areaPolygon and perimeter are now using that to compute these on a (WGS84) ellipsoid. New function distGeo (distance on an ellipsoid) also uses this. -- 17 January 2014, version 1.3-8 Fixed bug in destPointRhumb. Reported by Bart Kranstauber Fixed bug in bearingRhumb when using multiple points (incorrect results were returned). Reported by Bart Kranstauber Added 'ID' output to dist2Line distVincentyEllipsoid returned a distance of 0 between NA and NA; now it returns NA as intended (reported by Sebastian Luque) Fixed minor bug in distMeeus (using 'isTRUE' where 'which' is appropriate) (reported by Tyler Smith) -- 4 March 2012, version 1.2-27 Bug fix in antipodal; reported by Bart Kranstauber Fixed bug in distMeeus. It returned NaN when the distance between two points was 0. Reportedby Daniel Reidpath New function geomean to compute the mean of a set of coordinates (optionally with weights) -- 28 August 2011, version 1.2-25 Correction to gcIntermediate. Points were not exactly equidistant. Error spotted by Jason Davies. -- 18 May 2011, version 1.2-21 Normalizing longitude to -180..180 in midPoint. Suggested by Aaron Hardin -- 13 Jan 2011, version 1.2-18 New argument sepNA in gcIntermediate, suggested by Lee Butterman -- 1 Jan 2011, version 1.2-17 Merged functions distm and distm2 into distm Function with new distance algorithm "distMeeus" -- 4 Nov 2010 Additional options to gcIntermediate and greatCircle to return SpatialLines (that are cut in two when crossing the date line) -- 8 Sept 2010 bug fix in .pointsToMatrix (reported by Joe) -- 26 August 2010 New functions: dist2Line computes the (spherical) distance between points and polylines (and polygons borders). makeLine (like makePoly), adds intermediate great circle points to line segments, for better plotting onGreatCircle tests if points are on a specified great circle -- 25 August 2010 distVincentyEllipsoid now handles NA values. 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   *     |f|      error
   *     0.01     25 nm
   *     0.02     30 nm
   *     0.05     10 um
   *     0.1     1.5 mm
   *     0.2     300 mm
   *
* For very eccentric ellipsoids, use GeodesicExact instead. * * The algorithms are described in * - C. F. F. Karney, * * Algorithms for geodesics, * J. Geodesy 87, 43--55 (2013); * DOI: * 10.1007/s00190-012-0578-z; * addenda: * * geod-addenda.html. * . * For more information on geodesics see \ref geodesic. * * Example of use: * \include example-Geodesic.cpp * * GeodSolve is a command-line utility * providing access to the functionality of Geodesic and GeodesicLine. **********************************************************************/ class GEOGRAPHICLIB_EXPORT Geodesic { private: typedef Math::real real; friend class GeodesicLine; static const int nA1_ = GEOGRAPHICLIB_GEODESIC_ORDER; static const int nC1_ = GEOGRAPHICLIB_GEODESIC_ORDER; static const int nC1p_ = GEOGRAPHICLIB_GEODESIC_ORDER; static const int nA2_ = GEOGRAPHICLIB_GEODESIC_ORDER; static const int nC2_ = GEOGRAPHICLIB_GEODESIC_ORDER; static const int nA3_ = GEOGRAPHICLIB_GEODESIC_ORDER; static const int nA3x_ = nA3_; static const int nC3_ = GEOGRAPHICLIB_GEODESIC_ORDER; static const int nC3x_ = (nC3_ * (nC3_ - 1)) / 2; static const int nC4_ = GEOGRAPHICLIB_GEODESIC_ORDER; static const int nC4x_ = (nC4_ * (nC4_ + 1)) / 2; // Size for temporary array // nC = max(max(nC1_, nC1p_, nC2_) + 1, max(nC3_, nC4_)) static const int nC_ = GEOGRAPHICLIB_GEODESIC_ORDER + 1; static const unsigned maxit1_ = 20; unsigned maxit2_; real tiny_, tol0_, tol1_, tol2_, tolb_, xthresh_; enum captype { CAP_NONE = 0U, CAP_C1 = 1U<<0, CAP_C1p = 1U<<1, CAP_C2 = 1U<<2, CAP_C3 = 1U<<3, CAP_C4 = 1U<<4, CAP_ALL = 0x1FU, CAP_MASK = CAP_ALL, OUT_ALL = 0x7F80U, OUT_MASK = 0xFF80U, // Includes LONG_UNROLL }; static real SinCosSeries(bool sinp, real sinx, real cosx, const real c[], int n); static real Astroid(real x, real y); real _a, _f, _f1, _e2, _ep2, _n, _b, _c2, _etol2; real _aA3x[nA3x_], _cC3x[nC3x_], _cC4x[nC4x_]; void Lengths(real eps, real sig12, real ssig1, real csig1, real dn1, real ssig2, real csig2, real dn2, real cbet1, real cbet2, unsigned outmask, real& s12s, real& m12a, real& m0, real& M12, real& M21, real Ca[]) const; real InverseStart(real sbet1, real cbet1, real dn1, real sbet2, real cbet2, real dn2, real lam12, real slam12, real clam12, real& salp1, real& calp1, real& salp2, real& calp2, real& dnm, real Ca[]) const; real Lambda12(real sbet1, real cbet1, real dn1, real sbet2, real cbet2, real dn2, real salp1, real calp1, real slam120, real clam120, real& salp2, real& calp2, real& sig12, real& ssig1, real& csig1, real& ssig2, real& csig2, real& eps, real& domg12, bool diffp, real& dlam12, real Ca[]) const; real GenInverse(real lat1, real lon1, real lat2, real lon2, unsigned outmask, real& s12, real& salp1, real& calp1, real& salp2, real& calp2, real& m12, real& M12, real& M21, real& S12) const; // These are Maxima generated functions to provide series approximations to // the integrals for the ellipsoidal geodesic. static real A1m1f(real eps); static void C1f(real eps, real c[]); static void C1pf(real eps, real c[]); static real A2m1f(real eps); static void C2f(real eps, real c[]); void A3coeff(); real A3f(real eps) const; void C3coeff(); void C3f(real eps, real c[]) const; void C4coeff(); void C4f(real k2, real c[]) const; public: /** * Bit masks for what calculations to do. These masks do double duty. * They signify to the GeodesicLine constructor and to * Geodesic::Line what capabilities should be included in the GeodesicLine * object. They also specify which results to return in the general * routines Geodesic::GenDirect and Geodesic::GenInverse routines. * GeodesicLine::mask is a duplication of this enum. **********************************************************************/ enum mask { /** * No capabilities, no output. * @hideinitializer **********************************************************************/ NONE = 0U, /** * Calculate latitude \e lat2. (It's not necessary to include this as a * capability to GeodesicLine because this is included by default.) * @hideinitializer **********************************************************************/ LATITUDE = 1U<<7 | CAP_NONE, /** * Calculate longitude \e lon2. * @hideinitializer **********************************************************************/ LONGITUDE = 1U<<8 | CAP_C3, /** * Calculate azimuths \e azi1 and \e azi2. (It's not necessary to * include this as a capability to GeodesicLine because this is included * by default.) * @hideinitializer **********************************************************************/ AZIMUTH = 1U<<9 | CAP_NONE, /** * Calculate distance \e s12. * @hideinitializer **********************************************************************/ DISTANCE = 1U<<10 | CAP_C1, /** * A combination of the common capabilities: Geodesic::LATITUDE, * Geodesic::LONGITUDE, Geodesic::AZIMUTH, Geodesic::DISTANCE. * @hideinitializer **********************************************************************/ STANDARD = LATITUDE | LONGITUDE | AZIMUTH | DISTANCE, /** * Allow distance \e s12 to be used as input in the direct geodesic * problem. * @hideinitializer **********************************************************************/ DISTANCE_IN = 1U<<11 | CAP_C1 | CAP_C1p, /** * Calculate reduced length \e m12. * @hideinitializer **********************************************************************/ REDUCEDLENGTH = 1U<<12 | CAP_C1 | CAP_C2, /** * Calculate geodesic scales \e M12 and \e M21. * @hideinitializer **********************************************************************/ GEODESICSCALE = 1U<<13 | CAP_C1 | CAP_C2, /** * Calculate area \e S12. * @hideinitializer **********************************************************************/ AREA = 1U<<14 | CAP_C4, /** * Unroll \e lon2 in the direct calculation. * @hideinitializer **********************************************************************/ LONG_UNROLL = 1U<<15, /** * All capabilities, calculate everything. (Geodesic::LONG_UNROLL is not * included in this mask.) * @hideinitializer **********************************************************************/ ALL = OUT_ALL| CAP_ALL, }; /** \name Constructor **********************************************************************/ ///@{ /** * Constructor for an ellipsoid with * * @param[in] a equatorial radius (meters). * @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere. * Negative \e f gives a prolate ellipsoid. * @exception GeographicErr if \e a or (1 − \e f) \e a is not * positive. **********************************************************************/ Geodesic(real a, real f); ///@} /** \name Direct geodesic problem specified in terms of distance. **********************************************************************/ ///@{ /** * Solve the direct geodesic problem where the length of the geodesic * is specified in terms of distance. * * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] azi1 azimuth at point 1 (degrees). * @param[in] s12 distance between point 1 and point 2 (meters); it can be * negative. * @param[out] lat2 latitude of point 2 (degrees). * @param[out] lon2 longitude of point 2 (degrees). * @param[out] azi2 (forward) azimuth at point 2 (degrees). * @param[out] m12 reduced length of geodesic (meters). * @param[out] M12 geodesic scale of point 2 relative to point 1 * (dimensionless). * @param[out] M21 geodesic scale of point 1 relative to point 2 * (dimensionless). * @param[out] S12 area under the geodesic (meters2). * @return \e a12 arc length of between point 1 and point 2 (degrees). * * \e lat1 should be in the range [−90°, 90°]. The values of * \e lon2 and \e azi2 returned are in the range [−180°, * 180°]. * * If either point is at a pole, the azimuth is defined by keeping the * longitude fixed, writing \e lat = ±(90° − ε), * and taking the limit ε → 0+. An arc length greater that * 180° signifies a geodesic which is not a shortest path. (For a * prolate ellipsoid, an additional condition is necessary for a shortest * path: the longitudinal extent must not exceed of 180°.) * * The following functions are overloaded versions of Geodesic::Direct * which omit some of the output parameters. Note, however, that the arc * length is always computed and returned as the function value. **********************************************************************/ Math::real Direct(real lat1, real lon1, real azi1, real s12, real& lat2, real& lon2, real& azi2, real& m12, real& M12, real& M21, real& S12) const { real t; return GenDirect(lat1, lon1, azi1, false, s12, LATITUDE | LONGITUDE | AZIMUTH | REDUCEDLENGTH | GEODESICSCALE | AREA, lat2, lon2, azi2, t, m12, M12, M21, S12); } /** * See the documentation for Geodesic::Direct. **********************************************************************/ Math::real Direct(real lat1, real lon1, real azi1, real s12, real& lat2, real& lon2) const { real t; return GenDirect(lat1, lon1, azi1, false, s12, LATITUDE | LONGITUDE, lat2, lon2, t, t, t, t, t, t); } /** * See the documentation for Geodesic::Direct. **********************************************************************/ Math::real Direct(real lat1, real lon1, real azi1, real s12, real& lat2, real& lon2, real& azi2) const { real t; return GenDirect(lat1, lon1, azi1, false, s12, LATITUDE | LONGITUDE | AZIMUTH, lat2, lon2, azi2, t, t, t, t, t); } /** * See the documentation for Geodesic::Direct. **********************************************************************/ Math::real Direct(real lat1, real lon1, real azi1, real s12, real& lat2, real& lon2, real& azi2, real& m12) const { real t; return GenDirect(lat1, lon1, azi1, false, s12, LATITUDE | LONGITUDE | AZIMUTH | REDUCEDLENGTH, lat2, lon2, azi2, t, m12, t, t, t); } /** * See the documentation for Geodesic::Direct. **********************************************************************/ Math::real Direct(real lat1, real lon1, real azi1, real s12, real& lat2, real& lon2, real& azi2, real& M12, real& M21) const { real t; return GenDirect(lat1, lon1, azi1, false, s12, LATITUDE | LONGITUDE | AZIMUTH | GEODESICSCALE, lat2, lon2, azi2, t, t, M12, M21, t); } /** * See the documentation for Geodesic::Direct. **********************************************************************/ Math::real Direct(real lat1, real lon1, real azi1, real s12, real& lat2, real& lon2, real& azi2, real& m12, real& M12, real& M21) const { real t; return GenDirect(lat1, lon1, azi1, false, s12, LATITUDE | LONGITUDE | AZIMUTH | REDUCEDLENGTH | GEODESICSCALE, lat2, lon2, azi2, t, m12, M12, M21, t); } ///@} /** \name Direct geodesic problem specified in terms of arc length. **********************************************************************/ ///@{ /** * Solve the direct geodesic problem where the length of the geodesic * is specified in terms of arc length. * * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] azi1 azimuth at point 1 (degrees). * @param[in] a12 arc length between point 1 and point 2 (degrees); it can * be negative. * @param[out] lat2 latitude of point 2 (degrees). * @param[out] lon2 longitude of point 2 (degrees). * @param[out] azi2 (forward) azimuth at point 2 (degrees). * @param[out] s12 distance between point 1 and point 2 (meters). * @param[out] m12 reduced length of geodesic (meters). * @param[out] M12 geodesic scale of point 2 relative to point 1 * (dimensionless). * @param[out] M21 geodesic scale of point 1 relative to point 2 * (dimensionless). * @param[out] S12 area under the geodesic (meters2). * * \e lat1 should be in the range [−90°, 90°]. The values of * \e lon2 and \e azi2 returned are in the range [−180°, * 180°]. * * If either point is at a pole, the azimuth is defined by keeping the * longitude fixed, writing \e lat = ±(90° − ε), * and taking the limit ε → 0+. An arc length greater that * 180° signifies a geodesic which is not a shortest path. (For a * prolate ellipsoid, an additional condition is necessary for a shortest * path: the longitudinal extent must not exceed of 180°.) * * The following functions are overloaded versions of Geodesic::Direct * which omit some of the output parameters. **********************************************************************/ void ArcDirect(real lat1, real lon1, real azi1, real a12, real& lat2, real& lon2, real& azi2, real& s12, real& m12, real& M12, real& M21, real& S12) const { GenDirect(lat1, lon1, azi1, true, a12, LATITUDE | LONGITUDE | AZIMUTH | DISTANCE | REDUCEDLENGTH | GEODESICSCALE | AREA, lat2, lon2, azi2, s12, m12, M12, M21, S12); } /** * See the documentation for Geodesic::ArcDirect. **********************************************************************/ void ArcDirect(real lat1, real lon1, real azi1, real a12, real& lat2, real& lon2) const { real t; GenDirect(lat1, lon1, azi1, true, a12, LATITUDE | LONGITUDE, lat2, lon2, t, t, t, t, t, t); } /** * See the documentation for Geodesic::ArcDirect. **********************************************************************/ void ArcDirect(real lat1, real lon1, real azi1, real a12, real& lat2, real& lon2, real& azi2) const { real t; GenDirect(lat1, lon1, azi1, true, a12, LATITUDE | LONGITUDE | AZIMUTH, lat2, lon2, azi2, t, t, t, t, t); } /** * See the documentation for Geodesic::ArcDirect. **********************************************************************/ void ArcDirect(real lat1, real lon1, real azi1, real a12, real& lat2, real& lon2, real& azi2, real& s12) const { real t; GenDirect(lat1, lon1, azi1, true, a12, LATITUDE | LONGITUDE | AZIMUTH | DISTANCE, lat2, lon2, azi2, s12, t, t, t, t); } /** * See the documentation for Geodesic::ArcDirect. **********************************************************************/ void ArcDirect(real lat1, real lon1, real azi1, real a12, real& lat2, real& lon2, real& azi2, real& s12, real& m12) const { real t; GenDirect(lat1, lon1, azi1, true, a12, LATITUDE | LONGITUDE | AZIMUTH | DISTANCE | REDUCEDLENGTH, lat2, lon2, azi2, s12, m12, t, t, t); } /** * See the documentation for Geodesic::ArcDirect. **********************************************************************/ void ArcDirect(real lat1, real lon1, real azi1, real a12, real& lat2, real& lon2, real& azi2, real& s12, real& M12, real& M21) const { real t; GenDirect(lat1, lon1, azi1, true, a12, LATITUDE | LONGITUDE | AZIMUTH | DISTANCE | GEODESICSCALE, lat2, lon2, azi2, s12, t, M12, M21, t); } /** * See the documentation for Geodesic::ArcDirect. **********************************************************************/ void ArcDirect(real lat1, real lon1, real azi1, real a12, real& lat2, real& lon2, real& azi2, real& s12, real& m12, real& M12, real& M21) const { real t; GenDirect(lat1, lon1, azi1, true, a12, LATITUDE | LONGITUDE | AZIMUTH | DISTANCE | REDUCEDLENGTH | GEODESICSCALE, lat2, lon2, azi2, s12, m12, M12, M21, t); } ///@} /** \name General version of the direct geodesic solution. **********************************************************************/ ///@{ /** * The general direct geodesic problem. Geodesic::Direct and * Geodesic::ArcDirect are defined in terms of this function. * * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] azi1 azimuth at point 1 (degrees). * @param[in] arcmode boolean flag determining the meaning of the \e * s12_a12. * @param[in] s12_a12 if \e arcmode is false, this is the distance between * point 1 and point 2 (meters); otherwise it is the arc length between * point 1 and point 2 (degrees); it can be negative. * @param[in] outmask a bitor'ed combination of Geodesic::mask values * specifying which of the following parameters should be set. * @param[out] lat2 latitude of point 2 (degrees). * @param[out] lon2 longitude of point 2 (degrees). * @param[out] azi2 (forward) azimuth at point 2 (degrees). * @param[out] s12 distance between point 1 and point 2 (meters). * @param[out] m12 reduced length of geodesic (meters). * @param[out] M12 geodesic scale of point 2 relative to point 1 * (dimensionless). * @param[out] M21 geodesic scale of point 1 relative to point 2 * (dimensionless). * @param[out] S12 area under the geodesic (meters2). * @return \e a12 arc length of between point 1 and point 2 (degrees). * * The Geodesic::mask values possible for \e outmask are * - \e outmask |= Geodesic::LATITUDE for the latitude \e lat2; * - \e outmask |= Geodesic::LONGITUDE for the latitude \e lon2; * - \e outmask |= Geodesic::AZIMUTH for the latitude \e azi2; * - \e outmask |= Geodesic::DISTANCE for the distance \e s12; * - \e outmask |= Geodesic::REDUCEDLENGTH for the reduced length \e * m12; * - \e outmask |= Geodesic::GEODESICSCALE for the geodesic scales \e * M12 and \e M21; * - \e outmask |= Geodesic::AREA for the area \e S12; * - \e outmask |= Geodesic::ALL for all of the above; * - \e outmask |= Geodesic::LONG_UNROLL to unroll \e lon2 instead of * wrapping it into the range [−180°, 180°]. * . * The function value \e a12 is always computed and returned and this * equals \e s12_a12 is \e arcmode is true. If \e outmask includes * Geodesic::DISTANCE and \e arcmode is false, then \e s12 = \e s12_a12. * It is not necessary to include Geodesic::DISTANCE_IN in \e outmask; this * is automatically included is \e arcmode is false. * * With the Geodesic::LONG_UNROLL bit set, the quantity \e lon2 − \e * lon1 indicates how many times and in what sense the geodesic encircles * the ellipsoid. **********************************************************************/ Math::real GenDirect(real lat1, real lon1, real azi1, bool arcmode, real s12_a12, unsigned outmask, real& lat2, real& lon2, real& azi2, real& s12, real& m12, real& M12, real& M21, real& S12) const; ///@} /** \name Inverse geodesic problem. **********************************************************************/ ///@{ /** * Solve the inverse geodesic problem. * * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] lat2 latitude of point 2 (degrees). * @param[in] lon2 longitude of point 2 (degrees). * @param[out] s12 distance between point 1 and point 2 (meters). * @param[out] azi1 azimuth at point 1 (degrees). * @param[out] azi2 (forward) azimuth at point 2 (degrees). * @param[out] m12 reduced length of geodesic (meters). * @param[out] M12 geodesic scale of point 2 relative to point 1 * (dimensionless). * @param[out] M21 geodesic scale of point 1 relative to point 2 * (dimensionless). * @param[out] S12 area under the geodesic (meters2). * @return \e a12 arc length of between point 1 and point 2 (degrees). * * \e lat1 and \e lat2 should be in the range [−90°, 90°]. * The values of \e azi1 and \e azi2 returned are in the range * [−180°, 180°]. * * If either point is at a pole, the azimuth is defined by keeping the * longitude fixed, writing \e lat = ±(90° − ε), * and taking the limit ε → 0+. * * The solution to the inverse problem is found using Newton's method. If * this fails to converge (this is very unlikely in geodetic applications * but does occur for very eccentric ellipsoids), then the bisection method * is used to refine the solution. * * The following functions are overloaded versions of Geodesic::Inverse * which omit some of the output parameters. Note, however, that the arc * length is always computed and returned as the function value. **********************************************************************/ Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real& s12, real& azi1, real& azi2, real& m12, real& M12, real& M21, real& S12) const { return GenInverse(lat1, lon1, lat2, lon2, DISTANCE | AZIMUTH | REDUCEDLENGTH | GEODESICSCALE | AREA, s12, azi1, azi2, m12, M12, M21, S12); } /** * See the documentation for Geodesic::Inverse. **********************************************************************/ Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real& s12) const { real t; return GenInverse(lat1, lon1, lat2, lon2, DISTANCE, s12, t, t, t, t, t, t); } /** * See the documentation for Geodesic::Inverse. **********************************************************************/ Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real& azi1, real& azi2) const { real t; return GenInverse(lat1, lon1, lat2, lon2, AZIMUTH, t, azi1, azi2, t, t, t, t); } /** * See the documentation for Geodesic::Inverse. **********************************************************************/ Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real& s12, real& azi1, real& azi2) const { real t; return GenInverse(lat1, lon1, lat2, lon2, DISTANCE | AZIMUTH, s12, azi1, azi2, t, t, t, t); } /** * See the documentation for Geodesic::Inverse. **********************************************************************/ Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real& s12, real& azi1, real& azi2, real& m12) const { real t; return GenInverse(lat1, lon1, lat2, lon2, DISTANCE | AZIMUTH | REDUCEDLENGTH, s12, azi1, azi2, m12, t, t, t); } /** * See the documentation for Geodesic::Inverse. **********************************************************************/ Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real& s12, real& azi1, real& azi2, real& M12, real& M21) const { real t; return GenInverse(lat1, lon1, lat2, lon2, DISTANCE | AZIMUTH | GEODESICSCALE, s12, azi1, azi2, t, M12, M21, t); } /** * See the documentation for Geodesic::Inverse. **********************************************************************/ Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real& s12, real& azi1, real& azi2, real& m12, real& M12, real& M21) const { real t; return GenInverse(lat1, lon1, lat2, lon2, DISTANCE | AZIMUTH | REDUCEDLENGTH | GEODESICSCALE, s12, azi1, azi2, m12, M12, M21, t); } ///@} /** \name General version of inverse geodesic solution. **********************************************************************/ ///@{ /** * The general inverse geodesic calculation. Geodesic::Inverse is defined * in terms of this function. * * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] lat2 latitude of point 2 (degrees). * @param[in] lon2 longitude of point 2 (degrees). * @param[in] outmask a bitor'ed combination of Geodesic::mask values * specifying which of the following parameters should be set. * @param[out] s12 distance between point 1 and point 2 (meters). * @param[out] azi1 azimuth at point 1 (degrees). * @param[out] azi2 (forward) azimuth at point 2 (degrees). * @param[out] m12 reduced length of geodesic (meters). * @param[out] M12 geodesic scale of point 2 relative to point 1 * (dimensionless). * @param[out] M21 geodesic scale of point 1 relative to point 2 * (dimensionless). * @param[out] S12 area under the geodesic (meters2). * @return \e a12 arc length of between point 1 and point 2 (degrees). * * The Geodesic::mask values possible for \e outmask are * - \e outmask |= Geodesic::DISTANCE for the distance \e s12; * - \e outmask |= Geodesic::AZIMUTH for the latitude \e azi2; * - \e outmask |= Geodesic::REDUCEDLENGTH for the reduced length \e * m12; * - \e outmask |= Geodesic::GEODESICSCALE for the geodesic scales \e * M12 and \e M21; * - \e outmask |= Geodesic::AREA for the area \e S12; * - \e outmask |= Geodesic::ALL for all of the above. * . * The arc length is always computed and returned as the function value. **********************************************************************/ Math::real GenInverse(real lat1, real lon1, real lat2, real lon2, unsigned outmask, real& s12, real& azi1, real& azi2, real& m12, real& M12, real& M21, real& S12) const; ///@} /** \name Interface to GeodesicLine. **********************************************************************/ ///@{ /** * Set up to compute several points on a single geodesic. * * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] azi1 azimuth at point 1 (degrees). * @param[in] caps bitor'ed combination of Geodesic::mask values * specifying the capabilities the GeodesicLine object should possess, * i.e., which quantities can be returned in calls to * GeodesicLine::Position. * @return a GeodesicLine object. * * \e lat1 should be in the range [−90°, 90°]. * * The Geodesic::mask values are * - \e caps |= Geodesic::LATITUDE for the latitude \e lat2; this is * added automatically; * - \e caps |= Geodesic::LONGITUDE for the latitude \e lon2; * - \e caps |= Geodesic::AZIMUTH for the azimuth \e azi2; this is * added automatically; * - \e caps |= Geodesic::DISTANCE for the distance \e s12; * - \e caps |= Geodesic::REDUCEDLENGTH for the reduced length \e m12; * - \e caps |= Geodesic::GEODESICSCALE for the geodesic scales \e M12 * and \e M21; * - \e caps |= Geodesic::AREA for the area \e S12; * - \e caps |= Geodesic::DISTANCE_IN permits the length of the * geodesic to be given in terms of \e s12; without this capability the * length can only be specified in terms of arc length; * - \e caps |= Geodesic::ALL for all of the above. * . * The default value of \e caps is Geodesic::ALL. * * If the point is at a pole, the azimuth is defined by keeping \e lon1 * fixed, writing \e lat1 = ±(90 − ε), and taking the * limit ε → 0+. **********************************************************************/ GeodesicLine Line(real lat1, real lon1, real azi1, unsigned caps = ALL) const; /** * Define a GeodesicLine in terms of the inverse geodesic problem. * * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] lat2 latitude of point 2 (degrees). * @param[in] lon2 longitude of point 2 (degrees). * @param[in] caps bitor'ed combination of Geodesic::mask values * specifying the capabilities the GeodesicLine object should possess, * i.e., which quantities can be returned in calls to * GeodesicLine::Position. * @return a GeodesicLine object. * * This function sets point 3 of the GeodesicLine to correspond to point 2 * of the inverse geodesic problem. * * \e lat1 and \e lat2 should be in the range [−90°, 90°]. **********************************************************************/ GeodesicLine InverseLine(real lat1, real lon1, real lat2, real lon2, unsigned caps = ALL) const; /** * Define a GeodesicLine in terms of the direct geodesic problem specified * in terms of distance. * * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] azi1 azimuth at point 1 (degrees). * @param[in] s12 distance between point 1 and point 2 (meters); it can be * negative. * @param[in] caps bitor'ed combination of Geodesic::mask values * specifying the capabilities the GeodesicLine object should possess, * i.e., which quantities can be returned in calls to * GeodesicLine::Position. * @return a GeodesicLine object. * * This function sets point 3 of the GeodesicLine to correspond to point 2 * of the direct geodesic problem. * * \e lat1 should be in the range [−90°, 90°]. **********************************************************************/ GeodesicLine DirectLine(real lat1, real lon1, real azi1, real s12, unsigned caps = ALL) const; /** * Define a GeodesicLine in terms of the direct geodesic problem specified * in terms of arc length. * * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] azi1 azimuth at point 1 (degrees). * @param[in] a12 arc length between point 1 and point 2 (degrees); it can * be negative. * @param[in] caps bitor'ed combination of Geodesic::mask values * specifying the capabilities the GeodesicLine object should possess, * i.e., which quantities can be returned in calls to * GeodesicLine::Position. * @return a GeodesicLine object. * * This function sets point 3 of the GeodesicLine to correspond to point 2 * of the direct geodesic problem. * * \e lat1 should be in the range [−90°, 90°]. **********************************************************************/ GeodesicLine ArcDirectLine(real lat1, real lon1, real azi1, real a12, unsigned caps = ALL) const; /** * Define a GeodesicLine in terms of the direct geodesic problem specified * in terms of either distance or arc length. * * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] azi1 azimuth at point 1 (degrees). * @param[in] arcmode boolean flag determining the meaning of the \e * s12_a12. * @param[in] s12_a12 if \e arcmode is false, this is the distance between * point 1 and point 2 (meters); otherwise it is the arc length between * point 1 and point 2 (degrees); it can be negative. * @param[in] caps bitor'ed combination of Geodesic::mask values * specifying the capabilities the GeodesicLine object should possess, * i.e., which quantities can be returned in calls to * GeodesicLine::Position. * @return a GeodesicLine object. * * This function sets point 3 of the GeodesicLine to correspond to point 2 * of the direct geodesic problem. * * \e lat1 should be in the range [−90°, 90°]. **********************************************************************/ GeodesicLine GenDirectLine(real lat1, real lon1, real azi1, bool arcmode, real s12_a12, unsigned caps = ALL) const; ///@} /** \name Inspector functions. **********************************************************************/ ///@{ /** * @return \e a the equatorial radius of the ellipsoid (meters). This is * the value used in the constructor. **********************************************************************/ Math::real EquatorialRadius() const { return _a; } /** * @return \e f the flattening of the ellipsoid. This is the * value used in the constructor. **********************************************************************/ Math::real Flattening() const { return _f; } /** * @return total area of ellipsoid in meters2. The area of a * polygon encircling a pole can be found by adding * Geodesic::EllipsoidArea()/2 to the sum of \e S12 for each side of the * polygon. **********************************************************************/ Math::real EllipsoidArea() const { return 4 * Math::pi() * _c2; } ///@} /** * A global instantiation of Geodesic with the parameters for the WGS84 * ellipsoid. **********************************************************************/ static const Geodesic& WGS84(); }; } // namespace GeographicLib #endif // GEOGRAPHICLIB_GEODESIC_HPP geosphere/src/TransverseMercator.cpp0000644000176200001440000005450714323401055017367 0ustar liggesusers/** * \file TransverseMercator.cpp * \brief Implementation for GeographicLib::TransverseMercator class * * Copyright (c) Charles Karney (2008-2022) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ * * This implementation follows closely JHS 154, ETRS89 - * järjestelmään liittyvät karttaprojektiot, * tasokoordinaatistot ja karttalehtijako (Map projections, plane * coordinates, and map sheet index for ETRS89), published by JUHTA, Finnish * Geodetic Institute, and the National Land Survey of Finland (2006). * * The relevant section is available as the 2008 PDF file * http://docs.jhs-suositukset.fi/jhs-suositukset/JHS154/JHS154_liite1.pdf * * This is a straight transcription of the formulas in this paper with the * following exceptions: * - use of 6th order series instead of 4th order series. This reduces the * error to about 5nm for the UTM range of coordinates (instead of 200nm), * with a speed penalty of only 1%; * - use Newton's method instead of plain iteration to solve for latitude in * terms of isometric latitude in the Reverse method; * - use of Horner's representation for evaluating polynomials and Clenshaw's * method for summing trigonometric series; * - several modifications of the formulas to improve the numerical accuracy; * - evaluating the convergence and scale using the expression for the * projection or its inverse. * * If the preprocessor variable GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER is set * to an integer between 4 and 8, then this specifies the order of the series * used for the forward and reverse transformations. The default value is 6. * (The series accurate to 12th order is given in \ref tmseries.) **********************************************************************/ #include #include "TransverseMercator.h" #if defined(_MSC_VER) // Squelch warnings about enum-float expressions # pragma warning (disable: 5055) #endif namespace GeographicLib { using namespace std; TransverseMercator::TransverseMercator(real a, real f, real k0) : _a(a) , _f(f) , _k0(k0) , _e2(_f * (2 - _f)) , _es((_f < 0 ? -1 : 1) * sqrt(fabs(_e2))) , _e2m(1 - _e2) // _c = sqrt( pow(1 + _e, 1 + _e) * pow(1 - _e, 1 - _e) ) ) // See, for example, Lee (1976), p 100. , _c( sqrt(_e2m) * exp(Math::eatanhe(real(1), _es)) ) , _n(_f / (2 - _f)) { if (!(isfinite(_a) && _a > 0)) throw GeographicErr("Equatorial radius is not positive"); if (!(isfinite(_f) && _f < 1)) throw GeographicErr("Polar semi-axis is not positive"); if (!(isfinite(_k0) && _k0 > 0)) throw GeographicErr("Scale is not positive"); // Generated by Maxima on 2015-05-14 22:55:13-04:00 #if GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 2 static const real b1coeff[] = { // b1*(n+1), polynomial in n2 of order 2 1, 16, 64, 64, }; // count = 4 #elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 3 static const real b1coeff[] = { // b1*(n+1), polynomial in n2 of order 3 1, 4, 64, 256, 256, }; // count = 5 #elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 4 static const real b1coeff[] = { // b1*(n+1), polynomial in n2 of order 4 25, 64, 256, 4096, 16384, 16384, }; // count = 6 #else #error "Bad value for GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER" #endif #if GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4 static const real alpcoeff[] = { // alp[1]/n^1, polynomial in n of order 3 164, 225, -480, 360, 720, // alp[2]/n^2, polynomial in n of order 2 557, -864, 390, 1440, // alp[3]/n^3, polynomial in n of order 1 -1236, 427, 1680, // alp[4]/n^4, polynomial in n of order 0 49561, 161280, }; // count = 14 #elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5 static const real alpcoeff[] = { // alp[1]/n^1, polynomial in n of order 4 -635, 328, 450, -960, 720, 1440, // alp[2]/n^2, polynomial in n of order 3 4496, 3899, -6048, 2730, 10080, // alp[3]/n^3, polynomial in n of order 2 15061, -19776, 6832, 26880, // alp[4]/n^4, polynomial in n of order 1 -171840, 49561, 161280, // alp[5]/n^5, polynomial in n of order 0 34729, 80640, }; // count = 20 #elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6 static const real alpcoeff[] = { // alp[1]/n^1, polynomial in n of order 5 31564, -66675, 34440, 47250, -100800, 75600, 151200, // alp[2]/n^2, polynomial in n of order 4 -1983433, 863232, 748608, -1161216, 524160, 1935360, // alp[3]/n^3, polynomial in n of order 3 670412, 406647, -533952, 184464, 725760, // alp[4]/n^4, polynomial in n of order 2 6601661, -7732800, 2230245, 7257600, // alp[5]/n^5, polynomial in n of order 1 -13675556, 3438171, 7983360, // alp[6]/n^6, polynomial in n of order 0 212378941, 319334400, }; // count = 27 #elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7 static const real alpcoeff[] = { // alp[1]/n^1, polynomial in n of order 6 1804025, 2020096, -4267200, 2204160, 3024000, -6451200, 4838400, 9676800, // alp[2]/n^2, polynomial in n of order 5 4626384, -9917165, 4316160, 3743040, -5806080, 2620800, 9676800, // alp[3]/n^3, polynomial in n of order 4 -67102379, 26816480, 16265880, -21358080, 7378560, 29030400, // alp[4]/n^4, polynomial in n of order 3 155912000, 72618271, -85060800, 24532695, 79833600, // alp[5]/n^5, polynomial in n of order 2 102508609, -109404448, 27505368, 63866880, // alp[6]/n^6, polynomial in n of order 1 -12282192400LL, 2760926233LL, 4151347200LL, // alp[7]/n^7, polynomial in n of order 0 1522256789, 1383782400, }; // count = 35 #elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8 static const real alpcoeff[] = { // alp[1]/n^1, polynomial in n of order 7 -75900428, 37884525, 42422016, -89611200, 46287360, 63504000, -135475200, 101606400, 203212800, // alp[2]/n^2, polynomial in n of order 6 148003883, 83274912, -178508970, 77690880, 67374720, -104509440, 47174400, 174182400, // alp[3]/n^3, polynomial in n of order 5 318729724, -738126169, 294981280, 178924680, -234938880, 81164160, 319334400, // alp[4]/n^4, polynomial in n of order 4 -40176129013LL, 14967552000LL, 6971354016LL, -8165836800LL, 2355138720LL, 7664025600LL, // alp[5]/n^5, polynomial in n of order 3 10421654396LL, 3997835751LL, -4266773472LL, 1072709352, 2490808320LL, // alp[6]/n^6, polynomial in n of order 2 175214326799LL, -171950693600LL, 38652967262LL, 58118860800LL, // alp[7]/n^7, polynomial in n of order 1 -67039739596LL, 13700311101LL, 12454041600LL, // alp[8]/n^8, polynomial in n of order 0 1424729850961LL, 743921418240LL, }; // count = 44 #else #error "Bad value for GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER" #endif #if GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4 static const real betcoeff[] = { // bet[1]/n^1, polynomial in n of order 3 -4, 555, -960, 720, 1440, // bet[2]/n^2, polynomial in n of order 2 -437, 96, 30, 1440, // bet[3]/n^3, polynomial in n of order 1 -148, 119, 3360, // bet[4]/n^4, polynomial in n of order 0 4397, 161280, }; // count = 14 #elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5 static const real betcoeff[] = { // bet[1]/n^1, polynomial in n of order 4 -3645, -64, 8880, -15360, 11520, 23040, // bet[2]/n^2, polynomial in n of order 3 4416, -3059, 672, 210, 10080, // bet[3]/n^3, polynomial in n of order 2 -627, -592, 476, 13440, // bet[4]/n^4, polynomial in n of order 1 -3520, 4397, 161280, // bet[5]/n^5, polynomial in n of order 0 4583, 161280, }; // count = 20 #elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6 static const real betcoeff[] = { // bet[1]/n^1, polynomial in n of order 5 384796, -382725, -6720, 932400, -1612800, 1209600, 2419200, // bet[2]/n^2, polynomial in n of order 4 -1118711, 1695744, -1174656, 258048, 80640, 3870720, // bet[3]/n^3, polynomial in n of order 3 22276, -16929, -15984, 12852, 362880, // bet[4]/n^4, polynomial in n of order 2 -830251, -158400, 197865, 7257600, // bet[5]/n^5, polynomial in n of order 1 -435388, 453717, 15966720, // bet[6]/n^6, polynomial in n of order 0 20648693, 638668800, }; // count = 27 #elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7 static const real betcoeff[] = { // bet[1]/n^1, polynomial in n of order 6 -5406467, 6156736, -6123600, -107520, 14918400, -25804800, 19353600, 38707200, // bet[2]/n^2, polynomial in n of order 5 829456, -5593555, 8478720, -5873280, 1290240, 403200, 19353600, // bet[3]/n^3, polynomial in n of order 4 9261899, 3564160, -2708640, -2557440, 2056320, 58060800, // bet[4]/n^4, polynomial in n of order 3 14928352, -9132761, -1742400, 2176515, 79833600, // bet[5]/n^5, polynomial in n of order 2 -8005831, -1741552, 1814868, 63866880, // bet[6]/n^6, polynomial in n of order 1 -261810608, 268433009, 8302694400LL, // bet[7]/n^7, polynomial in n of order 0 219941297, 5535129600LL, }; // count = 35 #elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8 static const real betcoeff[] = { // bet[1]/n^1, polynomial in n of order 7 31777436, -37845269, 43097152, -42865200, -752640, 104428800, -180633600, 135475200, 270950400, // bet[2]/n^2, polynomial in n of order 6 24749483, 14930208, -100683990, 152616960, -105719040, 23224320, 7257600, 348364800, // bet[3]/n^3, polynomial in n of order 5 -232468668, 101880889, 39205760, -29795040, -28131840, 22619520, 638668800, // bet[4]/n^4, polynomial in n of order 4 324154477, 1433121792, -876745056, -167270400, 208945440, 7664025600LL, // bet[5]/n^5, polynomial in n of order 3 457888660, -312227409, -67920528, 70779852, 2490808320LL, // bet[6]/n^6, polynomial in n of order 2 -19841813847LL, -3665348512LL, 3758062126LL, 116237721600LL, // bet[7]/n^7, polynomial in n of order 1 -1989295244, 1979471673, 49816166400LL, // bet[8]/n^8, polynomial in n of order 0 191773887257LL, 3719607091200LL, }; // count = 44 #else #error "Bad value for GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER" #endif static_assert(sizeof(b1coeff) / sizeof(real) == maxpow_/2 + 2, "Coefficient array size mismatch for b1"); static_assert(sizeof(alpcoeff) / sizeof(real) == (maxpow_ * (maxpow_ + 3))/2, "Coefficient array size mismatch for alp"); static_assert(sizeof(betcoeff) / sizeof(real) == (maxpow_ * (maxpow_ + 3))/2, "Coefficient array size mismatch for bet"); int m = maxpow_/2; _b1 = Math::polyval(m, b1coeff, Math::sq(_n)) / (b1coeff[m + 1] * (1+_n)); // _a1 is the equivalent radius for computing the circumference of // ellipse. _a1 = _b1 * _a; int o = 0; real d = _n; for (int l = 1; l <= maxpow_; ++l) { m = maxpow_ - l; _alp[l] = d * Math::polyval(m, alpcoeff + o, _n) / alpcoeff[o + m + 1]; _bet[l] = d * Math::polyval(m, betcoeff + o, _n) / betcoeff[o + m + 1]; o += m + 2; d *= _n; } // Post condition: o == sizeof(alpcoeff) / sizeof(real) && // o == sizeof(betcoeff) / sizeof(real) } const TransverseMercator& TransverseMercator::UTM() { static const TransverseMercator utm(Constants::WGS84_a(), Constants::WGS84_f(), Constants::UTM_k0()); return utm; } // Engsager and Poder (2007) use trigonometric series to convert between phi // and phip. Here are the series... // // Conversion from phi to phip: // // phip = phi + sum(c[j] * sin(2*j*phi), j, 1, 6) // // c[1] = - 2 * n // + 2/3 * n^2 // + 4/3 * n^3 // - 82/45 * n^4 // + 32/45 * n^5 // + 4642/4725 * n^6; // c[2] = 5/3 * n^2 // - 16/15 * n^3 // - 13/9 * n^4 // + 904/315 * n^5 // - 1522/945 * n^6; // c[3] = - 26/15 * n^3 // + 34/21 * n^4 // + 8/5 * n^5 // - 12686/2835 * n^6; // c[4] = 1237/630 * n^4 // - 12/5 * n^5 // - 24832/14175 * n^6; // c[5] = - 734/315 * n^5 // + 109598/31185 * n^6; // c[6] = 444337/155925 * n^6; // // Conversion from phip to phi: // // phi = phip + sum(d[j] * sin(2*j*phip), j, 1, 6) // // d[1] = 2 * n // - 2/3 * n^2 // - 2 * n^3 // + 116/45 * n^4 // + 26/45 * n^5 // - 2854/675 * n^6; // d[2] = 7/3 * n^2 // - 8/5 * n^3 // - 227/45 * n^4 // + 2704/315 * n^5 // + 2323/945 * n^6; // d[3] = 56/15 * n^3 // - 136/35 * n^4 // - 1262/105 * n^5 // + 73814/2835 * n^6; // d[4] = 4279/630 * n^4 // - 332/35 * n^5 // - 399572/14175 * n^6; // d[5] = 4174/315 * n^5 // - 144838/6237 * n^6; // d[6] = 601676/22275 * n^6; // // In order to maintain sufficient relative accuracy close to the pole use // // S = sum(c[i]*sin(2*i*phi),i,1,6) // taup = (tau + tan(S)) / (1 - tau * tan(S)) // In Math::taupf and Math::tauf we evaluate the forward transform explicitly // and solve the reverse one by Newton's method. // // There are adapted from TransverseMercatorExact (taup and taupinv). tau = // tan(phi), taup = sinh(psi) void TransverseMercator::Forward(real lon0, real lat, real lon, real& x, real& y, real& gamma, real& k) const { lat = Math::LatFix(lat); lon = Math::AngDiff(lon0, lon); // Explicitly enforce the parity int latsign = signbit(lat) ? -1 : 1, lonsign = signbit(lon) ? -1 : 1; lon *= lonsign; lat *= latsign; bool backside = lon > Math::qd; if (backside) { if (lat == 0) latsign = -1; lon = Math::hd - lon; } real sphi, cphi, slam, clam; Math::sincosd(lat, sphi, cphi); Math::sincosd(lon, slam, clam); // phi = latitude // phi' = conformal latitude // psi = isometric latitude // tau = tan(phi) // tau' = tan(phi') // [xi', eta'] = Gauss-Schreiber TM coordinates // [xi, eta] = Gauss-Krueger TM coordinates // // We use // tan(phi') = sinh(psi) // sin(phi') = tanh(psi) // cos(phi') = sech(psi) // denom^2 = 1-cos(phi')^2*sin(lam)^2 = 1-sech(psi)^2*sin(lam)^2 // sin(xip) = sin(phi')/denom = tanh(psi)/denom // cos(xip) = cos(phi')*cos(lam)/denom = sech(psi)*cos(lam)/denom // cosh(etap) = 1/denom = 1/denom // sinh(etap) = cos(phi')*sin(lam)/denom = sech(psi)*sin(lam)/denom real etap, xip; if (lat != Math::qd) { real tau = sphi / cphi, taup = Math::taupf(tau, _es); xip = atan2(taup, clam); // Used to be // etap = Math::atanh(sin(lam) / cosh(psi)); etap = asinh(slam / hypot(taup, clam)); // convergence and scale for Gauss-Schreiber TM (xip, etap) -- gamma0 = // atan(tan(xip) * tanh(etap)) = atan(tan(lam) * sin(phi')); // sin(phi') = tau'/sqrt(1 + tau'^2) // Krueger p 22 (44) gamma = Math::atan2d(slam * taup, clam * hypot(real(1), taup)); // k0 = sqrt(1 - _e2 * sin(phi)^2) * (cos(phi') / cos(phi)) * cosh(etap) // Note 1/cos(phi) = cosh(psip); // and cos(phi') * cosh(etap) = 1/hypot(sinh(psi), cos(lam)) // // This form has cancelling errors. This property is lost if cosh(psip) // is replaced by 1/cos(phi), even though it's using "primary" data (phi // instead of psip). k = sqrt(_e2m + _e2 * Math::sq(cphi)) * hypot(real(1), tau) / hypot(taup, clam); } else { xip = Math::pi()/2; etap = 0; gamma = lon; k = _c; } // {xi',eta'} is {northing,easting} for Gauss-Schreiber transverse Mercator // (for eta' = 0, xi' = bet). {xi,eta} is {northing,easting} for transverse // Mercator with constant scale on the central meridian (for eta = 0, xip = // rectifying latitude). Define // // zeta = xi + i*eta // zeta' = xi' + i*eta' // // The conversion from conformal to rectifying latitude can be expressed as // a series in _n: // // zeta = zeta' + sum(h[j-1]' * sin(2 * j * zeta'), j = 1..maxpow_) // // where h[j]' = O(_n^j). The reversion of this series gives // // zeta' = zeta - sum(h[j-1] * sin(2 * j * zeta), j = 1..maxpow_) // // which is used in Reverse. // // Evaluate sums via Clenshaw method. See // https://en.wikipedia.org/wiki/Clenshaw_algorithm // // Let // // S = sum(a[k] * phi[k](x), k = 0..n) // phi[k+1](x) = alpha[k](x) * phi[k](x) + beta[k](x) * phi[k-1](x) // // Evaluate S with // // b[n+2] = b[n+1] = 0 // b[k] = alpha[k](x) * b[k+1] + beta[k+1](x) * b[k+2] + a[k] // S = (a[0] + beta[1](x) * b[2]) * phi[0](x) + b[1] * phi[1](x) // // Here we have // // x = 2 * zeta' // phi[k](x) = sin(k * x) // alpha[k](x) = 2 * cos(x) // beta[k](x) = -1 // [ sin(A+B) - 2*cos(B)*sin(A) + sin(A-B) = 0, A = k*x, B = x ] // n = maxpow_ // a[k] = _alp[k] // S = b[1] * sin(x) // // For the derivative we have // // x = 2 * zeta' // phi[k](x) = cos(k * x) // alpha[k](x) = 2 * cos(x) // beta[k](x) = -1 // [ cos(A+B) - 2*cos(B)*cos(A) + cos(A-B) = 0, A = k*x, B = x ] // a[0] = 1; a[k] = 2*k*_alp[k] // S = (a[0] - b[2]) + b[1] * cos(x) // // Matrix formulation (not used here): // phi[k](x) = [sin(k * x); k * cos(k * x)] // alpha[k](x) = 2 * [cos(x), 0; -sin(x), cos(x)] // beta[k](x) = -1 * [1, 0; 0, 1] // a[k] = _alp[k] * [1, 0; 0, 1] // b[n+2] = b[n+1] = [0, 0; 0, 0] // b[k] = alpha[k](x) * b[k+1] + beta[k+1](x) * b[k+2] + a[k] // N.B., for all k: b[k](1,2) = 0; b[k](1,1) = b[k](2,2) // S = (a[0] + beta[1](x) * b[2]) * phi[0](x) + b[1] * phi[1](x) // phi[0](x) = [0; 0] // phi[1](x) = [sin(x); cos(x)] real c0 = cos(2 * xip), ch0 = cosh(2 * etap), s0 = sin(2 * xip), sh0 = sinh(2 * etap); complex a(2 * c0 * ch0, -2 * s0 * sh0); // 2 * cos(2*zeta') int n = maxpow_; complex y0(n & 1 ? _alp[n] : 0), y1, // default initializer is 0+i0 z0(n & 1 ? 2*n * _alp[n] : 0), z1; if (n & 1) --n; while (n) { y1 = a * y0 - y1 + _alp[n]; z1 = a * z0 - z1 + 2*n * _alp[n]; --n; y0 = a * y1 - y0 + _alp[n]; z0 = a * z1 - z0 + 2*n * _alp[n]; --n; } a /= real(2); // cos(2*zeta') z1 = real(1) - z1 + a * z0; a = complex(s0 * ch0, c0 * sh0); // sin(2*zeta') y1 = complex(xip, etap) + a * y0; // Fold in change in convergence and scale for Gauss-Schreiber TM to // Gauss-Krueger TM. gamma -= Math::atan2d(z1.imag(), z1.real()); k *= _b1 * abs(z1); real xi = y1.real(), eta = y1.imag(); y = _a1 * _k0 * (backside ? Math::pi() - xi : xi) * latsign; x = _a1 * _k0 * eta * lonsign; if (backside) gamma = Math::hd - gamma; gamma *= latsign * lonsign; gamma = Math::AngNormalize(gamma); k *= _k0; } void TransverseMercator::Reverse(real lon0, real x, real y, real& lat, real& lon, real& gamma, real& k) const { // This undoes the steps in Forward. The wrinkles are: (1) Use of the // reverted series to express zeta' in terms of zeta. (2) Newton's method // to solve for phi in terms of tan(phi). real xi = y / (_a1 * _k0), eta = x / (_a1 * _k0); // Explicitly enforce the parity int xisign = signbit(xi) ? -1 : 1, etasign = signbit(eta) ? -1 : 1; xi *= xisign; eta *= etasign; bool backside = xi > Math::pi()/2; if (backside) xi = Math::pi() - xi; real c0 = cos(2 * xi), ch0 = cosh(2 * eta), s0 = sin(2 * xi), sh0 = sinh(2 * eta); complex a(2 * c0 * ch0, -2 * s0 * sh0); // 2 * cos(2*zeta) int n = maxpow_; complex y0(n & 1 ? -_bet[n] : 0), y1, // default initializer is 0+i0 z0(n & 1 ? -2*n * _bet[n] : 0), z1; if (n & 1) --n; while (n) { y1 = a * y0 - y1 - _bet[n]; z1 = a * z0 - z1 - 2*n * _bet[n]; --n; y0 = a * y1 - y0 - _bet[n]; z0 = a * z1 - z0 - 2*n * _bet[n]; --n; } a /= real(2); // cos(2*zeta) z1 = real(1) - z1 + a * z0; a = complex(s0 * ch0, c0 * sh0); // sin(2*zeta) y1 = complex(xi, eta) + a * y0; // Convergence and scale for Gauss-Schreiber TM to Gauss-Krueger TM. gamma = Math::atan2d(z1.imag(), z1.real()); k = _b1 / abs(z1); // JHS 154 has // // phi' = asin(sin(xi') / cosh(eta')) (Krueger p 17 (25)) // lam = asin(tanh(eta') / cos(phi') // psi = asinh(tan(phi')) real xip = y1.real(), etap = y1.imag(), s = sinh(etap), c = fmax(real(0), cos(xip)), // cos(pi/2) might be negative r = hypot(s, c); if (r != 0) { lon = Math::atan2d(s, c); // Krueger p 17 (25) // Use Newton's method to solve for tau real sxip = sin(xip), tau = Math::tauf(sxip/r, _es); gamma += Math::atan2d(sxip * tanh(etap), c); // Krueger p 19 (31) lat = Math::atand(tau); // Note cos(phi') * cosh(eta') = r k *= sqrt(_e2m + _e2 / (1 + Math::sq(tau))) * hypot(real(1), tau) * r; } else { lat = Math::qd; lon = 0; k *= _c; } lat *= xisign; if (backside) lon = Math::hd - lon; lon *= etasign; lon = Math::AngNormalize(lon + lon0); if (backside) gamma = Math::hd - gamma; gamma *= xisign * etasign; gamma = Math::AngNormalize(gamma); k *= _k0; } } // namespace GeographicLib geosphere/src/a_util.c0000644000176200001440000000116114161543551014447 0ustar liggesusers#include #ifndef M_PI #define M_PI (3.1415926535897932384626433) #endif #ifndef M_2PI #define M_2PI (3.1415926535897932384626433 * 2.0) #endif #ifndef M_PI_2 #define M_PI_2 (3.1415926535897932384626433 / 2) #endif double mod(double x, double n) { return(x - n * floor(x/n)); } double normalizeLonDeg(double lon) { return( mod( (lon + 180), 360 ) - 180 ); } double normalizeLonRad(double lon) { return( mod( (lon + M_PI), M_2PI) - M_PI); } /* Convert degrees to radians */ double toRad(double deg) { return deg * 0.0174532925199433; } double toDeg(double rad) { return rad * 57.2957795130823 ; } geosphere/src/GeodesicExact.cpp0000644000176200001440000016340114323377375016260 0ustar liggesusers/** * \file GeodesicExact.cpp * \brief Implementation for GeographicLib::GeodesicExact class * * Copyright (c) Charles Karney (2012-2022) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ * * This is a reformulation of the geodesic problem. The notation is as * follows: * - at a general point (no suffix or 1 or 2 as suffix) * - phi = latitude * - beta = latitude on auxiliary sphere * - omega = longitude on auxiliary sphere * - lambda = longitude * - alpha = azimuth of great circle * - sigma = arc length along great circle * - s = distance * - tau = scaled distance (= sigma at multiples of pi/2) * - at northwards equator crossing * - beta = phi = 0 * - omega = lambda = 0 * - alpha = alpha0 * - sigma = s = 0 * - a 12 suffix means a difference, e.g., s12 = s2 - s1. * - s and c prefixes mean sin and cos **********************************************************************/ #include "GeodesicExact.h" #include "GeodesicLineExact.h" #if defined(_MSC_VER) // Squelch warnings about potentially uninitialized local variables, // constant conditional and enum-float expressions and mixing enums # pragma warning (disable: 4701 4127 5055 5054) #endif namespace GeographicLib { using namespace std; GeodesicExact::GeodesicExact(real a, real f) : maxit2_(maxit1_ + Math::digits() + 10) // Underflow guard. We require // tiny_ * epsilon() > 0 // tiny_ + epsilon() == epsilon() , tiny_(sqrt(numeric_limits::min())) , tol0_(numeric_limits::epsilon()) // Increase multiplier in defn of tol1_ from 100 to 200 to fix inverse // case 52.784459512564 0 -52.784459512563990912 179.634407464943777557 // which otherwise failed for Visual Studio 10 (Release and Debug) , tol1_(200 * tol0_) , tol2_(sqrt(tol0_)) , tolb_(tol0_) // Check on bisection interval , xthresh_(1000 * tol2_) , _a(a) , _f(f) , _f1(1 - _f) , _e2(_f * (2 - _f)) , _ep2(_e2 / Math::sq(_f1)) // e2 / (1 - e2) , _n(_f / ( 2 - _f)) , _b(_a * _f1) // The Geodesic class substitutes atanh(sqrt(e2)) for asinh(sqrt(ep2)) in // the definition of _c2. The latter is more accurate for very oblate // ellipsoids (which the Geodesic class does not attempt to handle). , _c2((Math::sq(_a) + Math::sq(_b) * (_f == 0 ? 1 : (_f > 0 ? asinh(sqrt(_ep2)) : atan(sqrt(-_e2))) / sqrt(fabs(_e2))))/2) // authalic radius squared // The sig12 threshold for "really short". Using the auxiliary sphere // solution with dnm computed at (bet1 + bet2) / 2, the relative error in // the azimuth consistency check is sig12^2 * abs(f) * min(1, 1-f/2) / 2. // (Error measured for 1/100 < b/a < 100 and abs(f) >= 1/1000. For a // given f and sig12, the max error occurs for lines near the pole. If // the old rule for computing dnm = (dn1 + dn2)/2 is used, then the error // increases by a factor of 2.) Setting this equal to epsilon gives // sig12 = etol2. Here 0.1 is a safety factor (error decreased by 100) // and max(0.001, abs(f)) stops etol2 getting too large in the nearly // spherical case. , _etol2(real(0.1) * tol2_ / sqrt( fmax(real(0.001), fabs(_f)) * fmin(real(1), 1 - _f/2) / 2 )) { if (!(isfinite(_a) && _a > 0)) throw GeographicErr("Equatorial radius is not positive"); if (!(isfinite(_b) && _b > 0)) throw GeographicErr("Polar semi-axis is not positive"); // Required number of terms in DST for full accuracy for all precisions as // a function of n in [-0.99, 0.99]. Values determined by running // develop/AreaEst compiled with GEOGRAPHICLIB_PRECISION = 5. For // precision 4 and 5, GEOGRAPHICLIB_DIGITS was set to, resp., 384 and 768. // The error criterion is relative error less than or equal to epsilon/2 = // 0.5^digits, with digits = 24, 53, 64, 113, 256. The first 4 are the the // "standard" values for float, double, long double, and float128; the last // is the default for GeographicLib + mpfr. Also listed is the value of // alp0 resulting in the most error for the given N. // // float double long double quad mpfr // n N alp0 N alp0 N alp0 N alp0 N alp0 // -0.99 1024 0.09 3072 0.05 4096 0.04 8192 43.50 16384 22.01 // -0.98 512 0.18 1536 0.10 2048 0.09 4096 0.06 8192 0.04 // -0.97 384 0.25 1024 0.16 1536 0.13 3072 0.09 6144 0.06 // -0.96 256 0.36 768 0.21 1024 0.18 2048 0.13 4096 0.09 // -0.95 192 0.47 768 0.23 768 0.23 1536 0.17 4096 0.10 // -0.94 192 0.51 512 0.31 768 0.26 1536 0.18 3072 0.13 // -0.93 192 0.55 384 0.39 512 0.34 1024 0.24 3072 0.14 // -0.92 128 0.73 384 0.42 512 0.37 1024 0.26 2048 0.18 // -0.91 128 0.77 384 0.45 384 0.45 768 0.32 2048 0.19 // -0.90 96 0.94 256 0.58 384 0.47 768 0.34 2048 0.21 // -0.89 96 0.99 256 0.61 384 0.50 768 0.35 1536 0.25 // -0.88 96 1.04 256 0.64 384 0.52 768 0.37 1536 0.26 // -0.87 96 1.09 192 0.77 256 0.67 512 0.47 1536 0.27 // -0.86 64 1.38 192 0.80 256 0.69 512 0.49 1536 0.28 // -0.85 64 1.43 192 0.83 256 0.72 512 0.51 1024 0.36 // -0.84 64 1.49 192 0.86 256 0.75 384 0.61 1024 0.37 // -0.83 64 1.54 192 0.89 192 0.89 384 0.63 1024 0.39 // -0.82 48 1.82 192 0.92 192 0.92 384 0.65 1024 0.40 // -0.81 48 1.88 128 1.16 192 0.95 384 0.67 1024 0.41 // -0.80 48 1.94 128 1.19 192 0.97 384 0.69 768 0.49 // -0.79 48 1.99 128 1.23 192 1.00 384 0.71 768 0.50 // -0.78 48 2.04 128 1.26 192 1.03 384 0.73 768 0.51 // -0.77 48 2.10 128 1.29 192 1.05 256 0.91 768 0.53 // -0.76 48 2.15 128 1.32 128 1.32 256 0.93 768 0.54 // -0.75 48 2.20 96 1.56 128 1.35 256 0.96 768 0.55 // -0.74 32 2.74 96 1.60 128 1.38 256 0.98 768 0.57 // -0.73 32 2.81 96 1.63 128 1.41 256 1.00 768 0.58 // -0.72 32 2.87 96 1.67 128 1.44 256 1.02 512 0.72 // -0.71 32 2.93 96 1.70 128 1.47 192 1.20 512 0.74 // -0.70 32 2.99 96 1.73 96 1.73 192 1.23 512 0.75 // -0.69 32 3.05 96 1.77 96 1.77 192 1.25 512 0.77 // -0.68 32 3.11 96 1.80 96 1.80 192 1.28 512 0.78 // -0.67 24 3.64 96 1.84 96 1.84 192 1.30 512 0.80 // -0.66 24 3.71 96 1.87 96 1.87 192 1.32 512 0.81 // -0.65 24 3.77 64 2.33 96 1.90 192 1.35 384 0.95 // -0.64 24 3.84 64 2.37 96 1.93 192 1.37 384 0.97 // -0.63 24 3.90 64 2.41 96 1.97 192 1.39 384 0.98 // -0.62 24 3.97 64 2.45 96 2.00 192 1.42 384 1.00 // -0.61 24 4.04 64 2.49 96 2.03 192 1.44 384 1.02 // -0.60 24 4.10 64 2.53 96 2.06 192 1.46 384 1.03 // -0.59 24 4.16 64 2.57 64 2.57 128 1.82 384 1.05 // -0.58 24 4.23 64 2.60 64 2.60 128 1.84 384 1.07 // -0.57 24 4.29 48 3.05 64 2.64 128 1.87 384 1.08 // -0.56 24 4.36 48 3.10 64 2.68 128 1.90 384 1.10 // -0.55 16 5.38 48 3.14 64 2.72 128 1.93 384 1.11 // -0.54 16 5.46 48 3.19 64 2.76 128 1.96 384 1.13 // -0.53 16 5.54 48 3.23 64 2.80 128 1.98 256 1.40 // -0.52 16 5.61 48 3.27 64 2.84 128 2.01 256 1.42 // -0.51 16 5.69 48 3.32 64 2.88 128 2.04 256 1.44 // -0.50 16 5.77 48 3.36 64 2.92 96 2.38 256 1.46 // -0.49 16 5.85 48 3.41 48 3.41 96 2.42 256 1.48 // -0.48 16 5.92 48 3.45 48 3.45 96 2.45 256 1.50 // -0.47 16 6.00 48 3.50 48 3.50 96 2.48 256 1.52 // -0.46 16 6.08 48 3.54 48 3.54 96 2.51 256 1.54 // -0.45 12 7.06 48 3.59 48 3.59 96 2.54 256 1.56 // -0.44 12 7.15 48 3.63 48 3.63 96 2.57 256 1.58 // -0.43 12 7.24 32 4.49 48 3.68 96 2.61 256 1.60 // -0.42 12 7.33 32 4.55 48 3.72 96 2.64 192 1.87 // -0.41 12 7.42 32 4.60 48 3.77 96 2.67 192 1.89 // -0.40 12 7.51 32 4.66 48 3.81 96 2.70 192 1.91 // -0.39 12 7.60 32 4.71 48 3.86 96 2.73 192 1.94 // -0.38 12 7.69 32 4.77 48 3.90 96 2.77 192 1.96 // -0.37 12 7.77 32 4.82 48 3.95 96 2.80 192 1.98 // -0.36 12 7.86 32 4.88 48 3.99 96 2.83 192 2.00 // -0.35 12 7.95 32 4.94 32 4.94 64 3.50 192 2.03 // -0.34 12 8.04 32 4.99 32 4.99 64 3.54 192 2.05 // -0.33 12 8.13 24 5.81 32 5.05 64 3.58 192 2.07 // -0.32 12 8.22 24 5.88 32 5.10 64 3.62 192 2.10 // -0.31 12 8.31 24 5.94 32 5.16 64 3.66 192 2.12 // -0.30 8 10.16 24 6.01 32 5.22 64 3.70 192 2.14 // -0.29 8 10.27 24 6.07 32 5.27 64 3.74 192 2.17 // -0.28 8 10.38 24 6.14 32 5.33 64 3.78 128 2.68 // -0.27 8 10.49 24 6.20 32 5.39 64 3.82 128 2.71 // -0.26 8 10.60 24 6.27 32 5.45 64 3.87 128 2.74 // -0.25 8 10.72 24 6.34 32 5.50 48 4.51 128 2.77 // -0.24 8 10.83 24 6.40 24 6.40 48 4.55 128 2.80 // -0.23 8 10.94 24 6.47 24 6.47 48 4.60 128 2.83 // -0.22 8 11.05 24 6.54 24 6.54 48 4.65 128 2.86 // -0.21 6 12.72 24 6.60 24 6.60 48 4.70 128 2.89 // -0.20 6 12.85 24 6.67 24 6.67 48 4.75 128 2.92 // -0.19 6 12.97 16 8.21 24 6.74 48 4.80 128 2.95 // -0.18 6 13.10 16 8.29 24 6.81 48 4.85 96 3.44 // -0.17 6 13.23 16 8.37 24 6.88 48 4.89 96 3.47 // -0.16 6 13.36 16 8.46 24 6.95 48 4.94 96 3.51 // -0.15 6 13.49 16 8.54 24 7.02 48 5.00 96 3.54 // -0.14 6 13.62 16 8.62 24 7.09 48 5.05 96 3.58 // -0.13 6 13.75 16 8.71 24 7.16 48 5.10 96 3.62 // -0.12 6 13.88 16 8.80 16 8.80 32 6.28 96 3.65 // -0.11 6 14.01 12 10.19 16 8.88 32 6.35 96 3.69 // -0.10 4 16.85 12 10.28 16 8.97 32 6.41 96 3.73 // -0.09 4 17.01 12 10.38 16 9.06 32 6.47 96 3.76 // -0.08 4 17.17 12 10.48 16 9.14 32 6.54 96 3.80 // -0.07 4 17.32 12 10.58 16 9.23 32 6.60 64 4.69 // -0.06 4 17.48 12 10.69 12 10.69 24 7.67 64 4.74 // -0.05 4 17.64 12 10.79 12 10.79 24 7.75 64 4.79 // -0.04 4 17.80 12 10.89 12 10.89 24 7.82 64 4.84 // -0.03 4 17.96 8 13.26 12 10.99 24 7.90 48 5.63 // -0.02 4 18.11 8 13.38 12 11.10 24 7.97 48 5.68 // -0.01 4 18.27 6 15.36 8 13.51 16 9.78 48 5.74 // 0.00 4 1.00 4 1.00 4 1.00 4 1.00 4 1.00 // 0.01 4 18.57 6 15.62 8 13.75 16 9.96 48 5.85 // 0.02 4 18.70 8 13.86 12 11.51 24 8.28 48 5.91 // 0.03 4 18.83 8 13.97 12 11.61 24 8.36 48 5.97 // 0.04 4 18.96 12 11.71 12 11.71 24 8.44 64 5.23 // 0.05 4 19.09 12 11.81 12 11.81 24 8.52 64 5.28 // 0.06 4 19.22 12 11.92 12 11.92 24 8.60 64 5.33 // 0.07 4 19.36 12 12.02 16 10.52 32 7.55 64 5.39 // 0.08 4 19.49 12 12.13 16 10.61 32 7.63 64 5.44 // 0.09 4 19.62 12 12.23 16 10.71 32 7.70 96 4.50 // 0.10 4 19.76 12 12.34 16 10.80 32 7.77 96 4.54 // 0.11 4 19.89 12 12.45 16 10.90 32 7.85 96 4.59 // 0.12 6 17.01 16 11.00 16 11.00 32 7.92 96 4.63 // 0.13 6 17.14 16 11.10 16 11.10 32 8.00 96 4.68 // 0.14 6 17.27 16 11.20 24 9.26 48 6.64 96 4.73 // 0.15 6 17.40 16 11.30 24 9.35 48 6.70 96 4.77 // 0.16 6 17.53 16 11.40 24 9.44 48 6.77 96 4.82 // 0.17 6 17.67 16 11.51 24 9.53 48 6.84 96 4.87 // 0.18 6 17.80 16 11.61 24 9.62 48 6.90 96 4.92 // 0.19 6 17.94 16 11.72 24 9.71 48 6.97 128 4.31 // 0.20 6 18.07 16 11.83 24 9.80 48 7.04 128 4.36 // 0.21 6 18.21 24 9.90 24 9.90 48 7.11 128 4.40 // 0.22 6 18.35 24 9.99 24 9.99 48 7.18 128 4.45 // 0.23 6 18.49 24 10.09 24 10.09 48 7.26 128 4.49 // 0.24 6 18.63 24 10.19 24 10.19 48 7.33 128 4.54 // 0.25 6 18.77 24 10.28 24 10.28 48 7.41 128 4.59 // 0.26 6 18.92 24 10.39 24 10.39 48 7.48 128 4.64 // 0.27 8 17.00 24 10.49 32 9.17 64 6.58 128 4.69 // 0.28 8 17.14 24 10.59 32 9.26 64 6.65 128 4.74 // 0.29 8 17.28 24 10.69 32 9.35 64 6.72 192 3.92 // 0.30 8 17.43 24 10.80 32 9.45 64 6.79 192 3.96 // 0.31 8 17.57 24 10.91 32 9.54 64 6.86 192 4.01 // 0.32 8 17.72 24 11.02 32 9.64 64 6.93 192 4.05 // 0.33 8 17.87 24 11.13 32 9.74 64 7.01 192 4.09 // 0.34 8 18.02 24 11.24 32 9.84 64 7.08 192 4.14 // 0.35 8 18.17 24 11.36 32 9.95 64 7.16 192 4.19 // 0.36 8 18.32 24 11.47 32 10.05 64 7.24 192 4.23 // 0.37 8 18.48 32 10.16 32 10.16 64 7.32 192 4.28 // 0.38 8 18.63 32 10.27 32 10.27 96 6.08 192 4.33 // 0.39 8 18.79 32 10.38 48 8.58 96 6.15 192 4.38 // 0.40 8 18.95 32 10.49 48 8.68 96 6.22 192 4.43 // 0.41 8 19.11 32 10.60 48 8.78 96 6.30 192 4.49 // 0.42 12 16.45 32 10.72 48 8.88 96 6.37 192 4.54 // 0.43 12 16.61 32 10.84 48 8.98 96 6.45 192 4.59 // 0.44 12 16.77 32 10.96 48 9.08 96 6.52 256 4.04 // 0.45 12 16.93 32 11.09 48 9.19 96 6.60 256 4.09 // 0.46 12 17.10 32 11.21 48 9.30 96 6.68 256 4.14 // 0.47 12 17.26 32 11.34 48 9.41 96 6.77 256 4.19 // 0.48 12 17.44 32 11.47 48 9.52 96 6.85 256 4.24 // 0.49 12 17.61 48 9.64 48 9.64 96 6.94 256 4.30 // 0.50 12 17.79 48 9.76 48 9.76 96 7.03 256 4.35 // 0.51 12 17.97 48 9.88 48 9.88 96 7.12 256 4.41 // 0.52 12 18.15 48 10.00 48 10.00 96 7.21 256 4.47 // 0.53 12 18.34 48 10.13 48 10.13 128 6.36 256 4.53 // 0.54 12 18.53 48 10.26 48 10.26 128 6.45 256 4.59 // 0.55 12 18.72 48 10.40 64 9.10 128 6.53 384 3.82 // 0.56 12 18.92 48 10.53 64 9.22 128 6.63 384 3.87 // 0.57 12 19.12 48 10.68 64 9.35 128 6.72 384 3.93 // 0.58 12 19.33 48 10.82 64 9.48 128 6.82 384 3.98 // 0.59 12 19.54 48 10.97 64 9.61 128 6.92 384 4.05 // 0.60 12 19.75 48 11.13 64 9.75 128 7.02 384 4.11 // 0.61 12 19.97 48 11.28 64 9.89 128 7.13 384 4.17 // 0.62 12 20.20 48 11.45 64 10.04 128 7.24 384 4.24 // 0.63 16 18.39 48 11.62 64 10.19 192 6.05 384 4.31 // 0.64 16 18.62 64 10.35 64 10.35 192 6.15 384 4.38 // 0.65 16 18.86 64 10.51 96 8.71 192 6.25 384 4.45 // 0.66 16 19.10 64 10.68 96 8.85 192 6.36 384 4.53 // 0.67 16 19.35 64 10.86 96 9.00 192 6.47 512 4.00 // 0.68 16 19.60 64 11.04 96 9.16 192 6.58 512 4.08 // 0.69 16 19.87 64 11.23 96 9.32 192 6.70 512 4.15 // 0.70 16 20.14 64 11.43 96 9.49 192 6.83 512 4.23 // 0.71 16 20.42 64 11.63 96 9.67 192 6.96 512 4.32 // 0.72 16 20.71 64 11.85 96 9.85 192 7.10 512 4.40 // 0.73 16 21.01 96 10.04 96 10.04 192 7.25 512 4.50 // 0.74 16 21.32 96 10.25 96 10.25 256 6.44 768 3.76 // 0.75 16 21.65 96 10.46 96 10.46 256 6.58 768 3.84 // 0.76 16 21.99 96 10.68 128 9.36 256 6.73 768 3.93 // 0.77 24 19.41 96 10.92 128 9.57 256 6.89 768 4.03 // 0.78 24 19.76 96 11.17 128 9.79 256 7.06 768 4.13 // 0.79 24 20.13 96 11.44 128 10.03 256 7.24 768 4.24 // 0.80 24 20.51 96 11.72 128 10.29 384 6.11 768 4.35 // 0.81 24 20.92 96 12.02 128 10.56 384 6.28 768 4.48 // 0.82 24 21.35 96 12.34 192 8.99 384 6.46 1024 4.00 // 0.83 24 21.81 128 11.16 192 9.26 384 6.66 1024 4.13 // 0.84 24 22.29 128 11.50 192 9.55 384 6.88 1024 4.26 // 0.85 24 22.82 128 11.86 192 9.87 384 7.11 1024 4.41 // 0.86 24 23.38 128 12.26 192 10.21 384 7.37 1024 4.58 // 0.87 24 24.00 128 12.70 192 10.59 512 6.67 1536 3.90 // 0.88 24 24.67 192 11.01 192 11.01 512 6.95 1536 4.06 // 0.89 24 25.41 192 11.48 256 10.07 512 7.26 1536 4.25 // 0.90 24 26.24 192 12.00 256 10.54 768 6.27 1536 4.47 // 0.91 24 27.17 192 12.61 256 11.09 768 6.62 2048 4.10 // 0.92 24 28.23 192 13.30 384 9.74 768 7.02 2048 4.35 // 0.93 24 29.45 256 12.46 384 10.38 768 7.50 3072 3.82 // 0.94 24 30.86 256 13.36 384 11.16 1024 7.05 3072 4.13 // 0.95 24 32.53 384 12.14 512 10.67 1024 7.72 3072 4.53 // 0.96 24 34.51 384 13.42 512 11.83 1536 7.09 4096 4.40 // 0.97 24 36.88 512 13.45 768 11.24 2048 7.11 6144 4.16 // 0.98 16 41.78 768 13.48 1024 11.88 3072 7.12 8192 4.42 // 0.99 8 44.82 1024 16.00 1536 13.51 6144 7.14 16384 4.43 static const int ndiv = 100; // Encode N as small integer: 2,3,4,6,8,12... -> 0,1,2,3,4,5... // using this awk script // // { // n = $1; // if (n % 3 == 0) { // s = 1; // n = n/3; // } else { // s = 0; // n = n/2; // } // p = int( log(n)/log(2)+0.5 ); // printf "%d\n", 2*p+s; // } // // A couple of changes have been made: (1) the decrease in N for float and // n > 0.97 has been removed; (2) entrys of n=+/-1 have been included // (incrementing the previous code value by 1). #if GEOGRAPHICLIB_PRECISION == 1 static const unsigned char narr[2*ndiv+1] = { 19,18,16,15,14,13,13,13,12,12,11,11,11,11,10,10,10,10,9,9,9,9,9,9,9,9,8, 8,8,8,8,8,8,7,7,7,7,7,7,7,7,7,7,7,7,6,6,6,6,6,6,6,6,6,6,5,5,5,5,5,5,5,5, 5,5,5,5,5,5,5,4,4,4,4,4,4,4,4,4,3,3,3,3,3,3,3,3,3,3,3,2,2,2,2,2,2,2,2,2, 2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4, 4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6, 6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8 }; #elif GEOGRAPHICLIB_PRECISION == 2 static const unsigned char narr[2*ndiv+1] = { 22,21,19,18,17,17,16,15,15,15,14,14,14,13,13,13,13,13,13,12,12,12,12,12, 12,11,11,11,11,11,11,11,11,11,11,10,10,10,10,10,10,10,10,9,9,9,9,9,9,9,9, 9,9,9,9,9,9,8,8,8,8,8,8,8,8,8,8,7,7,7,7,7,7,7,7,7,7,7,7,7,7,6,6,6,6,6,6, 6,6,5,5,5,5,5,5,5,5,4,4,3,2,3,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7, 7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9, 9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,12,12, 12,12,12,13,13,13,13,13,14,14,15,15,16,17,18,19 }; #elif GEOGRAPHICLIB_PRECISION == 3 static const unsigned char narr[2*ndiv+1] = { 23,22,20,19,18,17,17,16,16,15,15,15,15,14,14,14,14,13,13,13,13,13,13,13, 12,12,12,12,12,12,11,11,11,11,11,11,11,11,11,11,11,10,10,10,10,10,10,10, 10,10,10,9,9,9,9,9,9,9,9,9,9,9,9,9,9,8,8,8,8,8,8,8,8,8,8,8,7,7,7,7,7,7,7, 7,7,7,7,7,6,6,6,6,6,6,5,5,5,5,5,4,2,4,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7, 7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9, 10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,12,12,12, 12,12,12,13,13,13,13,13,13,13,14,14,14,15,15,15,16,16,17,18,19,20 }; #elif GEOGRAPHICLIB_PRECISION == 4 static const unsigned char narr[2*ndiv+1] = { 25,24,22,21,20,19,19,18,18,17,17,17,17,16,16,16,15,15,15,15,15,15,15,14, 14,14,14,14,14,13,13,13,13,13,13,13,13,13,13,13,13,12,12,12,12,12,12,12, 12,12,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,10,10,10,10,10,10,10, 10,10,10,9,9,9,9,9,9,9,9,9,9,9,9,9,8,8,8,8,8,8,7,7,7,7,7,6,2,6,7,7,7,7,7, 8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10, 11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12, 12,13,13,13,13,13,13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15, 15,16,16,16,17,17,17,17,18,18,19,20,21,23,24 }; #elif GEOGRAPHICLIB_PRECISION == 5 static const unsigned char narr[2*ndiv+1] = { 27,26,24,23,22,22,21,21,20,20,20,19,19,19,19,18,18,18,18,18,17,17,17,17, 17,17,17,17,16,16,16,16,16,16,16,15,15,15,15,15,15,15,15,15,15,15,15,14, 14,14,14,14,14,14,14,14,14,14,13,13,13,13,13,13,13,13,13,13,13,13,13,13, 12,12,12,12,12,12,12,12,12,12,11,11,11,11,11,11,11,11,11,11,11,10,10,10, 10,9,9,9,2,9,9,9,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,12,12,12, 12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14, 14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,16,16,16, 16,16,16,16,17,17,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,20,20,21, 21,21,22,23,24,26,27 }; #else #error "Bad value for GEOGRAPHICLIB_PRECISION" #endif real n = ndiv * _n; // n in [-ndiv, ndiv] int j = ndiv + int(n < 0 ? floor(n) : ceil(n)); // j in [0, 2*ndiv] int N = int(narr[j]); // Decode 0,1,2,3,4,5... -> 2,3,4,6,8,12... N = (N % 2 == 0 ? 2 : 3) * (1 << (N/2)); #if GEOGRAPHICLIB_PRECISION == 5 if (Math::digits() > 256) { // Scale up N by the number of digits in the precision relative to // the number used for the test = 256. int M = (Math::digits() * N) / 256; while (N < M) N = N % 3 == 0 ? 4*N/3 : 3*N/2; } #endif _fft.reset(N); _nC4 = N; } const GeodesicExact& GeodesicExact::WGS84() { static const GeodesicExact wgs84(Constants::WGS84_a(), Constants::WGS84_f()); return wgs84; } GeodesicLineExact GeodesicExact::Line(real lat1, real lon1, real azi1, unsigned caps) const { return GeodesicLineExact(*this, lat1, lon1, azi1, caps); } Math::real GeodesicExact::GenDirect(real lat1, real lon1, real azi1, bool arcmode, real s12_a12, unsigned outmask, real& lat2, real& lon2, real& azi2, real& s12, real& m12, real& M12, real& M21, real& S12) const { // Automatically supply DISTANCE_IN if necessary if (!arcmode) outmask |= DISTANCE_IN; return GeodesicLineExact(*this, lat1, lon1, azi1, outmask) . // Note the dot! GenPosition(arcmode, s12_a12, outmask, lat2, lon2, azi2, s12, m12, M12, M21, S12); } GeodesicLineExact GeodesicExact::GenDirectLine(real lat1, real lon1, real azi1, bool arcmode, real s12_a12, unsigned caps) const { azi1 = Math::AngNormalize(azi1); real salp1, calp1; // Guard against underflow in salp0. Also -0 is converted to +0. Math::sincosd(Math::AngRound(azi1), salp1, calp1); // Automatically supply DISTANCE_IN if necessary if (!arcmode) caps |= DISTANCE_IN; return GeodesicLineExact(*this, lat1, lon1, azi1, salp1, calp1, caps, arcmode, s12_a12); } GeodesicLineExact GeodesicExact::DirectLine(real lat1, real lon1, real azi1, real s12, unsigned caps) const { return GenDirectLine(lat1, lon1, azi1, false, s12, caps); } GeodesicLineExact GeodesicExact::ArcDirectLine(real lat1, real lon1, real azi1, real a12, unsigned caps) const { return GenDirectLine(lat1, lon1, azi1, true, a12, caps); } Math::real GeodesicExact::GenInverse(real lat1, real lon1, real lat2, real lon2, unsigned outmask, real& s12, real& salp1, real& calp1, real& salp2, real& calp2, real& m12, real& M12, real& M21, real& S12) const { // Compute longitude difference (AngDiff does this carefully). Result is // in [-180, 180] but -180 is only for west-going geodesics. 180 is for // east-going and meridional geodesics. using std::isnan; // Needed for Centos 7, ubuntu 14 real lon12s, lon12 = Math::AngDiff(lon1, lon2, lon12s); // Make longitude difference positive. int lonsign = signbit(lon12) ? -1 : 1; lon12 *= lonsign; lon12s *= lonsign; real lam12 = lon12 * Math::degree(), slam12, clam12; // Calculate sincos of lon12 + error (this applies AngRound internally). Math::sincosde(lon12, lon12s, slam12, clam12); // the supplementary longitude difference lon12s = (Math::hd - lon12) - lon12s; // If really close to the equator, treat as on equator. lat1 = Math::AngRound(Math::LatFix(lat1)); lat2 = Math::AngRound(Math::LatFix(lat2)); // Swap points so that point with higher (abs) latitude is point 1 // If one latitude is a nan, then it becomes lat1. int swapp = fabs(lat1) < fabs(lat2) || isnan(lat2) ? -1 : 1; if (swapp < 0) { lonsign *= -1; swap(lat1, lat2); } // Make lat1 <= -0 int latsign = signbit(lat1) ? 1 : -1; lat1 *= latsign; lat2 *= latsign; // Now we have // // 0 <= lon12 <= 180 // -90 <= lat1 <= -0 // lat1 <= lat2 <= -lat1 // // longsign, swapp, latsign register the transformation to bring the // coordinates to this canonical form. In all cases, 1 means no change was // made. We make these transformations so that there are few cases to // check, e.g., on verifying quadrants in atan2. In addition, this // enforces some symmetries in the results returned. real sbet1, cbet1, sbet2, cbet2, s12x, m12x; // Initialize for the meridian. No longitude calculation is done in this // case to let the parameter default to 0. EllipticFunction E(-_ep2); Math::sincosd(lat1, sbet1, cbet1); sbet1 *= _f1; // Ensure cbet1 = +epsilon at poles; doing the fix on beta means that sig12 // will be <= 2*tiny for two points at the same pole. Math::norm(sbet1, cbet1); cbet1 = fmax(tiny_, cbet1); Math::sincosd(lat2, sbet2, cbet2); sbet2 *= _f1; // Ensure cbet2 = +epsilon at poles Math::norm(sbet2, cbet2); cbet2 = fmax(tiny_, cbet2); // If cbet1 < -sbet1, then cbet2 - cbet1 is a sensitive measure of the // |bet1| - |bet2|. Alternatively (cbet1 >= -sbet1), abs(sbet2) + sbet1 is // a better measure. This logic is used in assigning calp2 in Lambda12. // Sometimes these quantities vanish and in that case we force bet2 = +/- // bet1 exactly. An example where is is necessary is the inverse problem // 48.522876735459 0 -48.52287673545898293 179.599720456223079643 // which failed with Visual Studio 10 (Release and Debug) if (cbet1 < -sbet1) { if (cbet2 == cbet1) sbet2 = copysign(sbet1, sbet2); } else { if (fabs(sbet2) == -sbet1) cbet2 = cbet1; } real dn1 = (_f >= 0 ? sqrt(1 + _ep2 * Math::sq(sbet1)) : sqrt(1 - _e2 * Math::sq(cbet1)) / _f1), dn2 = (_f >= 0 ? sqrt(1 + _ep2 * Math::sq(sbet2)) : sqrt(1 - _e2 * Math::sq(cbet2)) / _f1); real a12, sig12; bool meridian = lat1 == -Math::qd || slam12 == 0; if (meridian) { // Endpoints are on a single full meridian, so the geodesic might lie on // a meridian. calp1 = clam12; salp1 = slam12; // Head to the target longitude calp2 = 1; salp2 = 0; // At the target we're heading north real // tan(bet) = tan(sig) * cos(alp) ssig1 = sbet1, csig1 = calp1 * cbet1, ssig2 = sbet2, csig2 = calp2 * cbet2; // sig12 = sig2 - sig1 sig12 = atan2(fmax(real(0), csig1 * ssig2 - ssig1 * csig2), csig1 * csig2 + ssig1 * ssig2); { real dummy; Lengths(E, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2, outmask | REDUCEDLENGTH, s12x, m12x, dummy, M12, M21); } // Add the check for sig12 since zero length geodesics might yield m12 < // 0. Test case was // // echo 20.001 0 20.001 0 | GeodSolve -i // // In fact, we will have sig12 > pi/2 for meridional geodesic which is // not a shortest path. if (sig12 < 1 || m12x >= 0) { // Need at least 2, to handle 90 0 90 180 if (sig12 < 3 * tiny_ || // Prevent negative s12 or m12 for short lines (sig12 < tol0_ && (s12x < 0 || m12x < 0))) sig12 = m12x = s12x = 0; m12x *= _b; s12x *= _b; a12 = sig12 / Math::degree(); } else // m12 < 0, i.e., prolate and too close to anti-podal meridian = false; } // somg12 == 2 marks that it needs to be calculated real omg12 = 0, somg12 = 2, comg12 = 0; if (!meridian && sbet1 == 0 && // and sbet2 == 0 (_f <= 0 || lon12s >= _f * Math::hd)) { // Geodesic runs along equator calp1 = calp2 = 0; salp1 = salp2 = 1; s12x = _a * lam12; sig12 = omg12 = lam12 / _f1; m12x = _b * sin(sig12); if (outmask & GEODESICSCALE) M12 = M21 = cos(sig12); a12 = lon12 / _f1; } else if (!meridian) { // Now point1 and point2 belong within a hemisphere bounded by a // meridian and geodesic is neither meridional or equatorial. // Figure a starting point for Newton's method real dnm; sig12 = InverseStart(E, sbet1, cbet1, dn1, sbet2, cbet2, dn2, lam12, slam12, clam12, salp1, calp1, salp2, calp2, dnm); if (sig12 >= 0) { // Short lines (InverseStart sets salp2, calp2, dnm) s12x = sig12 * _b * dnm; m12x = Math::sq(dnm) * _b * sin(sig12 / dnm); if (outmask & GEODESICSCALE) M12 = M21 = cos(sig12 / dnm); a12 = sig12 / Math::degree(); omg12 = lam12 / (_f1 * dnm); } else { // Newton's method. This is a straightforward solution of f(alp1) = // lambda12(alp1) - lam12 = 0 with one wrinkle. f(alp) has exactly one // root in the interval (0, pi) and its derivative is positive at the // root. Thus f(alp) is positive for alp > alp1 and negative for alp < // alp1. During the course of the iteration, a range (alp1a, alp1b) is // maintained which brackets the root and with each evaluation of // f(alp) the range is shrunk, if possible. Newton's method is // restarted whenever the derivative of f is negative (because the new // value of alp1 is then further from the solution) or if the new // estimate of alp1 lies outside (0,pi); in this case, the new starting // guess is taken to be (alp1a + alp1b) / 2. // // initial values to suppress warnings (if loop is executed 0 times) real ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0, domg12 = 0; unsigned numit = 0; // Bracketing range real salp1a = tiny_, calp1a = 1, salp1b = tiny_, calp1b = -1; for (bool tripn = false, tripb = false;; ++numit) { // 1/4 meridian = 10e6 m and random input. max err is estimated max // error in nm (checking solution of inverse problem by direct // solution). iter is mean and sd of number of iterations // // max iter // log2(b/a) err mean sd // -7 387 5.33 3.68 // -6 345 5.19 3.43 // -5 269 5.00 3.05 // -4 210 4.76 2.44 // -3 115 4.55 1.87 // -2 69 4.35 1.38 // -1 36 4.05 1.03 // 0 15 0.01 0.13 // 1 25 5.10 1.53 // 2 96 5.61 2.09 // 3 318 6.02 2.74 // 4 985 6.24 3.22 // 5 2352 6.32 3.44 // 6 6008 6.30 3.45 // 7 19024 6.19 3.30 real dv; real v = Lambda12(sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1, calp1, slam12, clam12, salp2, calp2, sig12, ssig1, csig1, ssig2, csig2, E, domg12, numit < maxit1_, dv); if (tripb || // Reversed test to allow escape with NaNs !(fabs(v) >= (tripn ? 8 : 1) * tol0_) || // Enough bisections to get accurate result numit == maxit2_) break; // Update bracketing values if (v > 0 && (numit > maxit1_ || calp1/salp1 > calp1b/salp1b)) { salp1b = salp1; calp1b = calp1; } else if (v < 0 && (numit > maxit1_ || calp1/salp1 < calp1a/salp1a)) { salp1a = salp1; calp1a = calp1; } if (numit < maxit1_ && dv > 0) { real dalp1 = -v/dv; // |dalp1| < pi test moved earlier because GEOGRAPHICLIB_PRECISION // = 5 can result in dalp1 = 10^(10^8). Then sin(dalp1) takes ages // (because of the need to do accurate range reduction). if (fabs(dalp1) < Math::pi()) { real sdalp1 = sin(dalp1), cdalp1 = cos(dalp1), nsalp1 = salp1 * cdalp1 + calp1 * sdalp1; if (nsalp1 > 0) { calp1 = calp1 * cdalp1 - salp1 * sdalp1; salp1 = nsalp1; Math::norm(salp1, calp1); // In some regimes we don't get quadratic convergence because // slope -> 0. So use convergence conditions based on epsilon // instead of sqrt(epsilon). tripn = fabs(v) <= 16 * tol0_; continue; } } } // Either dv was not positive or updated value was outside legal // range. Use the midpoint of the bracket as the next estimate. // This mechanism is not needed for the WGS84 ellipsoid, but it does // catch problems with more eccentric ellipsoids. Its efficacy is // such for the WGS84 test set with the starting guess set to alp1 = // 90deg: // the WGS84 test set: mean = 5.21, sd = 3.93, max = 24 // WGS84 and random input: mean = 4.74, sd = 0.99 salp1 = (salp1a + salp1b)/2; calp1 = (calp1a + calp1b)/2; Math::norm(salp1, calp1); tripn = false; tripb = (fabs(salp1a - salp1) + (calp1a - calp1) < tolb_ || fabs(salp1 - salp1b) + (calp1 - calp1b) < tolb_); } { real dummy; Lengths(E, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2, outmask, s12x, m12x, dummy, M12, M21); } m12x *= _b; s12x *= _b; a12 = sig12 / Math::degree(); if (outmask & AREA) { // omg12 = lam12 - domg12 real sdomg12 = sin(domg12), cdomg12 = cos(domg12); somg12 = slam12 * cdomg12 - clam12 * sdomg12; comg12 = clam12 * cdomg12 + slam12 * sdomg12; } } } if (outmask & DISTANCE) s12 = real(0) + s12x; // Convert -0 to 0 if (outmask & REDUCEDLENGTH) m12 = real(0) + m12x; // Convert -0 to 0 if (outmask & AREA) { real // From Lambda12: sin(alp1) * cos(bet1) = sin(alp0) salp0 = salp1 * cbet1, calp0 = hypot(calp1, salp1 * sbet1); // calp0 > 0 real alp12, // Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0). A4 = Math::sq(_a) * calp0 * salp0 * _e2; if (A4 != 0) { real k2 = Math::sq(calp0) * _ep2, // From Lambda12: tan(bet) = tan(sig) * cos(alp) ssig1 = sbet1, csig1 = calp1 * cbet1, ssig2 = sbet2, csig2 = calp2 * cbet2; Math::norm(ssig1, csig1); Math::norm(ssig2, csig2); I4Integrand i4(_ep2, k2); vector C4a(_nC4); _fft.transform(i4, C4a.data()); real B41 = DST::integral(ssig1, csig1, C4a.data(), _nC4), B42 = DST::integral(ssig2, csig2, C4a.data(), _nC4); S12 = A4 * (B42 - B41); } else // Avoid problems with indeterminate sig1, sig2 on equator S12 = 0; if (!meridian && somg12 == 2) { somg12 = sin(omg12); comg12 = cos(omg12); } if (!meridian && // omg12 < 3/4 * pi comg12 > -real(0.7071) && // Long difference not too big sbet2 - sbet1 < real(1.75)) { // Lat difference not too big // Use tan(Gamma/2) = tan(omg12/2) // * (tan(bet1/2)+tan(bet2/2))/(1+tan(bet1/2)*tan(bet2/2)) // with tan(x/2) = sin(x)/(1+cos(x)) real domg12 = 1 + comg12, dbet1 = 1 + cbet1, dbet2 = 1 + cbet2; alp12 = 2 * atan2( somg12 * ( sbet1 * dbet2 + sbet2 * dbet1 ), domg12 * ( sbet1 * sbet2 + dbet1 * dbet2 ) ); } else { // alp12 = alp2 - alp1, used in atan2 so no need to normalize real salp12 = salp2 * calp1 - calp2 * salp1, calp12 = calp2 * calp1 + salp2 * salp1; // The right thing appears to happen if alp1 = +/-180 and alp2 = 0, viz // salp12 = -0 and alp12 = -180. However this depends on the sign // being attached to 0 correctly. The following ensures the correct // behavior. if (salp12 == 0 && calp12 < 0) { salp12 = tiny_ * calp1; calp12 = -1; } alp12 = atan2(salp12, calp12); } S12 += _c2 * alp12; S12 *= swapp * lonsign * latsign; // Convert -0 to 0 S12 += 0; } // Convert calp, salp to azimuth accounting for lonsign, swapp, latsign. if (swapp < 0) { swap(salp1, salp2); swap(calp1, calp2); if (outmask & GEODESICSCALE) swap(M12, M21); } salp1 *= swapp * lonsign; calp1 *= swapp * latsign; salp2 *= swapp * lonsign; calp2 *= swapp * latsign; // Returned value in [0, 180] return a12; } Math::real GeodesicExact::GenInverse(real lat1, real lon1, real lat2, real lon2, unsigned outmask, real& s12, real& azi1, real& azi2, real& m12, real& M12, real& M21, real& S12) const { outmask &= OUT_MASK; real salp1, calp1, salp2, calp2, a12 = GenInverse(lat1, lon1, lat2, lon2, outmask, s12, salp1, calp1, salp2, calp2, m12, M12, M21, S12); if (outmask & AZIMUTH) { azi1 = Math::atan2d(salp1, calp1); azi2 = Math::atan2d(salp2, calp2); } return a12; } GeodesicLineExact GeodesicExact::InverseLine(real lat1, real lon1, real lat2, real lon2, unsigned caps) const { real t, salp1, calp1, salp2, calp2, a12 = GenInverse(lat1, lon1, lat2, lon2, // No need to specify AZIMUTH here 0u, t, salp1, calp1, salp2, calp2, t, t, t, t), azi1 = Math::atan2d(salp1, calp1); // Ensure that a12 can be converted to a distance if (caps & (OUT_MASK & DISTANCE_IN)) caps |= DISTANCE; return GeodesicLineExact(*this, lat1, lon1, azi1, salp1, calp1, caps, true, a12); } void GeodesicExact::Lengths(const EllipticFunction& E, real sig12, real ssig1, real csig1, real dn1, real ssig2, real csig2, real dn2, real cbet1, real cbet2, unsigned outmask, real& s12b, real& m12b, real& m0, real& M12, real& M21) const { // Return m12b = (reduced length)/_b; also calculate s12b = distance/_b, // and m0 = coefficient of secular term in expression for reduced length. outmask &= OUT_ALL; // outmask & DISTANCE: set s12b // outmask & REDUCEDLENGTH: set m12b & m0 // outmask & GEODESICSCALE: set M12 & M21 // It's OK to have repeated dummy arguments, // e.g., s12b = m0 = M12 = M21 = dummy if (outmask & DISTANCE) // Missing a factor of _b s12b = E.E() / (Math::pi() / 2) * (sig12 + (E.deltaE(ssig2, csig2, dn2) - E.deltaE(ssig1, csig1, dn1))); if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) { real m0x = - E.k2() * E.D() / (Math::pi() / 2), J12 = m0x * (sig12 + (E.deltaD(ssig2, csig2, dn2) - E.deltaD(ssig1, csig1, dn1))); if (outmask & REDUCEDLENGTH) { m0 = m0x; // Missing a factor of _b. Add parens around (csig1 * ssig2) and // (ssig1 * csig2) to ensure accurate cancellation in the case of // coincident points. m12b = dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) - csig1 * csig2 * J12; } if (outmask & GEODESICSCALE) { real csig12 = csig1 * csig2 + ssig1 * ssig2; real t = _ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2); M12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1; M21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2; } } } Math::real GeodesicExact::Astroid(real x, real y) { // Solve k^4+2*k^3-(x^2+y^2-1)*k^2-2*y^2*k-y^2 = 0 for positive root k. // This solution is adapted from Geocentric::Reverse. real k; real p = Math::sq(x), q = Math::sq(y), r = (p + q - 1) / 6; if ( !(q == 0 && r <= 0) ) { real // Avoid possible division by zero when r = 0 by multiplying equations // for s and t by r^3 and r, resp. S = p * q / 4, // S = r^3 * s r2 = Math::sq(r), r3 = r * r2, // The discriminant of the quadratic equation for T3. This is zero on // the evolute curve p^(1/3)+q^(1/3) = 1 disc = S * (S + 2 * r3); real u = r; if (disc >= 0) { real T3 = S + r3; // Pick the sign on the sqrt to maximize abs(T3). This minimizes loss // of precision due to cancellation. The result is unchanged because // of the way the T is used in definition of u. T3 += T3 < 0 ? -sqrt(disc) : sqrt(disc); // T3 = (r * t)^3 // N.B. cbrt always returns the real root. cbrt(-8) = -2. real T = cbrt(T3); // T = r * t // T can be zero; but then r2 / T -> 0. u += T + (T != 0 ? r2 / T : 0); } else { // T is complex, but the way u is defined the result is real. real ang = atan2(sqrt(-disc), -(S + r3)); // There are three possible cube roots. We choose the root which // avoids cancellation. Note that disc < 0 implies that r < 0. u += 2 * r * cos(ang / 3); } real v = sqrt(Math::sq(u) + q), // guaranteed positive // Avoid loss of accuracy when u < 0. uv = u < 0 ? q / (v - u) : u + v, // u+v, guaranteed positive w = (uv - q) / (2 * v); // positive? // Rearrange expression for k to avoid loss of accuracy due to // subtraction. Division by 0 not possible because uv > 0, w >= 0. k = uv / (sqrt(uv + Math::sq(w)) + w); // guaranteed positive } else { // q == 0 && r <= 0 // y = 0 with |x| <= 1. Handle this case directly. // for y small, positive root is k = abs(y)/sqrt(1-x^2) k = 0; } return k; } Math::real GeodesicExact::InverseStart(EllipticFunction& E, real sbet1, real cbet1, real dn1, real sbet2, real cbet2, real dn2, real lam12, real slam12, real clam12, real& salp1, real& calp1, // Only updated if return val >= 0 real& salp2, real& calp2, // Only updated for short lines real& dnm) const { // Return a starting point for Newton's method in salp1 and calp1 (function // value is -1). If Newton's method doesn't need to be used, return also // salp2 and calp2 and function value is sig12. real sig12 = -1, // Return value // bet12 = bet2 - bet1 in [0, pi); bet12a = bet2 + bet1 in (-pi, 0] sbet12 = sbet2 * cbet1 - cbet2 * sbet1, cbet12 = cbet2 * cbet1 + sbet2 * sbet1; real sbet12a = sbet2 * cbet1 + cbet2 * sbet1; bool shortline = cbet12 >= 0 && sbet12 < real(0.5) && cbet2 * lam12 < real(0.5); real somg12, comg12; if (shortline) { real sbetm2 = Math::sq(sbet1 + sbet2); // sin((bet1+bet2)/2)^2 // = (sbet1 + sbet2)^2 / ((sbet1 + sbet2)^2 + (cbet1 + cbet2)^2) sbetm2 /= sbetm2 + Math::sq(cbet1 + cbet2); dnm = sqrt(1 + _ep2 * sbetm2); real omg12 = lam12 / (_f1 * dnm); somg12 = sin(omg12); comg12 = cos(omg12); } else { somg12 = slam12; comg12 = clam12; } salp1 = cbet2 * somg12; calp1 = comg12 >= 0 ? sbet12 + cbet2 * sbet1 * Math::sq(somg12) / (1 + comg12) : sbet12a - cbet2 * sbet1 * Math::sq(somg12) / (1 - comg12); real ssig12 = hypot(salp1, calp1), csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12; if (shortline && ssig12 < _etol2) { // really short lines salp2 = cbet1 * somg12; calp2 = sbet12 - cbet1 * sbet2 * (comg12 >= 0 ? Math::sq(somg12) / (1 + comg12) : 1 - comg12); Math::norm(salp2, calp2); // Set return value sig12 = atan2(ssig12, csig12); } else if (fabs(_n) > real(0.1) || // Skip astroid calc if too eccentric csig12 >= 0 || ssig12 >= 6 * fabs(_n) * Math::pi() * Math::sq(cbet1)) { // Nothing to do, zeroth order spherical approximation is OK } else { // Scale lam12 and bet2 to x, y coordinate system where antipodal point // is at origin and singular point is at y = 0, x = -1. real x, y, lamscale, betscale; real lam12x = atan2(-slam12, -clam12); // lam12 - pi if (_f >= 0) { // In fact f == 0 does not get here // x = dlong, y = dlat { real k2 = Math::sq(sbet1) * _ep2; E.Reset(-k2, -_ep2, 1 + k2, 1 + _ep2); lamscale = _e2/_f1 * cbet1 * 2 * E.H(); } betscale = lamscale * cbet1; x = lam12x / lamscale; y = sbet12a / betscale; } else { // _f < 0 // x = dlat, y = dlong real cbet12a = cbet2 * cbet1 - sbet2 * sbet1, bet12a = atan2(sbet12a, cbet12a); real m12b, m0, dummy; // In the case of lon12 = 180, this repeats a calculation made in // Inverse. Lengths(E, Math::pi() + bet12a, sbet1, -cbet1, dn1, sbet2, cbet2, dn2, cbet1, cbet2, REDUCEDLENGTH, dummy, m12b, m0, dummy, dummy); x = -1 + m12b / (cbet1 * cbet2 * m0 * Math::pi()); betscale = x < -real(0.01) ? sbet12a / x : -_f * Math::sq(cbet1) * Math::pi(); lamscale = betscale / cbet1; y = lam12x / lamscale; } if (y > -tol1_ && x > -1 - xthresh_) { // strip near cut // Need real(x) here to cast away the volatility of x for min/max if (_f >= 0) { salp1 = fmin(real(1), -x); calp1 = - sqrt(1 - Math::sq(salp1)); } else { calp1 = fmax(real(x > -tol1_ ? 0 : -1), x); salp1 = sqrt(1 - Math::sq(calp1)); } } else { // Estimate alp1, by solving the astroid problem. // // Could estimate alpha1 = theta + pi/2, directly, i.e., // calp1 = y/k; salp1 = -x/(1+k); for _f >= 0 // calp1 = x/(1+k); salp1 = -y/k; for _f < 0 (need to check) // // However, it's better to estimate omg12 from astroid and use // spherical formula to compute alp1. This reduces the mean number of // Newton iterations for astroid cases from 2.24 (min 0, max 6) to 2.12 // (min 0 max 5). The changes in the number of iterations are as // follows: // // change percent // 1 5 // 0 78 // -1 16 // -2 0.6 // -3 0.04 // -4 0.002 // // The histogram of iterations is (m = number of iterations estimating // alp1 directly, n = number of iterations estimating via omg12, total // number of trials = 148605): // // iter m n // 0 148 186 // 1 13046 13845 // 2 93315 102225 // 3 36189 32341 // 4 5396 7 // 5 455 1 // 6 56 0 // // Because omg12 is near pi, estimate work with omg12a = pi - omg12 real k = Astroid(x, y); real omg12a = lamscale * ( _f >= 0 ? -x * k/(1 + k) : -y * (1 + k)/k ); somg12 = sin(omg12a); comg12 = -cos(omg12a); // Update spherical estimate of alp1 using omg12 instead of lam12 salp1 = cbet2 * somg12; calp1 = sbet12a - cbet2 * sbet1 * Math::sq(somg12) / (1 - comg12); } } // Sanity check on starting guess. Backwards check allows NaN through. if (!(salp1 <= 0)) Math::norm(salp1, calp1); else { salp1 = 1; calp1 = 0; } return sig12; } Math::real GeodesicExact::Lambda12(real sbet1, real cbet1, real dn1, real sbet2, real cbet2, real dn2, real salp1, real calp1, real slam120, real clam120, real& salp2, real& calp2, real& sig12, real& ssig1, real& csig1, real& ssig2, real& csig2, EllipticFunction& E, real& domg12, bool diffp, real& dlam12) const { if (sbet1 == 0 && calp1 == 0) // Break degeneracy of equatorial line. This case has already been // handled. calp1 = -tiny_; real // sin(alp1) * cos(bet1) = sin(alp0) salp0 = salp1 * cbet1, calp0 = hypot(calp1, salp1 * sbet1); // calp0 > 0 real somg1, comg1, somg2, comg2, somg12, comg12, cchi1, cchi2, lam12; // tan(bet1) = tan(sig1) * cos(alp1) // tan(omg1) = sin(alp0) * tan(sig1) = tan(omg1)=tan(alp1)*sin(bet1) ssig1 = sbet1; somg1 = salp0 * sbet1; csig1 = comg1 = calp1 * cbet1; // Without normalization we have schi1 = somg1. cchi1 = _f1 * dn1 * comg1; Math::norm(ssig1, csig1); // Math::norm(somg1, comg1); -- don't need to normalize! // Math::norm(schi1, cchi1); -- don't need to normalize! // Enforce symmetries in the case abs(bet2) = -bet1. Need to be careful // about this case, since this can yield singularities in the Newton // iteration. // sin(alp2) * cos(bet2) = sin(alp0) salp2 = cbet2 != cbet1 ? salp0 / cbet2 : salp1; // calp2 = sqrt(1 - sq(salp2)) // = sqrt(sq(calp0) - sq(sbet2)) / cbet2 // and subst for calp0 and rearrange to give (choose positive sqrt // to give alp2 in [0, pi/2]). calp2 = cbet2 != cbet1 || fabs(sbet2) != -sbet1 ? sqrt(Math::sq(calp1 * cbet1) + (cbet1 < -sbet1 ? (cbet2 - cbet1) * (cbet1 + cbet2) : (sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2 : fabs(calp1); // tan(bet2) = tan(sig2) * cos(alp2) // tan(omg2) = sin(alp0) * tan(sig2). ssig2 = sbet2; somg2 = salp0 * sbet2; csig2 = comg2 = calp2 * cbet2; // Without normalization we have schi2 = somg2. cchi2 = _f1 * dn2 * comg2; Math::norm(ssig2, csig2); // Math::norm(somg2, comg2); -- don't need to normalize! // Math::norm(schi2, cchi2); -- don't need to normalize! // sig12 = sig2 - sig1, limit to [0, pi] sig12 = atan2(fmax(real(0), csig1 * ssig2 - ssig1 * csig2), csig1 * csig2 + ssig1 * ssig2); // omg12 = omg2 - omg1, limit to [0, pi] somg12 = fmax(real(0), comg1 * somg2 - somg1 * comg2); comg12 = comg1 * comg2 + somg1 * somg2; real k2 = Math::sq(calp0) * _ep2; E.Reset(-k2, -_ep2, 1 + k2, 1 + _ep2); // chi12 = chi2 - chi1, limit to [0, pi] real schi12 = fmax(real(0), cchi1 * somg2 - somg1 * cchi2), cchi12 = cchi1 * cchi2 + somg1 * somg2; // eta = chi12 - lam120 real eta = atan2(schi12 * clam120 - cchi12 * slam120, cchi12 * clam120 + schi12 * slam120); real deta12 = -_e2/_f1 * salp0 * E.H() / (Math::pi() / 2) * (sig12 + (E.deltaH(ssig2, csig2, dn2) - E.deltaH(ssig1, csig1, dn1))); lam12 = eta + deta12; // domg12 = deta12 + chi12 - omg12 domg12 = deta12 + atan2(schi12 * comg12 - cchi12 * somg12, cchi12 * comg12 + schi12 * somg12); if (diffp) { if (calp2 == 0) dlam12 = - 2 * _f1 * dn1 / sbet1; else { real dummy; Lengths(E, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2, REDUCEDLENGTH, dummy, dlam12, dummy, dummy, dummy); dlam12 *= _f1 / (calp2 * cbet2); } } return lam12; } Math::real GeodesicExact::I4Integrand::asinhsqrt(real x) { // return asinh(sqrt(x))/sqrt(x) using std::sqrt; using std::asinh; using std::asin; return x == 0 ? 1 : (x > 0 ? asinh(sqrt(x))/sqrt(x) : asin(sqrt(-x))/sqrt(-x)); // NaNs end up here } Math::real GeodesicExact::I4Integrand::t(real x) { // This differs by from t as defined following Eq 61 in Karney (2013) by // the final subtraction of 1. This changes nothing since Eq 61 uses the // difference of two evaluations of t and improves the accuracy(?). using std::sqrt; // Group terms to minimize roundoff // with x = ep2, this is the same as // e2/(1-e2) + (atanh(e)/e - 1) return x + (sqrt(1 + x) * asinhsqrt(x) - 1); } Math::real GeodesicExact::I4Integrand::td(real x) { // d t(x) / dx using std::sqrt; return x == 0 ? 4/real(3) : // Group terms to minimize roundoff 1 + (1 - asinhsqrt(x) / sqrt(1+x)) / (2*x); } // Math::real GeodesicExact::I4Integrand::Dt(real x, real y) { // // ( t(x) - t(y) ) / (x - y) // using std::sqrt; using std::fabs; using std::asinh; using std::asin; // if (x == y) return td(x); // if (x * y <= 0) return ( t(x) - t(y) ) / (x - y); // real // sx = sqrt(fabs(x)), sx1 = sqrt(1 + x), // sy = sqrt(fabs(y)), sy1 = sqrt(1 + y), // z = (x - y) / (sx * sy1 + sy * sx1), // d1 = 2 * sx * sy, // d2 = 2 * (x * sy * sy1 + y * sx * sx1); // return x > 0 ? // ( 1 + (asinh(z)/z) / d1 - (asinh(sx) + asinh(sy)) / d2 ) : // // NaNs fall through to here // ( 1 - (asin (z)/z) / d1 - (asin (sx) + asin (sy)) / d2 ); // } Math::real GeodesicExact::I4Integrand::DtX(real y) const { // idiot version: // return ( tX - t(y) ) / (X - y); using std::sqrt; using std::fabs; using std::asinh; using std::asin; if (X == y) return tdX; if (X * y <= 0) return ( tX - t(y) ) / (X - y); real sy = sqrt(fabs(y)), sy1 = sqrt(1 + y), z = (X - y) / (sX * sy1 + sy * sX1), d1 = 2 * sX * sy, d2 = 2 * (X * sy * sy1 + y * sXX1); return X > 0 ? ( 1 + (asinh(z)/z) / d1 - (asinhsX + asinh(sy)) / d2 ) : // NaNs fall through to here ( 1 - (asin (z)/z) / d1 - (asinhsX + asin (sy)) / d2 ); } GeodesicExact::I4Integrand::I4Integrand(real ep2, real k2) : X( ep2 ) , tX( t(X) ) , tdX( td(X) ) , _k2( k2 ) { using std::fabs; using std::sqrt; using std::asinh; using std::asin; sX = sqrt(fabs(X)); // ep sX1 = sqrt(1 + X); // 1/(1-f) sXX1 = sX * sX1; asinhsX = X > 0 ? asinh(sX) : asin(sX); // atanh(e) } Math::real GeodesicExact::I4Integrand::operator()(real sig) const { using std::sin; real ssig = sin(sig); return - DtX(_k2 * Math::sq(ssig)) * ssig/2; } } // namespace GeographicLib geosphere/src/GeodesicLine.h0000644000176200001440000007401414323377037015544 0ustar liggesusers/** * \file GeodesicLine.hpp * \brief Header for GeographicLib::GeodesicLine class * * Copyright (c) Charles Karney (2009-2022) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ **********************************************************************/ #if !defined(GEOGRAPHICLIB_GEODESICLINE_HPP) #define GEOGRAPHICLIB_GEODESICLINE_HPP 1 #include "Constants.h" #include "Geodesic.h" namespace GeographicLib { /** * \brief A geodesic line * * GeodesicLine facilitates the determination of a series of points on a * single geodesic. The starting point (\e lat1, \e lon1) and the azimuth \e * azi1 are specified in the constructor; alternatively, the Geodesic::Line * method can be used to create a GeodesicLine. GeodesicLine.Position * returns the location of point 2 a distance \e s12 along the geodesic. In * addition, GeodesicLine.ArcPosition gives the position of point 2 an arc * length \e a12 along the geodesic. * * You can register the position of a reference point 3 a distance (arc * length), \e s13 (\e a13) along the geodesic with the * GeodesicLine.SetDistance (GeodesicLine.SetArc) functions. Points a * fractional distance along the line can be found by providing, for example, * 0.5 * Distance() as an argument to GeodesicLine.Position. The * Geodesic::InverseLine or Geodesic::DirectLine methods return GeodesicLine * objects with point 3 set to the point 2 of the corresponding geodesic * problem. GeodesicLine objects created with the public constructor or with * Geodesic::Line have \e s13 and \e a13 set to NaNs. * * The default copy constructor and assignment operators work with this * class. Similarly, a vector can be used to hold GeodesicLine objects. * * The calculations are accurate to better than 15 nm (15 nanometers). See * Sec. 9 of * arXiv:1102.1215v1 for * details. The algorithms used by this class are based on series expansions * using the flattening \e f as a small parameter. These are only accurate * for |f| < 0.02; however reasonably accurate results will be * obtained for |f| < 0.2. For very eccentric ellipsoids, use * GeodesicLineExact instead. * * The algorithms are described in * - C. F. F. Karney, * * Algorithms for geodesics, * J. Geodesy 87, 43--55 (2013); * DOI: * 10.1007/s00190-012-0578-z; * addenda: * * geod-addenda.html. * . * For more information on geodesics see \ref geodesic. * * Example of use: * \include example-GeodesicLine.cpp * * GeodSolve is a command-line utility * providing access to the functionality of Geodesic and GeodesicLine. **********************************************************************/ class GEOGRAPHICLIB_EXPORT GeodesicLine { private: typedef Math::real real; friend class Geodesic; static const int nC1_ = Geodesic::nC1_; static const int nC1p_ = Geodesic::nC1p_; static const int nC2_ = Geodesic::nC2_; static const int nC3_ = Geodesic::nC3_; static const int nC4_ = Geodesic::nC4_; real tiny_; real _lat1, _lon1, _azi1; real _a, _f, _b, _c2, _f1, _salp0, _calp0, _k2, _salp1, _calp1, _ssig1, _csig1, _dn1, _stau1, _ctau1, _somg1, _comg1, _aA1m1, _aA2m1, _aA3c, _bB11, _bB21, _bB31, _aA4, _bB41; real _a13, _s13; // index zero elements of _cC1a, _cC1pa, _cC2a, _cC3a are unused real _cC1a[nC1_ + 1], _cC1pa[nC1p_ + 1], _cC2a[nC2_ + 1], _cC3a[nC3_], _cC4a[nC4_]; // all the elements of _cC4a are used unsigned _caps; void LineInit(const Geodesic& g, real lat1, real lon1, real azi1, real salp1, real calp1, unsigned caps); GeodesicLine(const Geodesic& g, real lat1, real lon1, real azi1, real salp1, real calp1, unsigned caps, bool arcmode, real s13_a13); enum captype { CAP_NONE = Geodesic::CAP_NONE, CAP_C1 = Geodesic::CAP_C1, CAP_C1p = Geodesic::CAP_C1p, CAP_C2 = Geodesic::CAP_C2, CAP_C3 = Geodesic::CAP_C3, CAP_C4 = Geodesic::CAP_C4, CAP_ALL = Geodesic::CAP_ALL, CAP_MASK = Geodesic::CAP_MASK, OUT_ALL = Geodesic::OUT_ALL, OUT_MASK = Geodesic::OUT_MASK, }; public: /** * Bit masks for what calculations to do. They signify to the * GeodesicLine::GeodesicLine constructor and to Geodesic::Line what * capabilities should be included in the GeodesicLine object. This is * merely a duplication of Geodesic::mask. **********************************************************************/ enum mask { /** * No capabilities, no output. * @hideinitializer **********************************************************************/ NONE = Geodesic::NONE, /** * Calculate latitude \e lat2. (It's not necessary to include this as a * capability to GeodesicLine because this is included by default.) * @hideinitializer **********************************************************************/ LATITUDE = Geodesic::LATITUDE, /** * Calculate longitude \e lon2. * @hideinitializer **********************************************************************/ LONGITUDE = Geodesic::LONGITUDE, /** * Calculate azimuths \e azi1 and \e azi2. (It's not necessary to * include this as a capability to GeodesicLine because this is included * by default.) * @hideinitializer **********************************************************************/ AZIMUTH = Geodesic::AZIMUTH, /** * Calculate distance \e s12. * @hideinitializer **********************************************************************/ DISTANCE = Geodesic::DISTANCE, /** * A combination of the common capabilities: GeodesicLine::LATITUDE, * GeodesicLine::LONGITUDE, GeodesicLine::AZIMUTH, GeodesicLine::DISTANCE. * @hideinitializer **********************************************************************/ STANDARD = Geodesic::STANDARD, /** * Allow distance \e s12 to be used as input in the direct geodesic * problem. * @hideinitializer **********************************************************************/ DISTANCE_IN = Geodesic::DISTANCE_IN, /** * Calculate reduced length \e m12. * @hideinitializer **********************************************************************/ REDUCEDLENGTH = Geodesic::REDUCEDLENGTH, /** * Calculate geodesic scales \e M12 and \e M21. * @hideinitializer **********************************************************************/ GEODESICSCALE = Geodesic::GEODESICSCALE, /** * Calculate area \e S12. * @hideinitializer **********************************************************************/ AREA = Geodesic::AREA, /** * Unroll \e lon2 in the direct calculation. * @hideinitializer **********************************************************************/ LONG_UNROLL = Geodesic::LONG_UNROLL, /** * All capabilities, calculate everything. (GeodesicLine::LONG_UNROLL is * not included in this mask.) * @hideinitializer **********************************************************************/ ALL = Geodesic::ALL, }; /** \name Constructors **********************************************************************/ ///@{ /** * Constructor for a geodesic line staring at latitude \e lat1, longitude * \e lon1, and azimuth \e azi1 (all in degrees). * * @param[in] g A Geodesic object used to compute the necessary information * about the GeodesicLine. * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] azi1 azimuth at point 1 (degrees). * @param[in] caps bitor'ed combination of GeodesicLine::mask values * specifying the capabilities the GeodesicLine object should possess, * i.e., which quantities can be returned in calls to * GeodesicLine::Position. * * \e lat1 should be in the range [−90°, 90°]. * * The GeodesicLine::mask values are * - \e caps |= GeodesicLine::LATITUDE for the latitude \e lat2; this is * added automatically; * - \e caps |= GeodesicLine::LONGITUDE for the latitude \e lon2; * - \e caps |= GeodesicLine::AZIMUTH for the latitude \e azi2; this is * added automatically; * - \e caps |= GeodesicLine::DISTANCE for the distance \e s12; * - \e caps |= GeodesicLine::REDUCEDLENGTH for the reduced length \e m12; * - \e caps |= GeodesicLine::GEODESICSCALE for the geodesic scales \e M12 * and \e M21; * - \e caps |= GeodesicLine::AREA for the area \e S12; * - \e caps |= GeodesicLine::DISTANCE_IN permits the length of the * geodesic to be given in terms of \e s12; without this capability the * length can only be specified in terms of arc length; * - \e caps |= GeodesicLine::ALL for all of the above. * . * The default value of \e caps is GeodesicLine::ALL. * * If the point is at a pole, the azimuth is defined by keeping \e lon1 * fixed, writing \e lat1 = ±(90° − ε), and taking * the limit ε → 0+. **********************************************************************/ GeodesicLine(const Geodesic& g, real lat1, real lon1, real azi1, unsigned caps = ALL); /** * A default constructor. If GeodesicLine::Position is called on the * resulting object, it returns immediately (without doing any * calculations). The object can be set with a call to Geodesic::Line. * Use Init() to test whether object is still in this uninitialized state. **********************************************************************/ GeodesicLine() : _caps(0U) {} ///@} /** \name Position in terms of distance **********************************************************************/ ///@{ /** * Compute the position of point 2 which is a distance \e s12 (meters) from * point 1. * * @param[in] s12 distance from point 1 to point 2 (meters); it can be * negative. * @param[out] lat2 latitude of point 2 (degrees). * @param[out] lon2 longitude of point 2 (degrees); requires that the * GeodesicLine object was constructed with \e caps |= * GeodesicLine::LONGITUDE. * @param[out] azi2 (forward) azimuth at point 2 (degrees). * @param[out] m12 reduced length of geodesic (meters); requires that the * GeodesicLine object was constructed with \e caps |= * GeodesicLine::REDUCEDLENGTH. * @param[out] M12 geodesic scale of point 2 relative to point 1 * (dimensionless); requires that the GeodesicLine object was constructed * with \e caps |= GeodesicLine::GEODESICSCALE. * @param[out] M21 geodesic scale of point 1 relative to point 2 * (dimensionless); requires that the GeodesicLine object was constructed * with \e caps |= GeodesicLine::GEODESICSCALE. * @param[out] S12 area under the geodesic (meters2); requires * that the GeodesicLine object was constructed with \e caps |= * GeodesicLine::AREA. * @return \e a12 arc length from point 1 to point 2 (degrees). * * The values of \e lon2 and \e azi2 returned are in the range * [−180°, 180°]. * * The GeodesicLine object \e must have been constructed with \e caps |= * GeodesicLine::DISTANCE_IN; otherwise Math::NaN() is returned and no * parameters are set. Requesting a value which the GeodesicLine object is * not capable of computing is not an error; the corresponding argument * will not be altered. * * The following functions are overloaded versions of * GeodesicLine::Position which omit some of the output parameters. Note, * however, that the arc length is always computed and returned as the * function value. **********************************************************************/ Math::real Position(real s12, real& lat2, real& lon2, real& azi2, real& m12, real& M12, real& M21, real& S12) const { real t; return GenPosition(false, s12, LATITUDE | LONGITUDE | AZIMUTH | REDUCEDLENGTH | GEODESICSCALE | AREA, lat2, lon2, azi2, t, m12, M12, M21, S12); } /** * See the documentation for GeodesicLine::Position. **********************************************************************/ Math::real Position(real s12, real& lat2, real& lon2) const { real t; return GenPosition(false, s12, LATITUDE | LONGITUDE, lat2, lon2, t, t, t, t, t, t); } /** * See the documentation for GeodesicLine::Position. **********************************************************************/ Math::real Position(real s12, real& lat2, real& lon2, real& azi2) const { real t; return GenPosition(false, s12, LATITUDE | LONGITUDE | AZIMUTH, lat2, lon2, azi2, t, t, t, t, t); } /** * See the documentation for GeodesicLine::Position. **********************************************************************/ Math::real Position(real s12, real& lat2, real& lon2, real& azi2, real& m12) const { real t; return GenPosition(false, s12, LATITUDE | LONGITUDE | AZIMUTH | REDUCEDLENGTH, lat2, lon2, azi2, t, m12, t, t, t); } /** * See the documentation for GeodesicLine::Position. **********************************************************************/ Math::real Position(real s12, real& lat2, real& lon2, real& azi2, real& M12, real& M21) const { real t; return GenPosition(false, s12, LATITUDE | LONGITUDE | AZIMUTH | GEODESICSCALE, lat2, lon2, azi2, t, t, M12, M21, t); } /** * See the documentation for GeodesicLine::Position. **********************************************************************/ Math::real Position(real s12, real& lat2, real& lon2, real& azi2, real& m12, real& M12, real& M21) const { real t; return GenPosition(false, s12, LATITUDE | LONGITUDE | AZIMUTH | REDUCEDLENGTH | GEODESICSCALE, lat2, lon2, azi2, t, m12, M12, M21, t); } ///@} /** \name Position in terms of arc length **********************************************************************/ ///@{ /** * Compute the position of point 2 which is an arc length \e a12 (degrees) * from point 1. * * @param[in] a12 arc length from point 1 to point 2 (degrees); it can * be negative. * @param[out] lat2 latitude of point 2 (degrees). * @param[out] lon2 longitude of point 2 (degrees); requires that the * GeodesicLine object was constructed with \e caps |= * GeodesicLine::LONGITUDE. * @param[out] azi2 (forward) azimuth at point 2 (degrees). * @param[out] s12 distance from point 1 to point 2 (meters); requires * that the GeodesicLine object was constructed with \e caps |= * GeodesicLine::DISTANCE. * @param[out] m12 reduced length of geodesic (meters); requires that the * GeodesicLine object was constructed with \e caps |= * GeodesicLine::REDUCEDLENGTH. * @param[out] M12 geodesic scale of point 2 relative to point 1 * (dimensionless); requires that the GeodesicLine object was constructed * with \e caps |= GeodesicLine::GEODESICSCALE. * @param[out] M21 geodesic scale of point 1 relative to point 2 * (dimensionless); requires that the GeodesicLine object was constructed * with \e caps |= GeodesicLine::GEODESICSCALE. * @param[out] S12 area under the geodesic (meters2); requires * that the GeodesicLine object was constructed with \e caps |= * GeodesicLine::AREA. * * The values of \e lon2 and \e azi2 returned are in the range * [−180°, 180°]. * * Requesting a value which the GeodesicLine object is not capable of * computing is not an error; the corresponding argument will not be * altered. * * The following functions are overloaded versions of * GeodesicLine::ArcPosition which omit some of the output parameters. **********************************************************************/ void ArcPosition(real a12, real& lat2, real& lon2, real& azi2, real& s12, real& m12, real& M12, real& M21, real& S12) const { GenPosition(true, a12, LATITUDE | LONGITUDE | AZIMUTH | DISTANCE | REDUCEDLENGTH | GEODESICSCALE | AREA, lat2, lon2, azi2, s12, m12, M12, M21, S12); } /** * See the documentation for GeodesicLine::ArcPosition. **********************************************************************/ void ArcPosition(real a12, real& lat2, real& lon2) const { real t; GenPosition(true, a12, LATITUDE | LONGITUDE, lat2, lon2, t, t, t, t, t, t); } /** * See the documentation for GeodesicLine::ArcPosition. **********************************************************************/ void ArcPosition(real a12, real& lat2, real& lon2, real& azi2) const { real t; GenPosition(true, a12, LATITUDE | LONGITUDE | AZIMUTH, lat2, lon2, azi2, t, t, t, t, t); } /** * See the documentation for GeodesicLine::ArcPosition. **********************************************************************/ void ArcPosition(real a12, real& lat2, real& lon2, real& azi2, real& s12) const { real t; GenPosition(true, a12, LATITUDE | LONGITUDE | AZIMUTH | DISTANCE, lat2, lon2, azi2, s12, t, t, t, t); } /** * See the documentation for GeodesicLine::ArcPosition. **********************************************************************/ void ArcPosition(real a12, real& lat2, real& lon2, real& azi2, real& s12, real& m12) const { real t; GenPosition(true, a12, LATITUDE | LONGITUDE | AZIMUTH | DISTANCE | REDUCEDLENGTH, lat2, lon2, azi2, s12, m12, t, t, t); } /** * See the documentation for GeodesicLine::ArcPosition. **********************************************************************/ void ArcPosition(real a12, real& lat2, real& lon2, real& azi2, real& s12, real& M12, real& M21) const { real t; GenPosition(true, a12, LATITUDE | LONGITUDE | AZIMUTH | DISTANCE | GEODESICSCALE, lat2, lon2, azi2, s12, t, M12, M21, t); } /** * See the documentation for GeodesicLine::ArcPosition. **********************************************************************/ void ArcPosition(real a12, real& lat2, real& lon2, real& azi2, real& s12, real& m12, real& M12, real& M21) const { real t; GenPosition(true, a12, LATITUDE | LONGITUDE | AZIMUTH | DISTANCE | REDUCEDLENGTH | GEODESICSCALE, lat2, lon2, azi2, s12, m12, M12, M21, t); } ///@} /** \name The general position function. **********************************************************************/ ///@{ /** * The general position function. GeodesicLine::Position and * GeodesicLine::ArcPosition are defined in terms of this function. * * @param[in] arcmode boolean flag determining the meaning of the second * parameter; if \e arcmode is false, then the GeodesicLine object must * have been constructed with \e caps |= GeodesicLine::DISTANCE_IN. * @param[in] s12_a12 if \e arcmode is false, this is the distance between * point 1 and point 2 (meters); otherwise it is the arc length between * point 1 and point 2 (degrees); it can be negative. * @param[in] outmask a bitor'ed combination of GeodesicLine::mask values * specifying which of the following parameters should be set. * @param[out] lat2 latitude of point 2 (degrees). * @param[out] lon2 longitude of point 2 (degrees); requires that the * GeodesicLine object was constructed with \e caps |= * GeodesicLine::LONGITUDE. * @param[out] azi2 (forward) azimuth at point 2 (degrees). * @param[out] s12 distance from point 1 to point 2 (meters); requires * that the GeodesicLine object was constructed with \e caps |= * GeodesicLine::DISTANCE. * @param[out] m12 reduced length of geodesic (meters); requires that the * GeodesicLine object was constructed with \e caps |= * GeodesicLine::REDUCEDLENGTH. * @param[out] M12 geodesic scale of point 2 relative to point 1 * (dimensionless); requires that the GeodesicLine object was constructed * with \e caps |= GeodesicLine::GEODESICSCALE. * @param[out] M21 geodesic scale of point 1 relative to point 2 * (dimensionless); requires that the GeodesicLine object was constructed * with \e caps |= GeodesicLine::GEODESICSCALE. * @param[out] S12 area under the geodesic (meters2); requires * that the GeodesicLine object was constructed with \e caps |= * GeodesicLine::AREA. * @return \e a12 arc length from point 1 to point 2 (degrees). * * The GeodesicLine::mask values possible for \e outmask are * - \e outmask |= GeodesicLine::LATITUDE for the latitude \e lat2; * - \e outmask |= GeodesicLine::LONGITUDE for the latitude \e lon2; * - \e outmask |= GeodesicLine::AZIMUTH for the latitude \e azi2; * - \e outmask |= GeodesicLine::DISTANCE for the distance \e s12; * - \e outmask |= GeodesicLine::REDUCEDLENGTH for the reduced length \e * m12; * - \e outmask |= GeodesicLine::GEODESICSCALE for the geodesic scales \e * M12 and \e M21; * - \e outmask |= GeodesicLine::AREA for the area \e S12; * - \e outmask |= GeodesicLine::ALL for all of the above; * - \e outmask |= GeodesicLine::LONG_UNROLL to unroll \e lon2 instead of * reducing it into the range [−180°, 180°]. * . * Requesting a value which the GeodesicLine object is not capable of * computing is not an error; the corresponding argument will not be * altered. Note, however, that the arc length is always computed and * returned as the function value. * * With the GeodesicLine::LONG_UNROLL bit set, the quantity \e lon2 − * \e lon1 indicates how many times and in what sense the geodesic * encircles the ellipsoid. **********************************************************************/ Math::real GenPosition(bool arcmode, real s12_a12, unsigned outmask, real& lat2, real& lon2, real& azi2, real& s12, real& m12, real& M12, real& M21, real& S12) const; ///@} /** \name Setting point 3 **********************************************************************/ ///@{ /** * Specify position of point 3 in terms of distance. * * @param[in] s13 the distance from point 1 to point 3 (meters); it * can be negative. * * This is only useful if the GeodesicLine object has been constructed * with \e caps |= GeodesicLine::DISTANCE_IN. **********************************************************************/ void SetDistance(real s13); /** * Specify position of point 3 in terms of arc length. * * @param[in] a13 the arc length from point 1 to point 3 (degrees); it * can be negative. * * The distance \e s13 is only set if the GeodesicLine object has been * constructed with \e caps |= GeodesicLine::DISTANCE. **********************************************************************/ void SetArc(real a13); /** * Specify position of point 3 in terms of either distance or arc length. * * @param[in] arcmode boolean flag determining the meaning of the second * parameter; if \e arcmode is false, then the GeodesicLine object must * have been constructed with \e caps |= GeodesicLine::DISTANCE_IN. * @param[in] s13_a13 if \e arcmode is false, this is the distance from * point 1 to point 3 (meters); otherwise it is the arc length from * point 1 to point 3 (degrees); it can be negative. **********************************************************************/ void GenSetDistance(bool arcmode, real s13_a13); ///@} /** \name Inspector functions **********************************************************************/ ///@{ /** * @return true if the object has been initialized. **********************************************************************/ bool Init() const { return _caps != 0U; } /** * @return \e lat1 the latitude of point 1 (degrees). **********************************************************************/ Math::real Latitude() const { return Init() ? _lat1 : Math::NaN(); } /** * @return \e lon1 the longitude of point 1 (degrees). **********************************************************************/ Math::real Longitude() const { return Init() ? _lon1 : Math::NaN(); } /** * @return \e azi1 the azimuth (degrees) of the geodesic line at point 1. **********************************************************************/ Math::real Azimuth() const { return Init() ? _azi1 : Math::NaN(); } /** * The sine and cosine of \e azi1. * * @param[out] sazi1 the sine of \e azi1. * @param[out] cazi1 the cosine of \e azi1. **********************************************************************/ void Azimuth(real& sazi1, real& cazi1) const { if (Init()) { sazi1 = _salp1; cazi1 = _calp1; } } /** * @return \e azi0 the azimuth (degrees) of the geodesic line as it crosses * the equator in a northward direction. * * The result lies in [−90°, 90°]. **********************************************************************/ Math::real EquatorialAzimuth() const { return Init() ? Math::atan2d(_salp0, _calp0) : Math::NaN(); } /** * The sine and cosine of \e azi0. * * @param[out] sazi0 the sine of \e azi0. * @param[out] cazi0 the cosine of \e azi0. **********************************************************************/ void EquatorialAzimuth(real& sazi0, real& cazi0) const { if (Init()) { sazi0 = _salp0; cazi0 = _calp0; } } /** * @return \e a1 the arc length (degrees) between the northward equatorial * crossing and point 1. * * The result lies in [−180°, 180°]. **********************************************************************/ Math::real EquatorialArc() const { return Init() ? Math::atan2d(_ssig1, _csig1) : Math::NaN(); } /** * @return \e a the equatorial radius of the ellipsoid (meters). This is * the value inherited from the Geodesic object used in the constructor. **********************************************************************/ Math::real EquatorialRadius() const { return Init() ? _a : Math::NaN(); } /** * @return \e f the flattening of the ellipsoid. This is the value * inherited from the Geodesic object used in the constructor. **********************************************************************/ Math::real Flattening() const { return Init() ? _f : Math::NaN(); } /** * @return \e caps the computational capabilities that this object was * constructed with. LATITUDE and AZIMUTH are always included. **********************************************************************/ unsigned Capabilities() const { return _caps; } /** * Test what capabilities are available. * * @param[in] testcaps a set of bitor'ed GeodesicLine::mask values. * @return true if the GeodesicLine object has all these capabilities. **********************************************************************/ bool Capabilities(unsigned testcaps) const { testcaps &= OUT_ALL; return (_caps & testcaps) == testcaps; } /** * The distance or arc length to point 3. * * @param[in] arcmode boolean flag determining the meaning of returned * value. * @return \e s13 if \e arcmode is false; \e a13 if \e arcmode is true. **********************************************************************/ Math::real GenDistance(bool arcmode) const { return Init() ? (arcmode ? _a13 : _s13) : Math::NaN(); } /** * @return \e s13, the distance to point 3 (meters). **********************************************************************/ Math::real Distance() const { return GenDistance(false); } /** * @return \e a13, the arc length to point 3 (degrees). **********************************************************************/ Math::real Arc() const { return GenDistance(true); } ///@} }; } // namespace GeographicLib #endif // GEOGRAPHICLIB_GEODESICLINE_HPP geosphere/src/Ellipsoid.cpp0000644000176200001440000001033614323376011015455 0ustar liggesusers/** * \file Ellipsoid.cpp * \brief Implementation for GeographicLib::Ellipsoid class * * Copyright (c) Charles Karney (2012-2022) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ **********************************************************************/ #include "Ellipsoid.h" #if defined(_MSC_VER) // Squelch warnings about enum-float expressions # pragma warning (disable: 5055) #endif namespace GeographicLib { using namespace std; Ellipsoid::Ellipsoid(real a, real f) : stol_(real(0.01) * sqrt(numeric_limits::epsilon())) , _a(a) , _f(f) , _f1(1 - _f) , _f12(Math::sq(_f1)) , _e2(_f * (2 - _f)) , _es((_f < 0 ? -1 : 1) * sqrt(fabs(_e2))) , _e12(_e2 / (1 - _e2)) , _n(_f / (2 - _f)) , _b(_a * _f1) , _tm(_a, _f, real(1)) , _ell(-_e12) , _au(_a, _f, real(0), real(1), real(0), real(1), real(1)) {} const Ellipsoid& Ellipsoid::WGS84() { static const Ellipsoid wgs84(Constants::WGS84_a(), Constants::WGS84_f()); return wgs84; } Math::real Ellipsoid::QuarterMeridian() const { return _b * _ell.E(); } Math::real Ellipsoid::Area() const { return 4 * Math::pi() * ((Math::sq(_a) + Math::sq(_b) * (_e2 == 0 ? 1 : (_e2 > 0 ? atanh(sqrt(_e2)) : atan(sqrt(-_e2))) / sqrt(fabs(_e2))))/2); } Math::real Ellipsoid::ParametricLatitude(real phi) const { return Math::atand(_f1 * Math::tand(Math::LatFix(phi))); } Math::real Ellipsoid::InverseParametricLatitude(real beta) const { return Math::atand(Math::tand(Math::LatFix(beta)) / _f1); } Math::real Ellipsoid::GeocentricLatitude(real phi) const { return Math::atand(_f12 * Math::tand(Math::LatFix(phi))); } Math::real Ellipsoid::InverseGeocentricLatitude(real theta) const { return Math::atand(Math::tand(Math::LatFix(theta)) / _f12); } Math::real Ellipsoid::RectifyingLatitude(real phi) const { return fabs(phi) == Math::qd ? phi: Math::qd * MeridianDistance(phi) / QuarterMeridian(); } Math::real Ellipsoid::InverseRectifyingLatitude(real mu) const { if (fabs(mu) == Math::qd) return mu; return InverseParametricLatitude(_ell.Einv(mu * _ell.E() / Math::qd) / Math::degree()); } Math::real Ellipsoid::AuthalicLatitude(real phi) const { return Math::atand(_au.txif(Math::tand(Math::LatFix(phi)))); } Math::real Ellipsoid::InverseAuthalicLatitude(real xi) const { return Math::atand(_au.tphif(Math::tand(Math::LatFix(xi)))); } Math::real Ellipsoid::ConformalLatitude(real phi) const { return Math::atand(Math::taupf(Math::tand(Math::LatFix(phi)), _es)); } Math::real Ellipsoid::InverseConformalLatitude(real chi) const { return Math::atand(Math::tauf(Math::tand(Math::LatFix(chi)), _es)); } Math::real Ellipsoid::IsometricLatitude(real phi) const { return asinh(Math::taupf(Math::tand(Math::LatFix(phi)), _es)) / Math::degree(); } Math::real Ellipsoid::InverseIsometricLatitude(real psi) const { return Math::atand(Math::tauf(sinh(psi * Math::degree()), _es)); } Math::real Ellipsoid::CircleRadius(real phi) const { return fabs(phi) == Math::qd ? 0 : // a * cos(beta) _a / hypot(real(1), _f1 * Math::tand(Math::LatFix(phi))); } Math::real Ellipsoid::CircleHeight(real phi) const { real tbeta = _f1 * Math::tand(phi); // b * sin(beta) return _b * tbeta / hypot(real(1), _f1 * Math::tand(Math::LatFix(phi))); } Math::real Ellipsoid::MeridianDistance(real phi) const { return _b * _ell.Ed( ParametricLatitude(phi) ); } Math::real Ellipsoid::MeridionalCurvatureRadius(real phi) const { real v = 1 - _e2 * Math::sq(Math::sind(Math::LatFix(phi))); return _a * (1 - _e2) / (v * sqrt(v)); } Math::real Ellipsoid::TransverseCurvatureRadius(real phi) const { real v = 1 - _e2 * Math::sq(Math::sind(Math::LatFix(phi))); return _a / sqrt(v); } Math::real Ellipsoid::NormalCurvatureRadius(real phi, real azi) const { real calp, salp, v = 1 - _e2 * Math::sq(Math::sind(Math::LatFix(phi))); Math::sincosd(azi, salp, calp); return _a / (sqrt(v) * (Math::sq(calp) * v / (1 - _e2) + Math::sq(salp))); } } // namespace GeographicLib geosphere/src/PolygonArea.h0000644000176200001440000003234014323377037015426 0ustar liggesusers/** * \file PolygonArea.hpp * \brief Header for GeographicLib::PolygonAreaT class * * Copyright (c) Charles Karney (2010-2022) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ **********************************************************************/ #if !defined(GEOGRAPHICLIB_POLYGONAREA_HPP) #define GEOGRAPHICLIB_POLYGONAREA_HPP 1 #include "Geodesic.h" #include "GeodesicExact.h" #include "Rhumb.h" #include "Accumulator.h" namespace GeographicLib { /** * \brief Polygon areas * * This computes the area of a polygon whose edges are geodesics using the * method given in Section 6 of * - C. F. F. Karney, * * Algorithms for geodesics, * J. Geodesy 87, 43--55 (2013); * DOI: * 10.1007/s00190-012-0578-z; * addenda: * * geod-addenda.html. * * Arbitrarily complex polygons are allowed. In the case self-intersecting * of polygons the area is accumulated "algebraically", e.g., the areas of * the 2 loops in a figure-8 polygon will partially cancel. * * This class lets you add vertices and edges one at a time to the polygon. * The sequence must start with a vertex and thereafter vertices and edges * can be added in any order. Any vertex after the first creates a new edge * which is the \e shortest geodesic from the previous vertex. In some * cases there may be two or many such shortest geodesics and the area is * then not uniquely defined. In this case, either add an intermediate * vertex or add the edge \e as an edge (by defining its direction and * length). * * The area and perimeter are accumulated at two times the standard floating * point precision to guard against the loss of accuracy with many-sided * polygons. At any point you can ask for the perimeter and area so far. * There's an option to treat the points as defining a polyline instead of a * polygon; in that case, only the perimeter is computed. * * This is a templated class to allow it to be used with Geodesic, * GeodesicExact, and Rhumb. GeographicLib::PolygonArea, * GeographicLib::PolygonAreaExact, and GeographicLib::PolygonAreaRhumb are * typedefs for these cases. * * For GeographicLib::PolygonArea (edges defined by Geodesic), an upper bound * on the error is about 0.1 m2 per vertex. However this is a * wildly pessimistic estimate in most cases. A more realistic estimate of * the error is given by a test involving 107 approximately * regular polygons on the WGS84 ellipsoid. The centers and the orientations * of the polygons were uniformly distributed, the number of vertices was * log-uniformly distributed in [3, 300], and the center to vertex distance * log-uniformly distributed in [0.1 m, 9000 km]. * * Using double precision (the standard precision for GeographicLib), the * maximum error in the perimeter was 200 nm, and the maximum error in the * area was
   *     0.0013 m^2 for perimeter < 10 km
   *     0.0070 m^2 for perimeter < 100 km
   *     0.070 m^2 for perimeter < 1000 km
   *     0.11 m^2 for all perimeters
   *
* The errors are given in terms of the perimeter, because it is expected * that the errors depend mainly on the number of edges and the edge lengths. * * Using long doubles (GEOGRPAHICLIB_PRECISION = 3), the maximum error in the * perimeter was 200 pm, and the maximum error in the area was
   *     0.7 mm^2 for perim < 10 km
   *     3.2 mm^2 for perimeter < 100 km
   *     21 mm^2 for perimeter < 1000 km
   *     45 mm^2 for all perimeters
   *
* * @tparam GeodType the geodesic class to use. * * Example of use: * \include example-PolygonArea.cpp * * Planimeter is a command-line utility * providing access to the functionality of PolygonAreaT. **********************************************************************/ template class PolygonAreaT { private: typedef Math::real real; GeodType _earth; real _area0; // Full ellipsoid area bool _polyline; // Assume polyline (don't close and skip area) unsigned _mask; unsigned _num; int _crossings; Accumulator<> _areasum, _perimetersum; real _lat0, _lon0, _lat1, _lon1; static int transit(real lon1, real lon2); // an alternate version of transit to deal with longitudes in the direct // problem. static int transitdirect(real lon1, real lon2); void Remainder(Accumulator<>& a) const { a.remainder(_area0); } void Remainder(real& a) const { using std::remainder; a = remainder(a, _area0); } template void AreaReduce(T& area, int crossings, bool reverse, bool sign) const; public: /** * Constructor for PolygonAreaT. * * @param[in] earth the Geodesic object to use for geodesic calculations. * @param[in] polyline if true that treat the points as defining a polyline * instead of a polygon (default = false). **********************************************************************/ PolygonAreaT(const GeodType& earth, bool polyline = false) : _earth(earth) , _area0(_earth.EllipsoidArea()) , _polyline(polyline) , _mask(GeodType::LATITUDE | GeodType::LONGITUDE | GeodType::DISTANCE | (_polyline ? GeodType::NONE : GeodType::AREA | GeodType::LONG_UNROLL)) { Clear(); } /** * Clear PolygonAreaT, allowing a new polygon to be started. **********************************************************************/ void Clear() { _num = 0; _crossings = 0; _areasum = 0; _perimetersum = 0; _lat0 = _lon0 = _lat1 = _lon1 = Math::NaN(); } /** * Add a point to the polygon or polyline. * * @param[in] lat the latitude of the point (degrees). * @param[in] lon the longitude of the point (degrees). * * \e lat should be in the range [−90°, 90°]. **********************************************************************/ void AddPoint(real lat, real lon); /** * Add an edge to the polygon or polyline. * * @param[in] azi azimuth at current point (degrees). * @param[in] s distance from current point to next point (meters). * * This does nothing if no points have been added yet. Use * PolygonAreaT::CurrentPoint to determine the position of the new vertex. **********************************************************************/ void AddEdge(real azi, real s); /** * Return the results so far. * * @param[in] reverse if true then clockwise (instead of counter-clockwise) * traversal counts as a positive area. * @param[in] sign if true then return a signed result for the area if * the polygon is traversed in the "wrong" direction instead of returning * the area for the rest of the earth. * @param[out] perimeter the perimeter of the polygon or length of the * polyline (meters). * @param[out] area the area of the polygon (meters2); only set * if \e polyline is false in the constructor. * @return the number of points. * * More points can be added to the polygon after this call. **********************************************************************/ unsigned Compute(bool reverse, bool sign, real& perimeter, real& area) const; /** * Return the results assuming a tentative final test point is added; * however, the data for the test point is not saved. This lets you report * a running result for the perimeter and area as the user moves the mouse * cursor. Ordinary floating point arithmetic is used to accumulate the * data for the test point; thus the area and perimeter returned are less * accurate than if PolygonAreaT::AddPoint and PolygonAreaT::Compute are * used. * * @param[in] lat the latitude of the test point (degrees). * @param[in] lon the longitude of the test point (degrees). * @param[in] reverse if true then clockwise (instead of counter-clockwise) * traversal counts as a positive area. * @param[in] sign if true then return a signed result for the area if * the polygon is traversed in the "wrong" direction instead of returning * the area for the rest of the earth. * @param[out] perimeter the approximate perimeter of the polygon or length * of the polyline (meters). * @param[out] area the approximate area of the polygon * (meters2); only set if polyline is false in the * constructor. * @return the number of points. * * \e lat should be in the range [−90°, 90°]. **********************************************************************/ unsigned TestPoint(real lat, real lon, bool reverse, bool sign, real& perimeter, real& area) const; /** * Return the results assuming a tentative final test point is added via an * azimuth and distance; however, the data for the test point is not saved. * This lets you report a running result for the perimeter and area as the * user moves the mouse cursor. Ordinary floating point arithmetic is used * to accumulate the data for the test point; thus the area and perimeter * returned are less accurate than if PolygonAreaT::AddEdge and * PolygonAreaT::Compute are used. * * @param[in] azi azimuth at current point (degrees). * @param[in] s distance from current point to final test point (meters). * @param[in] reverse if true then clockwise (instead of counter-clockwise) * traversal counts as a positive area. * @param[in] sign if true then return a signed result for the area if * the polygon is traversed in the "wrong" direction instead of returning * the area for the rest of the earth. * @param[out] perimeter the approximate perimeter of the polygon or length * of the polyline (meters). * @param[out] area the approximate area of the polygon * (meters2); only set if polyline is false in the * constructor. * @return the number of points. **********************************************************************/ unsigned TestEdge(real azi, real s, bool reverse, bool sign, real& perimeter, real& area) const; /** \name Inspector functions **********************************************************************/ ///@{ /** * @return \e a the equatorial radius of the ellipsoid (meters). This is * the value inherited from the Geodesic object used in the constructor. **********************************************************************/ Math::real EquatorialRadius() const { return _earth.EquatorialRadius(); } /** * @return \e f the flattening of the ellipsoid. This is the value * inherited from the Geodesic object used in the constructor. **********************************************************************/ Math::real Flattening() const { return _earth.Flattening(); } /** * Report the previous vertex added to the polygon or polyline. * * @param[out] lat the latitude of the point (degrees). * @param[out] lon the longitude of the point (degrees). * * If no points have been added, then NaNs are returned. Otherwise, \e lon * will be in the range [−180°, 180°]. **********************************************************************/ void CurrentPoint(real& lat, real& lon) const { lat = _lat1; lon = _lon1; } /** * Report the number of points currently in the polygon or polyline. * * @return the number of points. * * If no points have been added, then 0 is returned. **********************************************************************/ unsigned NumberPoints() const { return _num; } /** * Report whether the current object is a polygon or a polyline. * * @return true if the object is a polyline. **********************************************************************/ bool Polyline() const { return _polyline; } ///@} }; /** * @relates PolygonAreaT * * Polygon areas using Geodesic. This should be used if the flattening is * small. **********************************************************************/ typedef PolygonAreaT PolygonArea; /** * @relates PolygonAreaT * * Polygon areas using GeodesicExact. (But note that the implementation of * areas in GeodesicExact uses a high order series and this is only accurate * for modest flattenings.) **********************************************************************/ typedef PolygonAreaT PolygonAreaExact; /** * @relates PolygonAreaT * * Polygon areas using Rhumb. **********************************************************************/ typedef PolygonAreaT PolygonAreaRhumb; } // namespace GeographicLib #endif // GEOGRAPHICLIB_POLYGONAREA_HPP geosphere/src/PolygonArea.cpp0000644000176200001440000001672214323376012015757 0ustar liggesusers/** * \file PolygonArea.cpp * \brief Implementation for GeographicLib::PolygonAreaT class * * Copyright (c) Charles Karney (2010-2022) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ **********************************************************************/ #include "PolygonArea.h" #if defined(_MSC_VER) // Squelch warnings about enum-float expressions # pragma warning (disable: 5055) #endif namespace GeographicLib { using namespace std; template int PolygonAreaT::transit(real lon1, real lon2) { // Return 1 or -1 if crossing prime meridian in east or west direction. // Otherwise return zero. longitude = +/-0 considered to be positive. // This is (should be?) compatible with transitdirect which computes // exactly the parity of // int(floor((lon1 + lon12) / 360)) - int(floor(lon1 / 360))) real lon12 = Math::AngDiff(lon1, lon2); lon1 = Math::AngNormalize(lon1); lon2 = Math::AngNormalize(lon2); // N.B. lon12 == 0 gives cross = 0 return // edge case lon1 = 180, lon2 = 360->0, lon12 = 180 to give 1 lon12 > 0 && ((lon1 < 0 && lon2 >= 0) || // lon12 > 0 && lon1 > 0 && lon2 == 0 implies lon1 == 180 (lon1 > 0 && lon2 == 0)) ? 1 : // non edge case lon1 = -180, lon2 = -360->-0, lon12 = -180 (lon12 < 0 && lon1 >= 0 && lon2 < 0 ? -1 : 0); // This was the old method (treating +/- 0 as negative). However, with the // new scheme for handling longitude differences this fails on: // lon1 = -180, lon2 = -360->-0, lon12 = -180 gives 0 not -1. // return // lon1 <= 0 && lon2 > 0 && lon12 > 0 ? 1 : // (lon2 <= 0 && lon1 > 0 && lon12 < 0 ? -1 : 0); } // an alternate version of transit to deal with longitudes in the direct // problem. template int PolygonAreaT::transitdirect(real lon1, real lon2) { // Compute exactly the parity of // int(floor(lon2 / 360)) - int(floor(lon1 / 360)) using std::remainder; // C++ C remainder -> [-360, 360] // Java % -> (-720, 720) switch to IEEEremainder -> [-360, 360] // JS % -> (-720, 720) // Python fmod -> (-720, 720) swith to Math.remainder // Fortran, Octave skip // If mod function gives result in [-360, 360] // [0, 360) -> 0; [-360, 0) or 360 -> 1 // If mod function gives result in (-720, 720) // [0, 360) or [-inf, -360) -> 0; [-360, 0) or [360, inf) -> 1 lon1 = remainder(lon1, real(2 * Math::td)); lon2 = remainder(lon2, real(2 * Math::td)); return ( (lon2 >= 0 && lon2 < Math::td ? 0 : 1) - (lon1 >= 0 && lon1 < Math::td ? 0 : 1) ); } template void PolygonAreaT::AddPoint(real lat, real lon) { if (_num == 0) { _lat0 = _lat1 = lat; _lon0 = _lon1 = lon; } else { real s12, S12, t; _earth.GenInverse(_lat1, _lon1, lat, lon, _mask, s12, t, t, t, t, t, S12); _perimetersum += s12; if (!_polyline) { _areasum += S12; _crossings += transit(_lon1, lon); } _lat1 = lat; _lon1 = lon; } ++_num; } template void PolygonAreaT::AddEdge(real azi, real s) { if (_num) { // Do nothing if _num is zero real lat, lon, S12, t; _earth.GenDirect(_lat1, _lon1, azi, false, s, _mask, lat, lon, t, t, t, t, t, S12); _perimetersum += s; if (!_polyline) { _areasum += S12; _crossings += transitdirect(_lon1, lon); } _lat1 = lat; _lon1 = lon; ++_num; } } template unsigned PolygonAreaT::Compute(bool reverse, bool sign, real& perimeter, real& area) const { real s12, S12, t; if (_num < 2) { perimeter = 0; if (!_polyline) area = 0; return _num; } if (_polyline) { perimeter = _perimetersum(); return _num; } _earth.GenInverse(_lat1, _lon1, _lat0, _lon0, _mask, s12, t, t, t, t, t, S12); perimeter = _perimetersum(s12); Accumulator<> tempsum(_areasum); tempsum += S12; int crossings = _crossings + transit(_lon1, _lon0); AreaReduce(tempsum, crossings, reverse, sign); area = real(0) + tempsum(); return _num; } template unsigned PolygonAreaT::TestPoint(real lat, real lon, bool reverse, bool sign, real& perimeter, real& area) const { if (_num == 0) { perimeter = 0; if (!_polyline) area = 0; return 1; } perimeter = _perimetersum(); real tempsum = _polyline ? 0 : _areasum(); int crossings = _crossings; unsigned num = _num + 1; for (int i = 0; i < (_polyline ? 1 : 2); ++i) { real s12, S12, t; _earth.GenInverse(i == 0 ? _lat1 : lat, i == 0 ? _lon1 : lon, i != 0 ? _lat0 : lat, i != 0 ? _lon0 : lon, _mask, s12, t, t, t, t, t, S12); perimeter += s12; if (!_polyline) { tempsum += S12; crossings += transit(i == 0 ? _lon1 : lon, i != 0 ? _lon0 : lon); } } if (_polyline) return num; AreaReduce(tempsum, crossings, reverse, sign); area = real(0) + tempsum; return num; } template unsigned PolygonAreaT::TestEdge(real azi, real s, bool reverse, bool sign, real& perimeter, real& area) const { if (_num == 0) { // we don't have a starting point! perimeter = Math::NaN(); if (!_polyline) area = Math::NaN(); return 0; } unsigned num = _num + 1; perimeter = _perimetersum() + s; if (_polyline) return num; real tempsum = _areasum(); int crossings = _crossings; { real lat, lon, s12, S12, t; _earth.GenDirect(_lat1, _lon1, azi, false, s, _mask, lat, lon, t, t, t, t, t, S12); tempsum += S12; crossings += transitdirect(_lon1, lon); _earth.GenInverse(lat, lon, _lat0, _lon0, _mask, s12, t, t, t, t, t, S12); perimeter += s12; tempsum += S12; crossings += transit(lon, _lon0); } AreaReduce(tempsum, crossings, reverse, sign); area = real(0) + tempsum; return num; } template template void PolygonAreaT::AreaReduce(T& area, int crossings, bool reverse, bool sign) const { Remainder(area); if (crossings & 1) area += (area < 0 ? 1 : -1) * _area0/2; // area is with the clockwise sense. If !reverse convert to // counter-clockwise convention. if (!reverse) area *= -1; // If sign put area in (-_area0/2, _area0/2], else put area in [0, _area0) if (sign) { if (area > _area0/2) area -= _area0; else if (area <= -_area0/2) area += _area0; } else { if (area >= _area0) area -= _area0; else if (area < 0) area += _area0; } } template class GEOGRAPHICLIB_EXPORT PolygonAreaT; template class GEOGRAPHICLIB_EXPORT PolygonAreaT; template class GEOGRAPHICLIB_EXPORT PolygonAreaT; } // namespace GeographicLib geosphere/src/TransverseMercator.h0000644000176200001440000002140014323377037017032 0ustar liggesusers/** * \file TransverseMercator.hpp * \brief Header for GeographicLib::TransverseMercator class * * Copyright (c) Charles Karney (2008-2022) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ **********************************************************************/ #if !defined(GEOGRAPHICLIB_TRANSVERSEMERCATOR_HPP) #define GEOGRAPHICLIB_TRANSVERSEMERCATOR_HPP 1 #include "Constants.h" #if !defined(GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER) /** * The order of the series approximation used in TransverseMercator. * GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER can be set to any integer in [4, 8]. **********************************************************************/ # define GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER \ (GEOGRAPHICLIB_PRECISION == 2 ? 6 : \ (GEOGRAPHICLIB_PRECISION == 1 ? 4 : 8)) #endif namespace GeographicLib { /** * \brief Transverse Mercator projection * * This uses Krüger's method which evaluates the projection and its * inverse in terms of a series. See * - L. Krüger, * Konforme * Abbildung des Erdellipsoids in der Ebene (Conformal mapping of the * ellipsoidal earth to the plane), Royal Prussian Geodetic Institute, New * Series 52, 172 pp. (1912). * - C. F. F. Karney, * * Transverse Mercator with an accuracy of a few nanometers, * J. Geodesy 85(8), 475--485 (Aug. 2011); * preprint * arXiv:1002.1417. * * Krüger's method has been extended from 4th to 6th order. The maximum * error is 5 nm (5 nanometers), ground distance, for all positions within 35 * degrees of the central meridian. The error in the convergence is 2 * × 10−15" and the relative error in the scale * is 6 × 10−12%%. See Sec. 4 of * arXiv:1002.1417 for details. * The speed penalty in going to 6th order is only about 1%. * * There's a singularity in the projection at φ = 0°, λ * − λ0 = ±(1 − \e e)90° (≈ * ±82.6° for the WGS84 ellipsoid), where \e e is the * eccentricity. Beyond this point, the series ceases to converge and the * results from this method will be garbage. To be on the safe side, don't * use this method if the angular distance from the central meridian exceeds * (1 − 2e)90° (≈ 75° for the WGS84 ellipsoid) * * TransverseMercatorExact is an alternative implementation of the projection * using exact formulas which yield accurate (to 8 nm) results over the * entire ellipsoid. * * The ellipsoid parameters and the central scale are set in the constructor. * The central meridian (which is a trivial shift of the longitude) is * specified as the \e lon0 argument of the TransverseMercator::Forward and * TransverseMercator::Reverse functions. The latitude of origin is taken to * be the equator. There is no provision in this class for specifying a * false easting or false northing or a different latitude of origin. * However these are can be simply included by the calling function. For * example, the UTMUPS class applies the false easting and false northing for * the UTM projections. A more complicated example is the British National * Grid ( * EPSG:7405) which requires the use of a latitude of origin. This is * implemented by the GeographicLib::OSGB class. * * This class also returns the meridian convergence \e gamma and scale \e k. * The meridian convergence is the bearing of grid north (the \e y axis) * measured clockwise from true north. * * See TransverseMercator.cpp for more information on the implementation. * * See \ref transversemercator for a discussion of this projection. * * Example of use: * \include example-TransverseMercator.cpp * * TransverseMercatorProj is a * command-line utility providing access to the functionality of * TransverseMercator and TransverseMercatorExact. **********************************************************************/ class GEOGRAPHICLIB_EXPORT TransverseMercator { private: typedef Math::real real; static const int maxpow_ = GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER; static const int numit_ = 5; real _a, _f, _k0, _e2, _es, _e2m, _c, _n; // _alp[0] and _bet[0] unused real _a1, _b1, _alp[maxpow_ + 1], _bet[maxpow_ + 1]; friend class Ellipsoid; // For access to taupf, tauf. public: /** * Constructor for an ellipsoid with * * @param[in] a equatorial radius (meters). * @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere. * Negative \e f gives a prolate ellipsoid. * @param[in] k0 central scale factor. * @exception GeographicErr if \e a, (1 − \e f) \e a, or \e k0 is * not positive. **********************************************************************/ TransverseMercator(real a, real f, real k0); /** * Forward projection, from geographic to transverse Mercator. * * @param[in] lon0 central meridian of the projection (degrees). * @param[in] lat latitude of point (degrees). * @param[in] lon longitude of point (degrees). * @param[out] x easting of point (meters). * @param[out] y northing of point (meters). * @param[out] gamma meridian convergence at point (degrees). * @param[out] k scale of projection at point. * * No false easting or northing is added. \e lat should be in the range * [−90°, 90°]. **********************************************************************/ void Forward(real lon0, real lat, real lon, real& x, real& y, real& gamma, real& k) const; /** * Reverse projection, from transverse Mercator to geographic. * * @param[in] lon0 central meridian of the projection (degrees). * @param[in] x easting of point (meters). * @param[in] y northing of point (meters). * @param[out] lat latitude of point (degrees). * @param[out] lon longitude of point (degrees). * @param[out] gamma meridian convergence at point (degrees). * @param[out] k scale of projection at point. * * No false easting or northing is added. The value of \e lon returned is * in the range [−180°, 180°]. **********************************************************************/ void Reverse(real lon0, real x, real y, real& lat, real& lon, real& gamma, real& k) const; /** * TransverseMercator::Forward without returning the convergence and scale. **********************************************************************/ void Forward(real lon0, real lat, real lon, real& x, real& y) const { real gamma, k; Forward(lon0, lat, lon, x, y, gamma, k); } /** * TransverseMercator::Reverse without returning the convergence and scale. **********************************************************************/ void Reverse(real lon0, real x, real y, real& lat, real& lon) const { real gamma, k; Reverse(lon0, x, y, lat, lon, gamma, k); } /** \name Inspector functions **********************************************************************/ ///@{ /** * @return \e a the equatorial radius of the ellipsoid (meters). This is * the value used in the constructor. **********************************************************************/ Math::real EquatorialRadius() const { return _a; } /** * @return \e f the flattening of the ellipsoid. This is the value used in * the constructor. **********************************************************************/ Math::real Flattening() const { return _f; } /** * @return \e k0 central scale for the projection. This is the value of \e * k0 used in the constructor and is the scale on the central meridian. **********************************************************************/ Math::real CentralScale() const { return _k0; } ///@} /** * A global instantiation of TransverseMercator with the WGS84 ellipsoid * and the UTM scale factor. However, unlike UTM, no false easting or * northing is added. **********************************************************************/ static const TransverseMercator& UTM(); }; } // namespace GeographicLib #endif // GEOGRAPHICLIB_TRANSVERSEMERCATOR_HPP geosphere/src/Geodesic.cpp0000644000176200001440000022103514323376011015253 0ustar liggesusers/** * \file Geodesic.cpp * \brief Implementation for GeographicLib::Geodesic class * * Copyright (c) Charles Karney (2009-2022) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ * * This is a reformulation of the geodesic problem. The notation is as * follows: * - at a general point (no suffix or 1 or 2 as suffix) * - phi = latitude * - beta = latitude on auxiliary sphere * - omega = longitude on auxiliary sphere * - lambda = longitude * - alpha = azimuth of great circle * - sigma = arc length along great circle * - s = distance * - tau = scaled distance (= sigma at multiples of pi/2) * - at northwards equator crossing * - beta = phi = 0 * - omega = lambda = 0 * - alpha = alpha0 * - sigma = s = 0 * - a 12 suffix means a difference, e.g., s12 = s2 - s1. * - s and c prefixes mean sin and cos **********************************************************************/ #include "Geodesic.h" #include "GeodesicLine.h" #if defined(_MSC_VER) // Squelch warnings about potentially uninitialized local variables, // constant conditional and enum-float expressions and mixing enums # pragma warning (disable: 4701 4127 5055 5054) #endif namespace GeographicLib { using namespace std; Geodesic::Geodesic(real a, real f) : maxit2_(maxit1_ + Math::digits() + 10) // Underflow guard. We require // tiny_ * epsilon() > 0 // tiny_ + epsilon() == epsilon() , tiny_(sqrt(numeric_limits::min())) , tol0_(numeric_limits::epsilon()) // Increase multiplier in defn of tol1_ from 100 to 200 to fix inverse // case 52.784459512564 0 -52.784459512563990912 179.634407464943777557 // which otherwise failed for Visual Studio 10 (Release and Debug) , tol1_(200 * tol0_) , tol2_(sqrt(tol0_)) , tolb_(tol0_) // Check on bisection interval , xthresh_(1000 * tol2_) , _a(a) , _f(f) , _f1(1 - _f) , _e2(_f * (2 - _f)) , _ep2(_e2 / Math::sq(_f1)) // e2 / (1 - e2) , _n(_f / ( 2 - _f)) , _b(_a * _f1) , _c2((Math::sq(_a) + Math::sq(_b) * (_e2 == 0 ? 1 : Math::eatanhe(real(1), (_f < 0 ? -1 : 1) * sqrt(fabs(_e2))) / _e2)) / 2) // authalic radius squared // The sig12 threshold for "really short". Using the auxiliary sphere // solution with dnm computed at (bet1 + bet2) / 2, the relative error in // the azimuth consistency check is sig12^2 * abs(f) * min(1, 1-f/2) / 2. // (Error measured for 1/100 < b/a < 100 and abs(f) >= 1/1000. For a // given f and sig12, the max error occurs for lines near the pole. If // the old rule for computing dnm = (dn1 + dn2)/2 is used, then the error // increases by a factor of 2.) Setting this equal to epsilon gives // sig12 = etol2. Here 0.1 is a safety factor (error decreased by 100) // and max(0.001, abs(f)) stops etol2 getting too large in the nearly // spherical case. , _etol2(real(0.1) * tol2_ / sqrt( fmax(real(0.001), fabs(_f)) * fmin(real(1), 1 - _f/2) / 2 )) { if (!(isfinite(_a) && _a > 0)) throw GeographicErr("Equatorial radius is not positive"); if (!(isfinite(_b) && _b > 0)) throw GeographicErr("Polar semi-axis is not positive"); A3coeff(); C3coeff(); C4coeff(); } const Geodesic& Geodesic::WGS84() { static const Geodesic wgs84(Constants::WGS84_a(), Constants::WGS84_f()); return wgs84; } Math::real Geodesic::SinCosSeries(bool sinp, real sinx, real cosx, const real c[], int n) { // Evaluate // y = sinp ? sum(c[i] * sin( 2*i * x), i, 1, n) : // sum(c[i] * cos((2*i+1) * x), i, 0, n-1) // using Clenshaw summation. N.B. c[0] is unused for sin series // Approx operation count = (n + 5) mult and (2 * n + 2) add c += (n + sinp); // Point to one beyond last element real ar = 2 * (cosx - sinx) * (cosx + sinx), // 2 * cos(2 * x) y0 = n & 1 ? *--c : 0, y1 = 0; // accumulators for sum // Now n is even n /= 2; while (n--) { // Unroll loop x 2, so accumulators return to their original role y1 = ar * y0 - y1 + *--c; y0 = ar * y1 - y0 + *--c; } return sinp ? 2 * sinx * cosx * y0 // sin(2 * x) * y0 : cosx * (y0 - y1); // cos(x) * (y0 - y1) } GeodesicLine Geodesic::Line(real lat1, real lon1, real azi1, unsigned caps) const { return GeodesicLine(*this, lat1, lon1, azi1, caps); } Math::real Geodesic::GenDirect(real lat1, real lon1, real azi1, bool arcmode, real s12_a12, unsigned outmask, real& lat2, real& lon2, real& azi2, real& s12, real& m12, real& M12, real& M21, real& S12) const { // Automatically supply DISTANCE_IN if necessary if (!arcmode) outmask |= DISTANCE_IN; return GeodesicLine(*this, lat1, lon1, azi1, outmask) . // Note the dot! GenPosition(arcmode, s12_a12, outmask, lat2, lon2, azi2, s12, m12, M12, M21, S12); } GeodesicLine Geodesic::GenDirectLine(real lat1, real lon1, real azi1, bool arcmode, real s12_a12, unsigned caps) const { azi1 = Math::AngNormalize(azi1); real salp1, calp1; // Guard against underflow in salp0. Also -0 is converted to +0. Math::sincosd(Math::AngRound(azi1), salp1, calp1); // Automatically supply DISTANCE_IN if necessary if (!arcmode) caps |= DISTANCE_IN; return GeodesicLine(*this, lat1, lon1, azi1, salp1, calp1, caps, arcmode, s12_a12); } GeodesicLine Geodesic::DirectLine(real lat1, real lon1, real azi1, real s12, unsigned caps) const { return GenDirectLine(lat1, lon1, azi1, false, s12, caps); } GeodesicLine Geodesic::ArcDirectLine(real lat1, real lon1, real azi1, real a12, unsigned caps) const { return GenDirectLine(lat1, lon1, azi1, true, a12, caps); } Math::real Geodesic::GenInverse(real lat1, real lon1, real lat2, real lon2, unsigned outmask, real& s12, real& salp1, real& calp1, real& salp2, real& calp2, real& m12, real& M12, real& M21, real& S12) const { // Compute longitude difference (AngDiff does this carefully). using std::isnan; // Needed for Centos 7, ubuntu 14 real lon12s, lon12 = Math::AngDiff(lon1, lon2, lon12s); // Make longitude difference positive. int lonsign = signbit(lon12) ? -1 : 1; lon12 *= lonsign; lon12s *= lonsign; real lam12 = lon12 * Math::degree(), slam12, clam12; // Calculate sincos of lon12 + error (this applies AngRound internally). Math::sincosde(lon12, lon12s, slam12, clam12); // the supplementary longitude difference lon12s = (Math::hd - lon12) - lon12s; // If really close to the equator, treat as on equator. lat1 = Math::AngRound(Math::LatFix(lat1)); lat2 = Math::AngRound(Math::LatFix(lat2)); // Swap points so that point with higher (abs) latitude is point 1. // If one latitude is a nan, then it becomes lat1. int swapp = fabs(lat1) < fabs(lat2) || isnan(lat2) ? -1 : 1; if (swapp < 0) { lonsign *= -1; swap(lat1, lat2); } // Make lat1 <= -0 int latsign = signbit(lat1) ? 1 : -1; lat1 *= latsign; lat2 *= latsign; // Now we have // // 0 <= lon12 <= 180 // -90 <= lat1 <= -0 // lat1 <= lat2 <= -lat1 // // longsign, swapp, latsign register the transformation to bring the // coordinates to this canonical form. In all cases, 1 means no change was // made. We make these transformations so that there are few cases to // check, e.g., on verifying quadrants in atan2. In addition, this // enforces some symmetries in the results returned. real sbet1, cbet1, sbet2, cbet2, s12x, m12x; Math::sincosd(lat1, sbet1, cbet1); sbet1 *= _f1; // Ensure cbet1 = +epsilon at poles; doing the fix on beta means that sig12 // will be <= 2*tiny for two points at the same pole. Math::norm(sbet1, cbet1); cbet1 = fmax(tiny_, cbet1); Math::sincosd(lat2, sbet2, cbet2); sbet2 *= _f1; // Ensure cbet2 = +epsilon at poles Math::norm(sbet2, cbet2); cbet2 = fmax(tiny_, cbet2); // If cbet1 < -sbet1, then cbet2 - cbet1 is a sensitive measure of the // |bet1| - |bet2|. Alternatively (cbet1 >= -sbet1), abs(sbet2) + sbet1 is // a better measure. This logic is used in assigning calp2 in Lambda12. // Sometimes these quantities vanish and in that case we force bet2 = +/- // bet1 exactly. An example where is is necessary is the inverse problem // 48.522876735459 0 -48.52287673545898293 179.599720456223079643 // which failed with Visual Studio 10 (Release and Debug) if (cbet1 < -sbet1) { if (cbet2 == cbet1) sbet2 = copysign(sbet1, sbet2); } else { if (fabs(sbet2) == -sbet1) cbet2 = cbet1; } real dn1 = sqrt(1 + _ep2 * Math::sq(sbet1)), dn2 = sqrt(1 + _ep2 * Math::sq(sbet2)); real a12, sig12; // index zero element of this array is unused real Ca[nC_]; bool meridian = lat1 == -Math::qd || slam12 == 0; if (meridian) { // Endpoints are on a single full meridian, so the geodesic might lie on // a meridian. calp1 = clam12; salp1 = slam12; // Head to the target longitude calp2 = 1; salp2 = 0; // At the target we're heading north real // tan(bet) = tan(sig) * cos(alp) ssig1 = sbet1, csig1 = calp1 * cbet1, ssig2 = sbet2, csig2 = calp2 * cbet2; // sig12 = sig2 - sig1 sig12 = atan2(fmax(real(0), csig1 * ssig2 - ssig1 * csig2) + real(0), csig1 * csig2 + ssig1 * ssig2); { real dummy; Lengths(_n, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2, outmask | DISTANCE | REDUCEDLENGTH, s12x, m12x, dummy, M12, M21, Ca); } // Add the check for sig12 since zero length geodesics might yield m12 < // 0. Test case was // // echo 20.001 0 20.001 0 | GeodSolve -i // // In fact, we will have sig12 > pi/2 for meridional geodesic which is // not a shortest path. // TODO: investigate m12 < 0 result for aarch/ppc (with -f -p 20) // 20.001000000000001 0.000000000000000 180.000000000000000 // 20.001000000000001 0.000000000000000 180.000000000000000 // 0.0000000002 0.000000000000001 -0.0000000001 // 0.99999999999999989 0.99999999999999989 0.000 if (sig12 < 1 || m12x >= 0) { // Need at least 2, to handle 90 0 90 180 if (sig12 < 3 * tiny_ || // Prevent negative s12 or m12 for short lines (sig12 < tol0_ && (s12x < 0 || m12x < 0))) sig12 = m12x = s12x = 0; m12x *= _b; s12x *= _b; a12 = sig12 / Math::degree(); } else // m12 < 0, i.e., prolate and too close to anti-podal meridian = false; } // somg12 == 2 marks that it needs to be calculated real omg12 = 0, somg12 = 2, comg12 = 0; if (!meridian && sbet1 == 0 && // and sbet2 == 0 (_f <= 0 || lon12s >= _f * Math::hd)) { // Geodesic runs along equator calp1 = calp2 = 0; salp1 = salp2 = 1; s12x = _a * lam12; sig12 = omg12 = lam12 / _f1; m12x = _b * sin(sig12); if (outmask & GEODESICSCALE) M12 = M21 = cos(sig12); a12 = lon12 / _f1; } else if (!meridian) { // Now point1 and point2 belong within a hemisphere bounded by a // meridian and geodesic is neither meridional or equatorial. // Figure a starting point for Newton's method real dnm; sig12 = InverseStart(sbet1, cbet1, dn1, sbet2, cbet2, dn2, lam12, slam12, clam12, salp1, calp1, salp2, calp2, dnm, Ca); if (sig12 >= 0) { // Short lines (InverseStart sets salp2, calp2, dnm) s12x = sig12 * _b * dnm; m12x = Math::sq(dnm) * _b * sin(sig12 / dnm); if (outmask & GEODESICSCALE) M12 = M21 = cos(sig12 / dnm); a12 = sig12 / Math::degree(); omg12 = lam12 / (_f1 * dnm); } else { // Newton's method. This is a straightforward solution of f(alp1) = // lambda12(alp1) - lam12 = 0 with one wrinkle. f(alp) has exactly one // root in the interval (0, pi) and its derivative is positive at the // root. Thus f(alp) is positive for alp > alp1 and negative for alp < // alp1. During the course of the iteration, a range (alp1a, alp1b) is // maintained which brackets the root and with each evaluation of // f(alp) the range is shrunk, if possible. Newton's method is // restarted whenever the derivative of f is negative (because the new // value of alp1 is then further from the solution) or if the new // estimate of alp1 lies outside (0,pi); in this case, the new starting // guess is taken to be (alp1a + alp1b) / 2. // // initial values to suppress warnings (if loop is executed 0 times) real ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0, eps = 0, domg12 = 0; unsigned numit = 0; // Bracketing range real salp1a = tiny_, calp1a = 1, salp1b = tiny_, calp1b = -1; for (bool tripn = false, tripb = false;; ++numit) { // the WGS84 test set: mean = 1.47, sd = 1.25, max = 16 // WGS84 and random input: mean = 2.85, sd = 0.60 real dv; real v = Lambda12(sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1, calp1, slam12, clam12, salp2, calp2, sig12, ssig1, csig1, ssig2, csig2, eps, domg12, numit < maxit1_, dv, Ca); if (tripb || // Reversed test to allow escape with NaNs !(fabs(v) >= (tripn ? 8 : 1) * tol0_) || // Enough bisections to get accurate result numit == maxit2_) break; // Update bracketing values if (v > 0 && (numit > maxit1_ || calp1/salp1 > calp1b/salp1b)) { salp1b = salp1; calp1b = calp1; } else if (v < 0 && (numit > maxit1_ || calp1/salp1 < calp1a/salp1a)) { salp1a = salp1; calp1a = calp1; } if (numit < maxit1_ && dv > 0) { real dalp1 = -v/dv; // |dalp1| < pi test moved earlier because GEOGRAPHICLIB_PRECISION // = 5 can result in dalp1 = 10^(10^8). Then sin(dalp1) takes ages // (because of the need to do accurate range reduction). if (fabs(dalp1) < Math::pi()) { real sdalp1 = sin(dalp1), cdalp1 = cos(dalp1), nsalp1 = salp1 * cdalp1 + calp1 * sdalp1; if (nsalp1 > 0) { calp1 = calp1 * cdalp1 - salp1 * sdalp1; salp1 = nsalp1; Math::norm(salp1, calp1); // In some regimes we don't get quadratic convergence because // slope -> 0. So use convergence conditions based on epsilon // instead of sqrt(epsilon). tripn = fabs(v) <= 16 * tol0_; continue; } } } // Either dv was not positive or updated value was outside legal // range. Use the midpoint of the bracket as the next estimate. // This mechanism is not needed for the WGS84 ellipsoid, but it does // catch problems with more eccentric ellipsoids. Its efficacy is // such for the WGS84 test set with the starting guess set to alp1 = // 90deg: // the WGS84 test set: mean = 5.21, sd = 3.93, max = 24 // WGS84 and random input: mean = 4.74, sd = 0.99 salp1 = (salp1a + salp1b)/2; calp1 = (calp1a + calp1b)/2; Math::norm(salp1, calp1); tripn = false; tripb = (fabs(salp1a - salp1) + (calp1a - calp1) < tolb_ || fabs(salp1 - salp1b) + (calp1 - calp1b) < tolb_); } { real dummy; // Ensure that the reduced length and geodesic scale are computed in // a "canonical" way, with the I2 integral. unsigned lengthmask = outmask | (outmask & (REDUCEDLENGTH | GEODESICSCALE) ? DISTANCE : NONE); Lengths(eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2, lengthmask, s12x, m12x, dummy, M12, M21, Ca); } m12x *= _b; s12x *= _b; a12 = sig12 / Math::degree(); if (outmask & AREA) { // omg12 = lam12 - domg12 real sdomg12 = sin(domg12), cdomg12 = cos(domg12); somg12 = slam12 * cdomg12 - clam12 * sdomg12; comg12 = clam12 * cdomg12 + slam12 * sdomg12; } } } if (outmask & DISTANCE) s12 = real(0) + s12x; // Convert -0 to 0 if (outmask & REDUCEDLENGTH) m12 = real(0) + m12x; // Convert -0 to 0 if (outmask & AREA) { real // From Lambda12: sin(alp1) * cos(bet1) = sin(alp0) salp0 = salp1 * cbet1, calp0 = hypot(calp1, salp1 * sbet1); // calp0 > 0 real alp12; if (calp0 != 0 && salp0 != 0) { real // From Lambda12: tan(bet) = tan(sig) * cos(alp) ssig1 = sbet1, csig1 = calp1 * cbet1, ssig2 = sbet2, csig2 = calp2 * cbet2, k2 = Math::sq(calp0) * _ep2, eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2), // Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0). A4 = Math::sq(_a) * calp0 * salp0 * _e2; Math::norm(ssig1, csig1); Math::norm(ssig2, csig2); C4f(eps, Ca); real B41 = SinCosSeries(false, ssig1, csig1, Ca, nC4_), B42 = SinCosSeries(false, ssig2, csig2, Ca, nC4_); S12 = A4 * (B42 - B41); } else // Avoid problems with indeterminate sig1, sig2 on equator S12 = 0; if (!meridian && somg12 == 2) { somg12 = sin(omg12); comg12 = cos(omg12); } if (!meridian && // omg12 < 3/4 * pi comg12 > -real(0.7071) && // Long difference not too big sbet2 - sbet1 < real(1.75)) { // Lat difference not too big // Use tan(Gamma/2) = tan(omg12/2) // * (tan(bet1/2)+tan(bet2/2))/(1+tan(bet1/2)*tan(bet2/2)) // with tan(x/2) = sin(x)/(1+cos(x)) real domg12 = 1 + comg12, dbet1 = 1 + cbet1, dbet2 = 1 + cbet2; alp12 = 2 * atan2( somg12 * ( sbet1 * dbet2 + sbet2 * dbet1 ), domg12 * ( sbet1 * sbet2 + dbet1 * dbet2 ) ); } else { // alp12 = alp2 - alp1, used in atan2 so no need to normalize real salp12 = salp2 * calp1 - calp2 * salp1, calp12 = calp2 * calp1 + salp2 * salp1; // The right thing appears to happen if alp1 = +/-180 and alp2 = 0, viz // salp12 = -0 and alp12 = -180. However this depends on the sign // being attached to 0 correctly. The following ensures the correct // behavior. if (salp12 == 0 && calp12 < 0) { salp12 = tiny_ * calp1; calp12 = -1; } alp12 = atan2(salp12, calp12); } S12 += _c2 * alp12; S12 *= swapp * lonsign * latsign; // Convert -0 to 0 S12 += 0; } // Convert calp, salp to azimuth accounting for lonsign, swapp, latsign. if (swapp < 0) { swap(salp1, salp2); swap(calp1, calp2); if (outmask & GEODESICSCALE) swap(M12, M21); } salp1 *= swapp * lonsign; calp1 *= swapp * latsign; salp2 *= swapp * lonsign; calp2 *= swapp * latsign; // Returned value in [0, 180] return a12; } Math::real Geodesic::GenInverse(real lat1, real lon1, real lat2, real lon2, unsigned outmask, real& s12, real& azi1, real& azi2, real& m12, real& M12, real& M21, real& S12) const { outmask &= OUT_MASK; real salp1, calp1, salp2, calp2, a12 = GenInverse(lat1, lon1, lat2, lon2, outmask, s12, salp1, calp1, salp2, calp2, m12, M12, M21, S12); if (outmask & AZIMUTH) { azi1 = Math::atan2d(salp1, calp1); azi2 = Math::atan2d(salp2, calp2); } return a12; } GeodesicLine Geodesic::InverseLine(real lat1, real lon1, real lat2, real lon2, unsigned caps) const { real t, salp1, calp1, salp2, calp2, a12 = GenInverse(lat1, lon1, lat2, lon2, // No need to specify AZIMUTH here 0u, t, salp1, calp1, salp2, calp2, t, t, t, t), azi1 = Math::atan2d(salp1, calp1); // Ensure that a12 can be converted to a distance if (caps & (OUT_MASK & DISTANCE_IN)) caps |= DISTANCE; return GeodesicLine(*this, lat1, lon1, azi1, salp1, calp1, caps, true, a12); } void Geodesic::Lengths(real eps, real sig12, real ssig1, real csig1, real dn1, real ssig2, real csig2, real dn2, real cbet1, real cbet2, unsigned outmask, real& s12b, real& m12b, real& m0, real& M12, real& M21, // Scratch area of the right size real Ca[]) const { // Return m12b = (reduced length)/_b; also calculate s12b = distance/_b, // and m0 = coefficient of secular term in expression for reduced length. outmask &= OUT_MASK; // outmask & DISTANCE: set s12b // outmask & REDUCEDLENGTH: set m12b & m0 // outmask & GEODESICSCALE: set M12 & M21 real m0x = 0, J12 = 0, A1 = 0, A2 = 0; real Cb[nC2_ + 1]; if (outmask & (DISTANCE | REDUCEDLENGTH | GEODESICSCALE)) { A1 = A1m1f(eps); C1f(eps, Ca); if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) { A2 = A2m1f(eps); C2f(eps, Cb); m0x = A1 - A2; A2 = 1 + A2; } A1 = 1 + A1; } if (outmask & DISTANCE) { real B1 = SinCosSeries(true, ssig2, csig2, Ca, nC1_) - SinCosSeries(true, ssig1, csig1, Ca, nC1_); // Missing a factor of _b s12b = A1 * (sig12 + B1); if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) { real B2 = SinCosSeries(true, ssig2, csig2, Cb, nC2_) - SinCosSeries(true, ssig1, csig1, Cb, nC2_); J12 = m0x * sig12 + (A1 * B1 - A2 * B2); } } else if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) { // Assume here that nC1_ >= nC2_ for (int l = 1; l <= nC2_; ++l) Cb[l] = A1 * Ca[l] - A2 * Cb[l]; J12 = m0x * sig12 + (SinCosSeries(true, ssig2, csig2, Cb, nC2_) - SinCosSeries(true, ssig1, csig1, Cb, nC2_)); } if (outmask & REDUCEDLENGTH) { m0 = m0x; // Missing a factor of _b. // Add parens around (csig1 * ssig2) and (ssig1 * csig2) to ensure // accurate cancellation in the case of coincident points. m12b = dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) - csig1 * csig2 * J12; } if (outmask & GEODESICSCALE) { real csig12 = csig1 * csig2 + ssig1 * ssig2; real t = _ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2); M12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1; M21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2; } } Math::real Geodesic::Astroid(real x, real y) { // Solve k^4+2*k^3-(x^2+y^2-1)*k^2-2*y^2*k-y^2 = 0 for positive root k. // This solution is adapted from Geocentric::Reverse. real k; real p = Math::sq(x), q = Math::sq(y), r = (p + q - 1) / 6; if ( !(q == 0 && r <= 0) ) { real // Avoid possible division by zero when r = 0 by multiplying equations // for s and t by r^3 and r, resp. S = p * q / 4, // S = r^3 * s r2 = Math::sq(r), r3 = r * r2, // The discriminant of the quadratic equation for T3. This is zero on // the evolute curve p^(1/3)+q^(1/3) = 1 disc = S * (S + 2 * r3); real u = r; if (disc >= 0) { real T3 = S + r3; // Pick the sign on the sqrt to maximize abs(T3). This minimizes loss // of precision due to cancellation. The result is unchanged because // of the way the T is used in definition of u. T3 += T3 < 0 ? -sqrt(disc) : sqrt(disc); // T3 = (r * t)^3 // N.B. cbrt always returns the real root. cbrt(-8) = -2. real T = cbrt(T3); // T = r * t // T can be zero; but then r2 / T -> 0. u += T + (T != 0 ? r2 / T : 0); } else { // T is complex, but the way u is defined the result is real. real ang = atan2(sqrt(-disc), -(S + r3)); // There are three possible cube roots. We choose the root which // avoids cancellation. Note that disc < 0 implies that r < 0. u += 2 * r * cos(ang / 3); } real v = sqrt(Math::sq(u) + q), // guaranteed positive // Avoid loss of accuracy when u < 0. uv = u < 0 ? q / (v - u) : u + v, // u+v, guaranteed positive w = (uv - q) / (2 * v); // positive? // Rearrange expression for k to avoid loss of accuracy due to // subtraction. Division by 0 not possible because uv > 0, w >= 0. k = uv / (sqrt(uv + Math::sq(w)) + w); // guaranteed positive } else { // q == 0 && r <= 0 // y = 0 with |x| <= 1. Handle this case directly. // for y small, positive root is k = abs(y)/sqrt(1-x^2) k = 0; } return k; } Math::real Geodesic::InverseStart(real sbet1, real cbet1, real dn1, real sbet2, real cbet2, real dn2, real lam12, real slam12, real clam12, real& salp1, real& calp1, // Only updated if return val >= 0 real& salp2, real& calp2, // Only updated for short lines real& dnm, // Scratch area of the right size real Ca[]) const { // Return a starting point for Newton's method in salp1 and calp1 (function // value is -1). If Newton's method doesn't need to be used, return also // salp2 and calp2 and function value is sig12. real sig12 = -1, // Return value // bet12 = bet2 - bet1 in [0, pi); bet12a = bet2 + bet1 in (-pi, 0] sbet12 = sbet2 * cbet1 - cbet2 * sbet1, cbet12 = cbet2 * cbet1 + sbet2 * sbet1; real sbet12a = sbet2 * cbet1 + cbet2 * sbet1; bool shortline = cbet12 >= 0 && sbet12 < real(0.5) && cbet2 * lam12 < real(0.5); real somg12, comg12; if (shortline) { real sbetm2 = Math::sq(sbet1 + sbet2); // sin((bet1+bet2)/2)^2 // = (sbet1 + sbet2)^2 / ((sbet1 + sbet2)^2 + (cbet1 + cbet2)^2) sbetm2 /= sbetm2 + Math::sq(cbet1 + cbet2); dnm = sqrt(1 + _ep2 * sbetm2); real omg12 = lam12 / (_f1 * dnm); somg12 = sin(omg12); comg12 = cos(omg12); } else { somg12 = slam12; comg12 = clam12; } salp1 = cbet2 * somg12; calp1 = comg12 >= 0 ? sbet12 + cbet2 * sbet1 * Math::sq(somg12) / (1 + comg12) : sbet12a - cbet2 * sbet1 * Math::sq(somg12) / (1 - comg12); real ssig12 = hypot(salp1, calp1), csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12; if (shortline && ssig12 < _etol2) { // really short lines salp2 = cbet1 * somg12; calp2 = sbet12 - cbet1 * sbet2 * (comg12 >= 0 ? Math::sq(somg12) / (1 + comg12) : 1 - comg12); Math::norm(salp2, calp2); // Set return value sig12 = atan2(ssig12, csig12); } else if (fabs(_n) > real(0.1) || // Skip astroid calc if too eccentric csig12 >= 0 || ssig12 >= 6 * fabs(_n) * Math::pi() * Math::sq(cbet1)) { // Nothing to do, zeroth order spherical approximation is OK } else { // Scale lam12 and bet2 to x, y coordinate system where antipodal point // is at origin and singular point is at y = 0, x = -1. real x, y, lamscale, betscale; real lam12x = atan2(-slam12, -clam12); // lam12 - pi if (_f >= 0) { // In fact f == 0 does not get here // x = dlong, y = dlat { real k2 = Math::sq(sbet1) * _ep2, eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2); lamscale = _f * cbet1 * A3f(eps) * Math::pi(); } betscale = lamscale * cbet1; x = lam12x / lamscale; y = sbet12a / betscale; } else { // _f < 0 // x = dlat, y = dlong real cbet12a = cbet2 * cbet1 - sbet2 * sbet1, bet12a = atan2(sbet12a, cbet12a); real m12b, m0, dummy; // In the case of lon12 = 180, this repeats a calculation made in // Inverse. Lengths(_n, Math::pi() + bet12a, sbet1, -cbet1, dn1, sbet2, cbet2, dn2, cbet1, cbet2, REDUCEDLENGTH, dummy, m12b, m0, dummy, dummy, Ca); x = -1 + m12b / (cbet1 * cbet2 * m0 * Math::pi()); betscale = x < -real(0.01) ? sbet12a / x : -_f * Math::sq(cbet1) * Math::pi(); lamscale = betscale / cbet1; y = lam12x / lamscale; } if (y > -tol1_ && x > -1 - xthresh_) { // strip near cut // Need real(x) here to cast away the volatility of x for min/max if (_f >= 0) { salp1 = fmin(real(1), -x); calp1 = - sqrt(1 - Math::sq(salp1)); } else { calp1 = fmax(real(x > -tol1_ ? 0 : -1), x); salp1 = sqrt(1 - Math::sq(calp1)); } } else { // Estimate alp1, by solving the astroid problem. // // Could estimate alpha1 = theta + pi/2, directly, i.e., // calp1 = y/k; salp1 = -x/(1+k); for _f >= 0 // calp1 = x/(1+k); salp1 = -y/k; for _f < 0 (need to check) // // However, it's better to estimate omg12 from astroid and use // spherical formula to compute alp1. This reduces the mean number of // Newton iterations for astroid cases from 2.24 (min 0, max 6) to 2.12 // (min 0 max 5). The changes in the number of iterations are as // follows: // // change percent // 1 5 // 0 78 // -1 16 // -2 0.6 // -3 0.04 // -4 0.002 // // The histogram of iterations is (m = number of iterations estimating // alp1 directly, n = number of iterations estimating via omg12, total // number of trials = 148605): // // iter m n // 0 148 186 // 1 13046 13845 // 2 93315 102225 // 3 36189 32341 // 4 5396 7 // 5 455 1 // 6 56 0 // // Because omg12 is near pi, estimate work with omg12a = pi - omg12 real k = Astroid(x, y); real omg12a = lamscale * ( _f >= 0 ? -x * k/(1 + k) : -y * (1 + k)/k ); somg12 = sin(omg12a); comg12 = -cos(omg12a); // Update spherical estimate of alp1 using omg12 instead of lam12 salp1 = cbet2 * somg12; calp1 = sbet12a - cbet2 * sbet1 * Math::sq(somg12) / (1 - comg12); } } // Sanity check on starting guess. Backwards check allows NaN through. if (!(salp1 <= 0)) Math::norm(salp1, calp1); else { salp1 = 1; calp1 = 0; } return sig12; } Math::real Geodesic::Lambda12(real sbet1, real cbet1, real dn1, real sbet2, real cbet2, real dn2, real salp1, real calp1, real slam120, real clam120, real& salp2, real& calp2, real& sig12, real& ssig1, real& csig1, real& ssig2, real& csig2, real& eps, real& domg12, bool diffp, real& dlam12, // Scratch area of the right size real Ca[]) const { if (sbet1 == 0 && calp1 == 0) // Break degeneracy of equatorial line. This case has already been // handled. calp1 = -tiny_; real // sin(alp1) * cos(bet1) = sin(alp0) salp0 = salp1 * cbet1, calp0 = hypot(calp1, salp1 * sbet1); // calp0 > 0 real somg1, comg1, somg2, comg2, somg12, comg12, lam12; // tan(bet1) = tan(sig1) * cos(alp1) // tan(omg1) = sin(alp0) * tan(sig1) = tan(omg1)=tan(alp1)*sin(bet1) ssig1 = sbet1; somg1 = salp0 * sbet1; csig1 = comg1 = calp1 * cbet1; Math::norm(ssig1, csig1); // Math::norm(somg1, comg1); -- don't need to normalize! // Enforce symmetries in the case abs(bet2) = -bet1. Need to be careful // about this case, since this can yield singularities in the Newton // iteration. // sin(alp2) * cos(bet2) = sin(alp0) salp2 = cbet2 != cbet1 ? salp0 / cbet2 : salp1; // calp2 = sqrt(1 - sq(salp2)) // = sqrt(sq(calp0) - sq(sbet2)) / cbet2 // and subst for calp0 and rearrange to give (choose positive sqrt // to give alp2 in [0, pi/2]). calp2 = cbet2 != cbet1 || fabs(sbet2) != -sbet1 ? sqrt(Math::sq(calp1 * cbet1) + (cbet1 < -sbet1 ? (cbet2 - cbet1) * (cbet1 + cbet2) : (sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2 : fabs(calp1); // tan(bet2) = tan(sig2) * cos(alp2) // tan(omg2) = sin(alp0) * tan(sig2). ssig2 = sbet2; somg2 = salp0 * sbet2; csig2 = comg2 = calp2 * cbet2; Math::norm(ssig2, csig2); // Math::norm(somg2, comg2); -- don't need to normalize! // sig12 = sig2 - sig1, limit to [0, pi] sig12 = atan2(fmax(real(0), csig1 * ssig2 - ssig1 * csig2) + real(0), csig1 * csig2 + ssig1 * ssig2); // omg12 = omg2 - omg1, limit to [0, pi] somg12 = fmax(real(0), comg1 * somg2 - somg1 * comg2) + real(0); comg12 = comg1 * comg2 + somg1 * somg2; // eta = omg12 - lam120 real eta = atan2(somg12 * clam120 - comg12 * slam120, comg12 * clam120 + somg12 * slam120); real B312; real k2 = Math::sq(calp0) * _ep2; eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2); C3f(eps, Ca); B312 = (SinCosSeries(true, ssig2, csig2, Ca, nC3_-1) - SinCosSeries(true, ssig1, csig1, Ca, nC3_-1)); domg12 = -_f * A3f(eps) * salp0 * (sig12 + B312); lam12 = eta + domg12; if (diffp) { if (calp2 == 0) dlam12 = - 2 * _f1 * dn1 / sbet1; else { real dummy; Lengths(eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2, REDUCEDLENGTH, dummy, dlam12, dummy, dummy, dummy, Ca); dlam12 *= _f1 / (calp2 * cbet2); } } return lam12; } Math::real Geodesic::A3f(real eps) const { // Evaluate A3 return Math::polyval(nA3_ - 1, _aA3x, eps); } void Geodesic::C3f(real eps, real c[]) const { // Evaluate C3 coeffs // Elements c[1] thru c[nC3_ - 1] are set real mult = 1; int o = 0; for (int l = 1; l < nC3_; ++l) { // l is index of C3[l] int m = nC3_ - l - 1; // order of polynomial in eps mult *= eps; c[l] = mult * Math::polyval(m, _cC3x + o, eps); o += m + 1; } // Post condition: o == nC3x_ } void Geodesic::C4f(real eps, real c[]) const { // Evaluate C4 coeffs // Elements c[0] thru c[nC4_ - 1] are set real mult = 1; int o = 0; for (int l = 0; l < nC4_; ++l) { // l is index of C4[l] int m = nC4_ - l - 1; // order of polynomial in eps c[l] = mult * Math::polyval(m, _cC4x + o, eps); o += m + 1; mult *= eps; } // Post condition: o == nC4x_ } // The static const coefficient arrays in the following functions are // generated by Maxima and give the coefficients of the Taylor expansions for // the geodesics. The convention on the order of these coefficients is as // follows: // // ascending order in the trigonometric expansion, // then powers of eps in descending order, // finally powers of n in descending order. // // (For some expansions, only a subset of levels occur.) For each polynomial // of order n at the lowest level, the (n+1) coefficients of the polynomial // are followed by a divisor which is applied to the whole polynomial. In // this way, the coefficients are expressible with no round off error. The // sizes of the coefficient arrays are: // // A1m1f, A2m1f = floor(N/2) + 2 // C1f, C1pf, C2f, A3coeff = (N^2 + 7*N - 2*floor(N/2)) / 4 // C3coeff = (N - 1) * (N^2 + 7*N - 2*floor(N/2)) / 8 // C4coeff = N * (N + 1) * (N + 5) / 6 // // where N = GEOGRAPHICLIB_GEODESIC_ORDER // = nA1 = nA2 = nC1 = nC1p = nA3 = nC4 // The scale factor A1-1 = mean value of (d/dsigma)I1 - 1 Math::real Geodesic::A1m1f(real eps) { // Generated by Maxima on 2015-05-05 18:08:12-04:00 #if GEOGRAPHICLIB_GEODESIC_ORDER/2 == 1 static const real coeff[] = { // (1-eps)*A1-1, polynomial in eps2 of order 1 1, 0, 4, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 2 static const real coeff[] = { // (1-eps)*A1-1, polynomial in eps2 of order 2 1, 16, 0, 64, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 3 static const real coeff[] = { // (1-eps)*A1-1, polynomial in eps2 of order 3 1, 4, 64, 0, 256, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 4 static const real coeff[] = { // (1-eps)*A1-1, polynomial in eps2 of order 4 25, 64, 256, 4096, 0, 16384, }; #else #error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER" #endif static_assert(sizeof(coeff) / sizeof(real) == nA1_/2 + 2, "Coefficient array size mismatch in A1m1f"); int m = nA1_/2; real t = Math::polyval(m, coeff, Math::sq(eps)) / coeff[m + 1]; return (t + eps) / (1 - eps); } // The coefficients C1[l] in the Fourier expansion of B1 void Geodesic::C1f(real eps, real c[]) { // Generated by Maxima on 2015-05-05 18:08:12-04:00 #if GEOGRAPHICLIB_GEODESIC_ORDER == 3 static const real coeff[] = { // C1[1]/eps^1, polynomial in eps2 of order 1 3, -8, 16, // C1[2]/eps^2, polynomial in eps2 of order 0 -1, 16, // C1[3]/eps^3, polynomial in eps2 of order 0 -1, 48, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 4 static const real coeff[] = { // C1[1]/eps^1, polynomial in eps2 of order 1 3, -8, 16, // C1[2]/eps^2, polynomial in eps2 of order 1 1, -2, 32, // C1[3]/eps^3, polynomial in eps2 of order 0 -1, 48, // C1[4]/eps^4, polynomial in eps2 of order 0 -5, 512, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 5 static const real coeff[] = { // C1[1]/eps^1, polynomial in eps2 of order 2 -1, 6, -16, 32, // C1[2]/eps^2, polynomial in eps2 of order 1 1, -2, 32, // C1[3]/eps^3, polynomial in eps2 of order 1 9, -16, 768, // C1[4]/eps^4, polynomial in eps2 of order 0 -5, 512, // C1[5]/eps^5, polynomial in eps2 of order 0 -7, 1280, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 6 static const real coeff[] = { // C1[1]/eps^1, polynomial in eps2 of order 2 -1, 6, -16, 32, // C1[2]/eps^2, polynomial in eps2 of order 2 -9, 64, -128, 2048, // C1[3]/eps^3, polynomial in eps2 of order 1 9, -16, 768, // C1[4]/eps^4, polynomial in eps2 of order 1 3, -5, 512, // C1[5]/eps^5, polynomial in eps2 of order 0 -7, 1280, // C1[6]/eps^6, polynomial in eps2 of order 0 -7, 2048, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 7 static const real coeff[] = { // C1[1]/eps^1, polynomial in eps2 of order 3 19, -64, 384, -1024, 2048, // C1[2]/eps^2, polynomial in eps2 of order 2 -9, 64, -128, 2048, // C1[3]/eps^3, polynomial in eps2 of order 2 -9, 72, -128, 6144, // C1[4]/eps^4, polynomial in eps2 of order 1 3, -5, 512, // C1[5]/eps^5, polynomial in eps2 of order 1 35, -56, 10240, // C1[6]/eps^6, polynomial in eps2 of order 0 -7, 2048, // C1[7]/eps^7, polynomial in eps2 of order 0 -33, 14336, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 8 static const real coeff[] = { // C1[1]/eps^1, polynomial in eps2 of order 3 19, -64, 384, -1024, 2048, // C1[2]/eps^2, polynomial in eps2 of order 3 7, -18, 128, -256, 4096, // C1[3]/eps^3, polynomial in eps2 of order 2 -9, 72, -128, 6144, // C1[4]/eps^4, polynomial in eps2 of order 2 -11, 96, -160, 16384, // C1[5]/eps^5, polynomial in eps2 of order 1 35, -56, 10240, // C1[6]/eps^6, polynomial in eps2 of order 1 9, -14, 4096, // C1[7]/eps^7, polynomial in eps2 of order 0 -33, 14336, // C1[8]/eps^8, polynomial in eps2 of order 0 -429, 262144, }; #else #error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER" #endif static_assert(sizeof(coeff) / sizeof(real) == (nC1_*nC1_ + 7*nC1_ - 2*(nC1_/2)) / 4, "Coefficient array size mismatch in C1f"); real eps2 = Math::sq(eps), d = eps; int o = 0; for (int l = 1; l <= nC1_; ++l) { // l is index of C1p[l] int m = (nC1_ - l) / 2; // order of polynomial in eps^2 c[l] = d * Math::polyval(m, coeff + o, eps2) / coeff[o + m + 1]; o += m + 2; d *= eps; } // Post condition: o == sizeof(coeff) / sizeof(real) } // The coefficients C1p[l] in the Fourier expansion of B1p void Geodesic::C1pf(real eps, real c[]) { // Generated by Maxima on 2015-05-05 18:08:12-04:00 #if GEOGRAPHICLIB_GEODESIC_ORDER == 3 static const real coeff[] = { // C1p[1]/eps^1, polynomial in eps2 of order 1 -9, 16, 32, // C1p[2]/eps^2, polynomial in eps2 of order 0 5, 16, // C1p[3]/eps^3, polynomial in eps2 of order 0 29, 96, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 4 static const real coeff[] = { // C1p[1]/eps^1, polynomial in eps2 of order 1 -9, 16, 32, // C1p[2]/eps^2, polynomial in eps2 of order 1 -37, 30, 96, // C1p[3]/eps^3, polynomial in eps2 of order 0 29, 96, // C1p[4]/eps^4, polynomial in eps2 of order 0 539, 1536, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 5 static const real coeff[] = { // C1p[1]/eps^1, polynomial in eps2 of order 2 205, -432, 768, 1536, // C1p[2]/eps^2, polynomial in eps2 of order 1 -37, 30, 96, // C1p[3]/eps^3, polynomial in eps2 of order 1 -225, 116, 384, // C1p[4]/eps^4, polynomial in eps2 of order 0 539, 1536, // C1p[5]/eps^5, polynomial in eps2 of order 0 3467, 7680, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 6 static const real coeff[] = { // C1p[1]/eps^1, polynomial in eps2 of order 2 205, -432, 768, 1536, // C1p[2]/eps^2, polynomial in eps2 of order 2 4005, -4736, 3840, 12288, // C1p[3]/eps^3, polynomial in eps2 of order 1 -225, 116, 384, // C1p[4]/eps^4, polynomial in eps2 of order 1 -7173, 2695, 7680, // C1p[5]/eps^5, polynomial in eps2 of order 0 3467, 7680, // C1p[6]/eps^6, polynomial in eps2 of order 0 38081, 61440, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 7 static const real coeff[] = { // C1p[1]/eps^1, polynomial in eps2 of order 3 -4879, 9840, -20736, 36864, 73728, // C1p[2]/eps^2, polynomial in eps2 of order 2 4005, -4736, 3840, 12288, // C1p[3]/eps^3, polynomial in eps2 of order 2 8703, -7200, 3712, 12288, // C1p[4]/eps^4, polynomial in eps2 of order 1 -7173, 2695, 7680, // C1p[5]/eps^5, polynomial in eps2 of order 1 -141115, 41604, 92160, // C1p[6]/eps^6, polynomial in eps2 of order 0 38081, 61440, // C1p[7]/eps^7, polynomial in eps2 of order 0 459485, 516096, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 8 static const real coeff[] = { // C1p[1]/eps^1, polynomial in eps2 of order 3 -4879, 9840, -20736, 36864, 73728, // C1p[2]/eps^2, polynomial in eps2 of order 3 -86171, 120150, -142080, 115200, 368640, // C1p[3]/eps^3, polynomial in eps2 of order 2 8703, -7200, 3712, 12288, // C1p[4]/eps^4, polynomial in eps2 of order 2 1082857, -688608, 258720, 737280, // C1p[5]/eps^5, polynomial in eps2 of order 1 -141115, 41604, 92160, // C1p[6]/eps^6, polynomial in eps2 of order 1 -2200311, 533134, 860160, // C1p[7]/eps^7, polynomial in eps2 of order 0 459485, 516096, // C1p[8]/eps^8, polynomial in eps2 of order 0 109167851, 82575360, }; #else #error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER" #endif static_assert(sizeof(coeff) / sizeof(real) == (nC1p_*nC1p_ + 7*nC1p_ - 2*(nC1p_/2)) / 4, "Coefficient array size mismatch in C1pf"); real eps2 = Math::sq(eps), d = eps; int o = 0; for (int l = 1; l <= nC1p_; ++l) { // l is index of C1p[l] int m = (nC1p_ - l) / 2; // order of polynomial in eps^2 c[l] = d * Math::polyval(m, coeff + o, eps2) / coeff[o + m + 1]; o += m + 2; d *= eps; } // Post condition: o == sizeof(coeff) / sizeof(real) } // The scale factor A2-1 = mean value of (d/dsigma)I2 - 1 Math::real Geodesic::A2m1f(real eps) { // Generated by Maxima on 2015-05-29 08:09:47-04:00 #if GEOGRAPHICLIB_GEODESIC_ORDER/2 == 1 static const real coeff[] = { // (eps+1)*A2-1, polynomial in eps2 of order 1 -3, 0, 4, }; // count = 3 #elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 2 static const real coeff[] = { // (eps+1)*A2-1, polynomial in eps2 of order 2 -7, -48, 0, 64, }; // count = 4 #elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 3 static const real coeff[] = { // (eps+1)*A2-1, polynomial in eps2 of order 3 -11, -28, -192, 0, 256, }; // count = 5 #elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 4 static const real coeff[] = { // (eps+1)*A2-1, polynomial in eps2 of order 4 -375, -704, -1792, -12288, 0, 16384, }; // count = 6 #else #error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER" #endif static_assert(sizeof(coeff) / sizeof(real) == nA2_/2 + 2, "Coefficient array size mismatch in A2m1f"); int m = nA2_/2; real t = Math::polyval(m, coeff, Math::sq(eps)) / coeff[m + 1]; return (t - eps) / (1 + eps); } // The coefficients C2[l] in the Fourier expansion of B2 void Geodesic::C2f(real eps, real c[]) { // Generated by Maxima on 2015-05-05 18:08:12-04:00 #if GEOGRAPHICLIB_GEODESIC_ORDER == 3 static const real coeff[] = { // C2[1]/eps^1, polynomial in eps2 of order 1 1, 8, 16, // C2[2]/eps^2, polynomial in eps2 of order 0 3, 16, // C2[3]/eps^3, polynomial in eps2 of order 0 5, 48, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 4 static const real coeff[] = { // C2[1]/eps^1, polynomial in eps2 of order 1 1, 8, 16, // C2[2]/eps^2, polynomial in eps2 of order 1 1, 6, 32, // C2[3]/eps^3, polynomial in eps2 of order 0 5, 48, // C2[4]/eps^4, polynomial in eps2 of order 0 35, 512, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 5 static const real coeff[] = { // C2[1]/eps^1, polynomial in eps2 of order 2 1, 2, 16, 32, // C2[2]/eps^2, polynomial in eps2 of order 1 1, 6, 32, // C2[3]/eps^3, polynomial in eps2 of order 1 15, 80, 768, // C2[4]/eps^4, polynomial in eps2 of order 0 35, 512, // C2[5]/eps^5, polynomial in eps2 of order 0 63, 1280, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 6 static const real coeff[] = { // C2[1]/eps^1, polynomial in eps2 of order 2 1, 2, 16, 32, // C2[2]/eps^2, polynomial in eps2 of order 2 35, 64, 384, 2048, // C2[3]/eps^3, polynomial in eps2 of order 1 15, 80, 768, // C2[4]/eps^4, polynomial in eps2 of order 1 7, 35, 512, // C2[5]/eps^5, polynomial in eps2 of order 0 63, 1280, // C2[6]/eps^6, polynomial in eps2 of order 0 77, 2048, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 7 static const real coeff[] = { // C2[1]/eps^1, polynomial in eps2 of order 3 41, 64, 128, 1024, 2048, // C2[2]/eps^2, polynomial in eps2 of order 2 35, 64, 384, 2048, // C2[3]/eps^3, polynomial in eps2 of order 2 69, 120, 640, 6144, // C2[4]/eps^4, polynomial in eps2 of order 1 7, 35, 512, // C2[5]/eps^5, polynomial in eps2 of order 1 105, 504, 10240, // C2[6]/eps^6, polynomial in eps2 of order 0 77, 2048, // C2[7]/eps^7, polynomial in eps2 of order 0 429, 14336, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 8 static const real coeff[] = { // C2[1]/eps^1, polynomial in eps2 of order 3 41, 64, 128, 1024, 2048, // C2[2]/eps^2, polynomial in eps2 of order 3 47, 70, 128, 768, 4096, // C2[3]/eps^3, polynomial in eps2 of order 2 69, 120, 640, 6144, // C2[4]/eps^4, polynomial in eps2 of order 2 133, 224, 1120, 16384, // C2[5]/eps^5, polynomial in eps2 of order 1 105, 504, 10240, // C2[6]/eps^6, polynomial in eps2 of order 1 33, 154, 4096, // C2[7]/eps^7, polynomial in eps2 of order 0 429, 14336, // C2[8]/eps^8, polynomial in eps2 of order 0 6435, 262144, }; #else #error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER" #endif static_assert(sizeof(coeff) / sizeof(real) == (nC2_*nC2_ + 7*nC2_ - 2*(nC2_/2)) / 4, "Coefficient array size mismatch in C2f"); real eps2 = Math::sq(eps), d = eps; int o = 0; for (int l = 1; l <= nC2_; ++l) { // l is index of C2[l] int m = (nC2_ - l) / 2; // order of polynomial in eps^2 c[l] = d * Math::polyval(m, coeff + o, eps2) / coeff[o + m + 1]; o += m + 2; d *= eps; } // Post condition: o == sizeof(coeff) / sizeof(real) } // The scale factor A3 = mean value of (d/dsigma)I3 void Geodesic::A3coeff() { // Generated by Maxima on 2015-05-05 18:08:13-04:00 #if GEOGRAPHICLIB_GEODESIC_ORDER == 3 static const real coeff[] = { // A3, coeff of eps^2, polynomial in n of order 0 -1, 4, // A3, coeff of eps^1, polynomial in n of order 1 1, -1, 2, // A3, coeff of eps^0, polynomial in n of order 0 1, 1, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 4 static const real coeff[] = { // A3, coeff of eps^3, polynomial in n of order 0 -1, 16, // A3, coeff of eps^2, polynomial in n of order 1 -1, -2, 8, // A3, coeff of eps^1, polynomial in n of order 1 1, -1, 2, // A3, coeff of eps^0, polynomial in n of order 0 1, 1, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 5 static const real coeff[] = { // A3, coeff of eps^4, polynomial in n of order 0 -3, 64, // A3, coeff of eps^3, polynomial in n of order 1 -3, -1, 16, // A3, coeff of eps^2, polynomial in n of order 2 3, -1, -2, 8, // A3, coeff of eps^1, polynomial in n of order 1 1, -1, 2, // A3, coeff of eps^0, polynomial in n of order 0 1, 1, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 6 static const real coeff[] = { // A3, coeff of eps^5, polynomial in n of order 0 -3, 128, // A3, coeff of eps^4, polynomial in n of order 1 -2, -3, 64, // A3, coeff of eps^3, polynomial in n of order 2 -1, -3, -1, 16, // A3, coeff of eps^2, polynomial in n of order 2 3, -1, -2, 8, // A3, coeff of eps^1, polynomial in n of order 1 1, -1, 2, // A3, coeff of eps^0, polynomial in n of order 0 1, 1, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 7 static const real coeff[] = { // A3, coeff of eps^6, polynomial in n of order 0 -5, 256, // A3, coeff of eps^5, polynomial in n of order 1 -5, -3, 128, // A3, coeff of eps^4, polynomial in n of order 2 -10, -2, -3, 64, // A3, coeff of eps^3, polynomial in n of order 3 5, -1, -3, -1, 16, // A3, coeff of eps^2, polynomial in n of order 2 3, -1, -2, 8, // A3, coeff of eps^1, polynomial in n of order 1 1, -1, 2, // A3, coeff of eps^0, polynomial in n of order 0 1, 1, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 8 static const real coeff[] = { // A3, coeff of eps^7, polynomial in n of order 0 -25, 2048, // A3, coeff of eps^6, polynomial in n of order 1 -15, -20, 1024, // A3, coeff of eps^5, polynomial in n of order 2 -5, -10, -6, 256, // A3, coeff of eps^4, polynomial in n of order 3 -5, -20, -4, -6, 128, // A3, coeff of eps^3, polynomial in n of order 3 5, -1, -3, -1, 16, // A3, coeff of eps^2, polynomial in n of order 2 3, -1, -2, 8, // A3, coeff of eps^1, polynomial in n of order 1 1, -1, 2, // A3, coeff of eps^0, polynomial in n of order 0 1, 1, }; #else #error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER" #endif static_assert(sizeof(coeff) / sizeof(real) == (nA3_*nA3_ + 7*nA3_ - 2*(nA3_/2)) / 4, "Coefficient array size mismatch in A3f"); int o = 0, k = 0; for (int j = nA3_ - 1; j >= 0; --j) { // coeff of eps^j int m = min(nA3_ - j - 1, j); // order of polynomial in n _aA3x[k++] = Math::polyval(m, coeff + o, _n) / coeff[o + m + 1]; o += m + 2; } // Post condition: o == sizeof(coeff) / sizeof(real) && k == nA3x_ } // The coefficients C3[l] in the Fourier expansion of B3 void Geodesic::C3coeff() { // Generated by Maxima on 2015-05-05 18:08:13-04:00 #if GEOGRAPHICLIB_GEODESIC_ORDER == 3 static const real coeff[] = { // C3[1], coeff of eps^2, polynomial in n of order 0 1, 8, // C3[1], coeff of eps^1, polynomial in n of order 1 -1, 1, 4, // C3[2], coeff of eps^2, polynomial in n of order 0 1, 16, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 4 static const real coeff[] = { // C3[1], coeff of eps^3, polynomial in n of order 0 3, 64, // C3[1], coeff of eps^2, polynomial in n of order 1 // This is a case where a leading 0 term has been inserted to maintain the // pattern in the orders of the polynomials. 0, 1, 8, // C3[1], coeff of eps^1, polynomial in n of order 1 -1, 1, 4, // C3[2], coeff of eps^3, polynomial in n of order 0 3, 64, // C3[2], coeff of eps^2, polynomial in n of order 1 -3, 2, 32, // C3[3], coeff of eps^3, polynomial in n of order 0 5, 192, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 5 static const real coeff[] = { // C3[1], coeff of eps^4, polynomial in n of order 0 5, 128, // C3[1], coeff of eps^3, polynomial in n of order 1 3, 3, 64, // C3[1], coeff of eps^2, polynomial in n of order 2 -1, 0, 1, 8, // C3[1], coeff of eps^1, polynomial in n of order 1 -1, 1, 4, // C3[2], coeff of eps^4, polynomial in n of order 0 3, 128, // C3[2], coeff of eps^3, polynomial in n of order 1 -2, 3, 64, // C3[2], coeff of eps^2, polynomial in n of order 2 1, -3, 2, 32, // C3[3], coeff of eps^4, polynomial in n of order 0 3, 128, // C3[3], coeff of eps^3, polynomial in n of order 1 -9, 5, 192, // C3[4], coeff of eps^4, polynomial in n of order 0 7, 512, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 6 static const real coeff[] = { // C3[1], coeff of eps^5, polynomial in n of order 0 3, 128, // C3[1], coeff of eps^4, polynomial in n of order 1 2, 5, 128, // C3[1], coeff of eps^3, polynomial in n of order 2 -1, 3, 3, 64, // C3[1], coeff of eps^2, polynomial in n of order 2 -1, 0, 1, 8, // C3[1], coeff of eps^1, polynomial in n of order 1 -1, 1, 4, // C3[2], coeff of eps^5, polynomial in n of order 0 5, 256, // C3[2], coeff of eps^4, polynomial in n of order 1 1, 3, 128, // C3[2], coeff of eps^3, polynomial in n of order 2 -3, -2, 3, 64, // C3[2], coeff of eps^2, polynomial in n of order 2 1, -3, 2, 32, // C3[3], coeff of eps^5, polynomial in n of order 0 7, 512, // C3[3], coeff of eps^4, polynomial in n of order 1 -10, 9, 384, // C3[3], coeff of eps^3, polynomial in n of order 2 5, -9, 5, 192, // C3[4], coeff of eps^5, polynomial in n of order 0 7, 512, // C3[4], coeff of eps^4, polynomial in n of order 1 -14, 7, 512, // C3[5], coeff of eps^5, polynomial in n of order 0 21, 2560, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 7 static const real coeff[] = { // C3[1], coeff of eps^6, polynomial in n of order 0 21, 1024, // C3[1], coeff of eps^5, polynomial in n of order 1 11, 12, 512, // C3[1], coeff of eps^4, polynomial in n of order 2 2, 2, 5, 128, // C3[1], coeff of eps^3, polynomial in n of order 3 -5, -1, 3, 3, 64, // C3[1], coeff of eps^2, polynomial in n of order 2 -1, 0, 1, 8, // C3[1], coeff of eps^1, polynomial in n of order 1 -1, 1, 4, // C3[2], coeff of eps^6, polynomial in n of order 0 27, 2048, // C3[2], coeff of eps^5, polynomial in n of order 1 1, 5, 256, // C3[2], coeff of eps^4, polynomial in n of order 2 -9, 2, 6, 256, // C3[2], coeff of eps^3, polynomial in n of order 3 2, -3, -2, 3, 64, // C3[2], coeff of eps^2, polynomial in n of order 2 1, -3, 2, 32, // C3[3], coeff of eps^6, polynomial in n of order 0 3, 256, // C3[3], coeff of eps^5, polynomial in n of order 1 -4, 21, 1536, // C3[3], coeff of eps^4, polynomial in n of order 2 -6, -10, 9, 384, // C3[3], coeff of eps^3, polynomial in n of order 3 -1, 5, -9, 5, 192, // C3[4], coeff of eps^6, polynomial in n of order 0 9, 1024, // C3[4], coeff of eps^5, polynomial in n of order 1 -10, 7, 512, // C3[4], coeff of eps^4, polynomial in n of order 2 10, -14, 7, 512, // C3[5], coeff of eps^6, polynomial in n of order 0 9, 1024, // C3[5], coeff of eps^5, polynomial in n of order 1 -45, 21, 2560, // C3[6], coeff of eps^6, polynomial in n of order 0 11, 2048, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 8 static const real coeff[] = { // C3[1], coeff of eps^7, polynomial in n of order 0 243, 16384, // C3[1], coeff of eps^6, polynomial in n of order 1 10, 21, 1024, // C3[1], coeff of eps^5, polynomial in n of order 2 3, 11, 12, 512, // C3[1], coeff of eps^4, polynomial in n of order 3 -2, 2, 2, 5, 128, // C3[1], coeff of eps^3, polynomial in n of order 3 -5, -1, 3, 3, 64, // C3[1], coeff of eps^2, polynomial in n of order 2 -1, 0, 1, 8, // C3[1], coeff of eps^1, polynomial in n of order 1 -1, 1, 4, // C3[2], coeff of eps^7, polynomial in n of order 0 187, 16384, // C3[2], coeff of eps^6, polynomial in n of order 1 69, 108, 8192, // C3[2], coeff of eps^5, polynomial in n of order 2 -2, 1, 5, 256, // C3[2], coeff of eps^4, polynomial in n of order 3 -6, -9, 2, 6, 256, // C3[2], coeff of eps^3, polynomial in n of order 3 2, -3, -2, 3, 64, // C3[2], coeff of eps^2, polynomial in n of order 2 1, -3, 2, 32, // C3[3], coeff of eps^7, polynomial in n of order 0 139, 16384, // C3[3], coeff of eps^6, polynomial in n of order 1 -1, 12, 1024, // C3[3], coeff of eps^5, polynomial in n of order 2 -77, -8, 42, 3072, // C3[3], coeff of eps^4, polynomial in n of order 3 10, -6, -10, 9, 384, // C3[3], coeff of eps^3, polynomial in n of order 3 -1, 5, -9, 5, 192, // C3[4], coeff of eps^7, polynomial in n of order 0 127, 16384, // C3[4], coeff of eps^6, polynomial in n of order 1 -43, 72, 8192, // C3[4], coeff of eps^5, polynomial in n of order 2 -7, -40, 28, 2048, // C3[4], coeff of eps^4, polynomial in n of order 3 -7, 20, -28, 14, 1024, // C3[5], coeff of eps^7, polynomial in n of order 0 99, 16384, // C3[5], coeff of eps^6, polynomial in n of order 1 -15, 9, 1024, // C3[5], coeff of eps^5, polynomial in n of order 2 75, -90, 42, 5120, // C3[6], coeff of eps^7, polynomial in n of order 0 99, 16384, // C3[6], coeff of eps^6, polynomial in n of order 1 -99, 44, 8192, // C3[7], coeff of eps^7, polynomial in n of order 0 429, 114688, }; #else #error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER" #endif static_assert(sizeof(coeff) / sizeof(real) == ((nC3_-1)*(nC3_*nC3_ + 7*nC3_ - 2*(nC3_/2)))/8, "Coefficient array size mismatch in C3coeff"); int o = 0, k = 0; for (int l = 1; l < nC3_; ++l) { // l is index of C3[l] for (int j = nC3_ - 1; j >= l; --j) { // coeff of eps^j int m = min(nC3_ - j - 1, j); // order of polynomial in n _cC3x[k++] = Math::polyval(m, coeff + o, _n) / coeff[o + m + 1]; o += m + 2; } } // Post condition: o == sizeof(coeff) / sizeof(real) && k == nC3x_ } void Geodesic::C4coeff() { // Generated by Maxima on 2015-05-05 18:08:13-04:00 #if GEOGRAPHICLIB_GEODESIC_ORDER == 3 static const real coeff[] = { // C4[0], coeff of eps^2, polynomial in n of order 0 -2, 105, // C4[0], coeff of eps^1, polynomial in n of order 1 16, -7, 35, // C4[0], coeff of eps^0, polynomial in n of order 2 8, -28, 70, 105, // C4[1], coeff of eps^2, polynomial in n of order 0 -2, 105, // C4[1], coeff of eps^1, polynomial in n of order 1 -16, 7, 315, // C4[2], coeff of eps^2, polynomial in n of order 0 4, 525, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 4 static const real coeff[] = { // C4[0], coeff of eps^3, polynomial in n of order 0 11, 315, // C4[0], coeff of eps^2, polynomial in n of order 1 -32, -6, 315, // C4[0], coeff of eps^1, polynomial in n of order 2 -32, 48, -21, 105, // C4[0], coeff of eps^0, polynomial in n of order 3 4, 24, -84, 210, 315, // C4[1], coeff of eps^3, polynomial in n of order 0 -1, 105, // C4[1], coeff of eps^2, polynomial in n of order 1 64, -18, 945, // C4[1], coeff of eps^1, polynomial in n of order 2 32, -48, 21, 945, // C4[2], coeff of eps^3, polynomial in n of order 0 -8, 1575, // C4[2], coeff of eps^2, polynomial in n of order 1 -32, 12, 1575, // C4[3], coeff of eps^3, polynomial in n of order 0 8, 2205, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 5 static const real coeff[] = { // C4[0], coeff of eps^4, polynomial in n of order 0 4, 1155, // C4[0], coeff of eps^3, polynomial in n of order 1 -368, 121, 3465, // C4[0], coeff of eps^2, polynomial in n of order 2 1088, -352, -66, 3465, // C4[0], coeff of eps^1, polynomial in n of order 3 48, -352, 528, -231, 1155, // C4[0], coeff of eps^0, polynomial in n of order 4 16, 44, 264, -924, 2310, 3465, // C4[1], coeff of eps^4, polynomial in n of order 0 4, 1155, // C4[1], coeff of eps^3, polynomial in n of order 1 80, -99, 10395, // C4[1], coeff of eps^2, polynomial in n of order 2 -896, 704, -198, 10395, // C4[1], coeff of eps^1, polynomial in n of order 3 -48, 352, -528, 231, 10395, // C4[2], coeff of eps^4, polynomial in n of order 0 -8, 1925, // C4[2], coeff of eps^3, polynomial in n of order 1 384, -88, 17325, // C4[2], coeff of eps^2, polynomial in n of order 2 320, -352, 132, 17325, // C4[3], coeff of eps^4, polynomial in n of order 0 -16, 8085, // C4[3], coeff of eps^3, polynomial in n of order 1 -256, 88, 24255, // C4[4], coeff of eps^4, polynomial in n of order 0 64, 31185, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 6 static const real coeff[] = { // C4[0], coeff of eps^5, polynomial in n of order 0 97, 15015, // C4[0], coeff of eps^4, polynomial in n of order 1 1088, 156, 45045, // C4[0], coeff of eps^3, polynomial in n of order 2 -224, -4784, 1573, 45045, // C4[0], coeff of eps^2, polynomial in n of order 3 -10656, 14144, -4576, -858, 45045, // C4[0], coeff of eps^1, polynomial in n of order 4 64, 624, -4576, 6864, -3003, 15015, // C4[0], coeff of eps^0, polynomial in n of order 5 100, 208, 572, 3432, -12012, 30030, 45045, // C4[1], coeff of eps^5, polynomial in n of order 0 1, 9009, // C4[1], coeff of eps^4, polynomial in n of order 1 -2944, 468, 135135, // C4[1], coeff of eps^3, polynomial in n of order 2 5792, 1040, -1287, 135135, // C4[1], coeff of eps^2, polynomial in n of order 3 5952, -11648, 9152, -2574, 135135, // C4[1], coeff of eps^1, polynomial in n of order 4 -64, -624, 4576, -6864, 3003, 135135, // C4[2], coeff of eps^5, polynomial in n of order 0 8, 10725, // C4[2], coeff of eps^4, polynomial in n of order 1 1856, -936, 225225, // C4[2], coeff of eps^3, polynomial in n of order 2 -8448, 4992, -1144, 225225, // C4[2], coeff of eps^2, polynomial in n of order 3 -1440, 4160, -4576, 1716, 225225, // C4[3], coeff of eps^5, polynomial in n of order 0 -136, 63063, // C4[3], coeff of eps^4, polynomial in n of order 1 1024, -208, 105105, // C4[3], coeff of eps^3, polynomial in n of order 2 3584, -3328, 1144, 315315, // C4[4], coeff of eps^5, polynomial in n of order 0 -128, 135135, // C4[4], coeff of eps^4, polynomial in n of order 1 -2560, 832, 405405, // C4[5], coeff of eps^5, polynomial in n of order 0 128, 99099, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 7 static const real coeff[] = { // C4[0], coeff of eps^6, polynomial in n of order 0 10, 9009, // C4[0], coeff of eps^5, polynomial in n of order 1 -464, 291, 45045, // C4[0], coeff of eps^4, polynomial in n of order 2 -4480, 1088, 156, 45045, // C4[0], coeff of eps^3, polynomial in n of order 3 10736, -224, -4784, 1573, 45045, // C4[0], coeff of eps^2, polynomial in n of order 4 1664, -10656, 14144, -4576, -858, 45045, // C4[0], coeff of eps^1, polynomial in n of order 5 16, 64, 624, -4576, 6864, -3003, 15015, // C4[0], coeff of eps^0, polynomial in n of order 6 56, 100, 208, 572, 3432, -12012, 30030, 45045, // C4[1], coeff of eps^6, polynomial in n of order 0 10, 9009, // C4[1], coeff of eps^5, polynomial in n of order 1 112, 15, 135135, // C4[1], coeff of eps^4, polynomial in n of order 2 3840, -2944, 468, 135135, // C4[1], coeff of eps^3, polynomial in n of order 3 -10704, 5792, 1040, -1287, 135135, // C4[1], coeff of eps^2, polynomial in n of order 4 -768, 5952, -11648, 9152, -2574, 135135, // C4[1], coeff of eps^1, polynomial in n of order 5 -16, -64, -624, 4576, -6864, 3003, 135135, // C4[2], coeff of eps^6, polynomial in n of order 0 -4, 25025, // C4[2], coeff of eps^5, polynomial in n of order 1 -1664, 168, 225225, // C4[2], coeff of eps^4, polynomial in n of order 2 1664, 1856, -936, 225225, // C4[2], coeff of eps^3, polynomial in n of order 3 6784, -8448, 4992, -1144, 225225, // C4[2], coeff of eps^2, polynomial in n of order 4 128, -1440, 4160, -4576, 1716, 225225, // C4[3], coeff of eps^6, polynomial in n of order 0 64, 315315, // C4[3], coeff of eps^5, polynomial in n of order 1 1792, -680, 315315, // C4[3], coeff of eps^4, polynomial in n of order 2 -2048, 1024, -208, 105105, // C4[3], coeff of eps^3, polynomial in n of order 3 -1792, 3584, -3328, 1144, 315315, // C4[4], coeff of eps^6, polynomial in n of order 0 -512, 405405, // C4[4], coeff of eps^5, polynomial in n of order 1 2048, -384, 405405, // C4[4], coeff of eps^4, polynomial in n of order 2 3072, -2560, 832, 405405, // C4[5], coeff of eps^6, polynomial in n of order 0 -256, 495495, // C4[5], coeff of eps^5, polynomial in n of order 1 -2048, 640, 495495, // C4[6], coeff of eps^6, polynomial in n of order 0 512, 585585, }; #elif GEOGRAPHICLIB_GEODESIC_ORDER == 8 static const real coeff[] = { // C4[0], coeff of eps^7, polynomial in n of order 0 193, 85085, // C4[0], coeff of eps^6, polynomial in n of order 1 4192, 850, 765765, // C4[0], coeff of eps^5, polynomial in n of order 2 20960, -7888, 4947, 765765, // C4[0], coeff of eps^4, polynomial in n of order 3 12480, -76160, 18496, 2652, 765765, // C4[0], coeff of eps^3, polynomial in n of order 4 -154048, 182512, -3808, -81328, 26741, 765765, // C4[0], coeff of eps^2, polynomial in n of order 5 3232, 28288, -181152, 240448, -77792, -14586, 765765, // C4[0], coeff of eps^1, polynomial in n of order 6 96, 272, 1088, 10608, -77792, 116688, -51051, 255255, // C4[0], coeff of eps^0, polynomial in n of order 7 588, 952, 1700, 3536, 9724, 58344, -204204, 510510, 765765, // C4[1], coeff of eps^7, polynomial in n of order 0 349, 2297295, // C4[1], coeff of eps^6, polynomial in n of order 1 -1472, 510, 459459, // C4[1], coeff of eps^5, polynomial in n of order 2 -39840, 1904, 255, 2297295, // C4[1], coeff of eps^4, polynomial in n of order 3 52608, 65280, -50048, 7956, 2297295, // C4[1], coeff of eps^3, polynomial in n of order 4 103744, -181968, 98464, 17680, -21879, 2297295, // C4[1], coeff of eps^2, polynomial in n of order 5 -1344, -13056, 101184, -198016, 155584, -43758, 2297295, // C4[1], coeff of eps^1, polynomial in n of order 6 -96, -272, -1088, -10608, 77792, -116688, 51051, 2297295, // C4[2], coeff of eps^7, polynomial in n of order 0 464, 1276275, // C4[2], coeff of eps^6, polynomial in n of order 1 -928, -612, 3828825, // C4[2], coeff of eps^5, polynomial in n of order 2 64256, -28288, 2856, 3828825, // C4[2], coeff of eps^4, polynomial in n of order 3 -126528, 28288, 31552, -15912, 3828825, // C4[2], coeff of eps^3, polynomial in n of order 4 -41472, 115328, -143616, 84864, -19448, 3828825, // C4[2], coeff of eps^2, polynomial in n of order 5 160, 2176, -24480, 70720, -77792, 29172, 3828825, // C4[3], coeff of eps^7, polynomial in n of order 0 -16, 97461, // C4[3], coeff of eps^6, polynomial in n of order 1 -16384, 1088, 5360355, // C4[3], coeff of eps^5, polynomial in n of order 2 -2560, 30464, -11560, 5360355, // C4[3], coeff of eps^4, polynomial in n of order 3 35840, -34816, 17408, -3536, 1786785, // C4[3], coeff of eps^3, polynomial in n of order 4 7168, -30464, 60928, -56576, 19448, 5360355, // C4[4], coeff of eps^7, polynomial in n of order 0 128, 2297295, // C4[4], coeff of eps^6, polynomial in n of order 1 26624, -8704, 6891885, // C4[4], coeff of eps^5, polynomial in n of order 2 -77824, 34816, -6528, 6891885, // C4[4], coeff of eps^4, polynomial in n of order 3 -32256, 52224, -43520, 14144, 6891885, // C4[5], coeff of eps^7, polynomial in n of order 0 -6784, 8423415, // C4[5], coeff of eps^6, polynomial in n of order 1 24576, -4352, 8423415, // C4[5], coeff of eps^5, polynomial in n of order 2 45056, -34816, 10880, 8423415, // C4[6], coeff of eps^7, polynomial in n of order 0 -1024, 3318315, // C4[6], coeff of eps^6, polynomial in n of order 1 -28672, 8704, 9954945, // C4[7], coeff of eps^7, polynomial in n of order 0 1024, 1640925, }; #else #error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER" #endif static_assert(sizeof(coeff) / sizeof(real) == (nC4_ * (nC4_ + 1) * (nC4_ + 5)) / 6, "Coefficient array size mismatch in C4coeff"); int o = 0, k = 0; for (int l = 0; l < nC4_; ++l) { // l is index of C4[l] for (int j = nC4_ - 1; j >= l; --j) { // coeff of eps^j int m = nC4_ - j - 1; // order of polynomial in n _cC4x[k++] = Math::polyval(m, coeff + o, _n) / coeff[o + m + 1]; o += m + 2; } } // Post condition: o == sizeof(coeff) / sizeof(real) && k == nC4x_ } } // namespace GeographicLib geosphere/src/EllipticFunction.h0000644000176200001440000006570614323377037016475 0ustar liggesusers/** * \file EllipticFunction.hpp * \brief Header for GeographicLib::EllipticFunction class * * Copyright (c) Charles Karney (2008-2022) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ **********************************************************************/ #if !defined(GEOGRAPHICLIB_ELLIPTICFUNCTION_HPP) #define GEOGRAPHICLIB_ELLIPTICFUNCTION_HPP 1 #include "Constants.h" namespace GeographicLib { /** * \brief Elliptic integrals and functions * * This provides the elliptic functions and integrals needed for Ellipsoid, * GeodesicExact, and TransverseMercatorExact. Two categories of function * are provided: * - \e static functions to compute symmetric elliptic integrals * (https://dlmf.nist.gov/19.16.i) * - \e member functions to compute Legrendre's elliptic * integrals (https://dlmf.nist.gov/19.2.ii) and the * Jacobi elliptic functions (https://dlmf.nist.gov/22.2). * . * In the latter case, an object is constructed giving the modulus \e k (and * optionally the parameter α2). The modulus is always * passed as its square k2 which allows \e k to be pure * imaginary (k2 < 0). (Confusingly, Abramowitz and * Stegun call \e m = k2 the "parameter" and \e n = * α2 the "characteristic".) * * In geodesic applications, it is convenient to separate the incomplete * integrals into secular and periodic components, e.g., * \f[ * E(\phi, k) = (2 E(k) / \pi) [ \phi + \delta E(\phi, k) ] * \f] * where δ\e E(φ, \e k) is an odd periodic function with period * π. * * The computation of the elliptic integrals uses the algorithms given in * - B. C. Carlson, * Computation of real or * complex elliptic integrals, Numerical Algorithms 10, 13--26 (1995); * preprint. * . * with the additional optimizations given in https://dlmf.nist.gov/19.36.i. * The computation of the Jacobi elliptic functions uses the algorithm given * in * - R. Bulirsch, * Numerical Calculation of * Elliptic Integrals and Elliptic Functions, Numericshe Mathematik 7, * 78--90 (1965). * . * The notation follows https://dlmf.nist.gov/19 and https://dlmf.nist.gov/22 * * Example of use: * \include example-EllipticFunction.cpp **********************************************************************/ class GEOGRAPHICLIB_EXPORT EllipticFunction { private: typedef Math::real real; enum { num_ = 13 }; // Max depth required for sncndn; probably 5 is enough. real _k2, _kp2, _alpha2, _alphap2, _eps; real _kKc, _eEc, _dDc, _pPic, _gGc, _hHc; public: /** \name Constructor **********************************************************************/ ///@{ /** * Constructor specifying the modulus and parameter. * * @param[in] k2 the square of the modulus k2. * k2 must lie in (−∞, 1]. * @param[in] alpha2 the parameter α2. * α2 must lie in (−∞, 1]. * @exception GeographicErr if \e k2 or \e alpha2 is out of its legal * range. * * If only elliptic integrals of the first and second kinds are needed, * then set α2 = 0 (the default value); in this case, we * have Π(φ, 0, \e k) = \e F(φ, \e k), \e G(φ, 0, \e k) = \e * E(φ, \e k), and \e H(φ, 0, \e k) = \e F(φ, \e k) - \e * D(φ, \e k). **********************************************************************/ EllipticFunction(real k2 = 0, real alpha2 = 0) { Reset(k2, alpha2); } /** * Constructor specifying the modulus and parameter and their complements. * * @param[in] k2 the square of the modulus k2. * k2 must lie in (−∞, 1]. * @param[in] alpha2 the parameter α2. * α2 must lie in (−∞, 1]. * @param[in] kp2 the complementary modulus squared k'2 = * 1 − k2. This must lie in [0, ∞). * @param[in] alphap2 the complementary parameter α'2 = 1 * − α2. This must lie in [0, ∞). * @exception GeographicErr if \e k2, \e alpha2, \e kp2, or \e alphap2 is * out of its legal range. * * The arguments must satisfy \e k2 + \e kp2 = 1 and \e alpha2 + \e alphap2 * = 1. (No checking is done that these conditions are met.) This * constructor is provided to enable accuracy to be maintained, e.g., when * \e k is very close to unity. **********************************************************************/ EllipticFunction(real k2, real alpha2, real kp2, real alphap2) { Reset(k2, alpha2, kp2, alphap2); } /** * Reset the modulus and parameter. * * @param[in] k2 the new value of square of the modulus * k2 which must lie in (−∞, ]. * done.) * @param[in] alpha2 the new value of parameter α2. * α2 must lie in (−∞, 1]. * @exception GeographicErr if \e k2 or \e alpha2 is out of its legal * range. **********************************************************************/ void Reset(real k2 = 0, real alpha2 = 0) { Reset(k2, alpha2, 1 - k2, 1 - alpha2); } /** * Reset the modulus and parameter supplying also their complements. * * @param[in] k2 the square of the modulus k2. * k2 must lie in (−∞, 1]. * @param[in] alpha2 the parameter α2. * α2 must lie in (−∞, 1]. * @param[in] kp2 the complementary modulus squared k'2 = * 1 − k2. This must lie in [0, ∞). * @param[in] alphap2 the complementary parameter α'2 = 1 * − α2. This must lie in [0, ∞). * @exception GeographicErr if \e k2, \e alpha2, \e kp2, or \e alphap2 is * out of its legal range. * * The arguments must satisfy \e k2 + \e kp2 = 1 and \e alpha2 + \e alphap2 * = 1. (No checking is done that these conditions are met.) This * constructor is provided to enable accuracy to be maintained, e.g., when * is very small. **********************************************************************/ void Reset(real k2, real alpha2, real kp2, real alphap2); ///@} /** \name Inspector functions. **********************************************************************/ ///@{ /** * @return the square of the modulus k2. **********************************************************************/ Math::real k2() const { return _k2; } /** * @return the square of the complementary modulus k'2 = * 1 − k2. **********************************************************************/ Math::real kp2() const { return _kp2; } /** * @return the parameter α2. **********************************************************************/ Math::real alpha2() const { return _alpha2; } /** * @return the complementary parameter α'2 = 1 − * α2. **********************************************************************/ Math::real alphap2() const { return _alphap2; } ///@} /** \name Complete elliptic integrals. **********************************************************************/ ///@{ /** * The complete integral of the first kind. * * @return \e K(\e k). * * \e K(\e k) is defined in https://dlmf.nist.gov/19.2.E4 * \f[ * K(k) = \int_0^{\pi/2} \frac1{\sqrt{1-k^2\sin^2\phi}}\,d\phi. * \f] **********************************************************************/ Math::real K() const { return _kKc; } /** * The complete integral of the second kind. * * @return \e E(\e k). * * \e E(\e k) is defined in https://dlmf.nist.gov/19.2.E5 * \f[ * E(k) = \int_0^{\pi/2} \sqrt{1-k^2\sin^2\phi}\,d\phi. * \f] **********************************************************************/ Math::real E() const { return _eEc; } /** * Jahnke's complete integral. * * @return \e D(\e k). * * \e D(\e k) is defined in https://dlmf.nist.gov/19.2.E6 * \f[ * D(k) = * \int_0^{\pi/2} \frac{\sin^2\phi}{\sqrt{1-k^2\sin^2\phi}}\,d\phi. * \f] **********************************************************************/ Math::real D() const { return _dDc; } /** * The difference between the complete integrals of the first and second * kinds. * * @return \e K(\e k) − \e E(\e k). **********************************************************************/ Math::real KE() const { return _k2 * _dDc; } /** * The complete integral of the third kind. * * @return Π(α2, \e k). * * Π(α2, \e k) is defined in * https://dlmf.nist.gov/19.2.E7 * \f[ * \Pi(\alpha^2, k) = \int_0^{\pi/2} * \frac1{\sqrt{1-k^2\sin^2\phi}(1 - \alpha^2\sin^2\phi)}\,d\phi. * \f] **********************************************************************/ Math::real Pi() const { return _pPic; } /** * Legendre's complete geodesic longitude integral. * * @return \e G(α2, \e k). * * \e G(α2, \e k) is given by * \f[ * G(\alpha^2, k) = \int_0^{\pi/2} * \frac{\sqrt{1-k^2\sin^2\phi}}{1 - \alpha^2\sin^2\phi}\,d\phi. * \f] **********************************************************************/ Math::real G() const { return _gGc; } /** * Cayley's complete geodesic longitude difference integral. * * @return \e H(α2, \e k). * * \e H(α2, \e k) is given by * \f[ * H(\alpha^2, k) = \int_0^{\pi/2} * \frac{\cos^2\phi}{(1-\alpha^2\sin^2\phi)\sqrt{1-k^2\sin^2\phi}} * \,d\phi. * \f] **********************************************************************/ Math::real H() const { return _hHc; } ///@} /** \name Incomplete elliptic integrals. **********************************************************************/ ///@{ /** * The incomplete integral of the first kind. * * @param[in] phi * @return \e F(φ, \e k). * * \e F(φ, \e k) is defined in https://dlmf.nist.gov/19.2.E4 * \f[ * F(\phi, k) = \int_0^\phi \frac1{\sqrt{1-k^2\sin^2\theta}}\,d\theta. * \f] **********************************************************************/ Math::real F(real phi) const; /** * The incomplete integral of the second kind. * * @param[in] phi * @return \e E(φ, \e k). * * \e E(φ, \e k) is defined in https://dlmf.nist.gov/19.2.E5 * \f[ * E(\phi, k) = \int_0^\phi \sqrt{1-k^2\sin^2\theta}\,d\theta. * \f] **********************************************************************/ Math::real E(real phi) const; /** * The incomplete integral of the second kind with the argument given in * degrees. * * @param[in] ang in degrees. * @return \e E(π ang/180, \e k). **********************************************************************/ Math::real Ed(real ang) const; /** * The inverse of the incomplete integral of the second kind. * * @param[in] x * @return φ = E−1(\e x, \e k); i.e., the * solution of such that \e E(φ, \e k) = \e x. **********************************************************************/ Math::real Einv(real x) const; /** * The incomplete integral of the third kind. * * @param[in] phi * @return Π(φ, α2, \e k). * * Π(φ, α2, \e k) is defined in * https://dlmf.nist.gov/19.2.E7 * \f[ * \Pi(\phi, \alpha^2, k) = \int_0^\phi * \frac1{\sqrt{1-k^2\sin^2\theta}(1 - \alpha^2\sin^2\theta)}\,d\theta. * \f] **********************************************************************/ Math::real Pi(real phi) const; /** * Jahnke's incomplete elliptic integral. * * @param[in] phi * @return \e D(φ, \e k). * * \e D(φ, \e k) is defined in https://dlmf.nist.gov/19.2.E4 * \f[ * D(\phi, k) = \int_0^\phi * \frac{\sin^2\theta}{\sqrt{1-k^2\sin^2\theta}}\,d\theta. * \f] **********************************************************************/ Math::real D(real phi) const; /** * Legendre's geodesic longitude integral. * * @param[in] phi * @return \e G(φ, α2, \e k). * * \e G(φ, α2, \e k) is defined by * \f[ * \begin{align} * G(\phi, \alpha^2, k) &= * \frac{k^2}{\alpha^2} F(\phi, k) + * \biggl(1 - \frac{k^2}{\alpha^2}\biggr) \Pi(\phi, \alpha^2, k) \\ * &= \int_0^\phi * \frac{\sqrt{1-k^2\sin^2\theta}}{1 - \alpha^2\sin^2\theta}\,d\theta. * \end{align} * \f] * * Legendre expresses the longitude of a point on the geodesic in terms of * this combination of elliptic integrals in Exercices de Calcul * Intégral, Vol. 1 (1811), p. 181, * https://books.google.com/books?id=riIOAAAAQAAJ&pg=PA181. * * See \ref geodellip for the expression for the longitude in terms of this * function. **********************************************************************/ Math::real G(real phi) const; /** * Cayley's geodesic longitude difference integral. * * @param[in] phi * @return \e H(φ, α2, \e k). * * \e H(φ, α2, \e k) is defined by * \f[ * \begin{align} * H(\phi, \alpha^2, k) &= * \frac1{\alpha^2} F(\phi, k) + * \biggl(1 - \frac1{\alpha^2}\biggr) \Pi(\phi, \alpha^2, k) \\ * &= \int_0^\phi * \frac{\cos^2\theta} * {(1-\alpha^2\sin^2\theta)\sqrt{1-k^2\sin^2\theta}} * \,d\theta. * \end{align} * \f] * * Cayley expresses the longitude difference of a point on the geodesic in * terms of this combination of elliptic integrals in Phil. Mag. 40 * (1870), p. 333, https://books.google.com/books?id=Zk0wAAAAIAAJ&pg=PA333. * * See \ref geodellip for the expression for the longitude in terms of this * function. **********************************************************************/ Math::real H(real phi) const; ///@} /** \name Incomplete integrals in terms of Jacobi elliptic functions. **********************************************************************/ ///@{ /** * The incomplete integral of the first kind in terms of Jacobi elliptic * functions. * * @param[in] sn = sinφ. * @param[in] cn = cosφ. * @param[in] dn = sqrt(1 − k2 * sin2φ). * @return \e F(φ, \e k) as though φ ∈ (−π, π]. **********************************************************************/ Math::real F(real sn, real cn, real dn) const; /** * The incomplete integral of the second kind in terms of Jacobi elliptic * functions. * * @param[in] sn = sinφ. * @param[in] cn = cosφ. * @param[in] dn = sqrt(1 − k2 * sin2φ). * @return \e E(φ, \e k) as though φ ∈ (−π, π]. **********************************************************************/ Math::real E(real sn, real cn, real dn) const; /** * The incomplete integral of the third kind in terms of Jacobi elliptic * functions. * * @param[in] sn = sinφ. * @param[in] cn = cosφ. * @param[in] dn = sqrt(1 − k2 * sin2φ). * @return Π(φ, α2, \e k) as though φ ∈ * (−π, π]. **********************************************************************/ Math::real Pi(real sn, real cn, real dn) const; /** * Jahnke's incomplete elliptic integral in terms of Jacobi elliptic * functions. * * @param[in] sn = sinφ. * @param[in] cn = cosφ. * @param[in] dn = sqrt(1 − k2 * sin2φ). * @return \e D(φ, \e k) as though φ ∈ (−π, π]. **********************************************************************/ Math::real D(real sn, real cn, real dn) const; /** * Legendre's geodesic longitude integral in terms of Jacobi elliptic * functions. * * @param[in] sn = sinφ. * @param[in] cn = cosφ. * @param[in] dn = sqrt(1 − k2 * sin2φ). * @return \e G(φ, α2, \e k) as though φ ∈ * (−π, π]. **********************************************************************/ Math::real G(real sn, real cn, real dn) const; /** * Cayley's geodesic longitude difference integral in terms of Jacobi * elliptic functions. * * @param[in] sn = sinφ. * @param[in] cn = cosφ. * @param[in] dn = sqrt(1 − k2 * sin2φ). * @return \e H(φ, α2, \e k) as though φ ∈ * (−π, π]. **********************************************************************/ Math::real H(real sn, real cn, real dn) const; ///@} /** \name Periodic versions of incomplete elliptic integrals. **********************************************************************/ ///@{ /** * The periodic incomplete integral of the first kind. * * @param[in] sn = sinφ. * @param[in] cn = cosφ. * @param[in] dn = sqrt(1 − k2 * sin2φ). * @return the periodic function π \e F(φ, \e k) / (2 \e K(\e k)) - * φ. **********************************************************************/ Math::real deltaF(real sn, real cn, real dn) const; /** * The periodic incomplete integral of the second kind. * * @param[in] sn = sinφ. * @param[in] cn = cosφ. * @param[in] dn = sqrt(1 − k2 * sin2φ). * @return the periodic function π \e E(φ, \e k) / (2 \e E(\e k)) - * φ. **********************************************************************/ Math::real deltaE(real sn, real cn, real dn) const; /** * The periodic inverse of the incomplete integral of the second kind. * * @param[in] stau = sinτ. * @param[in] ctau = sinτ. * @return the periodic function E−1(τ (2 \e * E(\e k)/π), \e k) - τ. **********************************************************************/ Math::real deltaEinv(real stau, real ctau) const; /** * The periodic incomplete integral of the third kind. * * @param[in] sn = sinφ. * @param[in] cn = cosφ. * @param[in] dn = sqrt(1 − k2 * sin2φ). * @return the periodic function π Π(φ, α2, * \e k) / (2 Π(α2, \e k)) - φ. **********************************************************************/ Math::real deltaPi(real sn, real cn, real dn) const; /** * The periodic Jahnke's incomplete elliptic integral. * * @param[in] sn = sinφ. * @param[in] cn = cosφ. * @param[in] dn = sqrt(1 − k2 * sin2φ). * @return the periodic function π \e D(φ, \e k) / (2 \e D(\e k)) - * φ. **********************************************************************/ Math::real deltaD(real sn, real cn, real dn) const; /** * Legendre's periodic geodesic longitude integral. * * @param[in] sn = sinφ. * @param[in] cn = cosφ. * @param[in] dn = sqrt(1 − k2 * sin2φ). * @return the periodic function π \e G(φ, \e k) / (2 \e G(\e k)) - * φ. **********************************************************************/ Math::real deltaG(real sn, real cn, real dn) const; /** * Cayley's periodic geodesic longitude difference integral. * * @param[in] sn = sinφ. * @param[in] cn = cosφ. * @param[in] dn = sqrt(1 − k2 * sin2φ). * @return the periodic function π \e H(φ, \e k) / (2 \e H(\e k)) - * φ. **********************************************************************/ Math::real deltaH(real sn, real cn, real dn) const; ///@} /** \name Elliptic functions. **********************************************************************/ ///@{ /** * The Jacobi elliptic functions. * * @param[in] x the argument. * @param[out] sn sn(\e x, \e k). * @param[out] cn cn(\e x, \e k). * @param[out] dn dn(\e x, \e k). **********************************************************************/ void sncndn(real x, real& sn, real& cn, real& dn) const; /** * The Δ amplitude function. * * @param[in] sn sinφ. * @param[in] cn cosφ. * @return Δ = sqrt(1 − k2 * sin2φ). **********************************************************************/ Math::real Delta(real sn, real cn) const { using std::sqrt; return sqrt(_k2 < 0 ? 1 - _k2 * sn*sn : _kp2 + _k2 * cn*cn); } ///@} /** \name Symmetric elliptic integrals. **********************************************************************/ ///@{ /** * Symmetric integral of the first kind RF. * * @param[in] x * @param[in] y * @param[in] z * @return RF(\e x, \e y, \e z). * * RF is defined in https://dlmf.nist.gov/19.16.E1 * \f[ R_F(x, y, z) = \frac12 * \int_0^\infty\frac1{\sqrt{(t + x) (t + y) (t + z)}}\, dt, \f] * where at most one of arguments, \e x, \e y, \e z, can be zero and those * arguments that are nonzero must be positive. If one of the arguments is * zero, it is more efficient to call the two-argument version of this * function with the non-zero arguments. **********************************************************************/ static real RF(real x, real y, real z); /** * Complete symmetric integral of the first kind, * RF with one argument zero. * * @param[in] x * @param[in] y * @return RF(\e x, \e y, 0). * * The arguments \e x and \e y must be positive. **********************************************************************/ static real RF(real x, real y); /** * Degenerate symmetric integral of the first kind * RC. * * @param[in] x * @param[in] y * @return RC(\e x, \e y) = * RF(\e x, \e y, \e y). * * RC is defined in https://dlmf.nist.gov/19.2.E17 * \f[ R_C(x, y) = \frac12 * \int_0^\infty\frac1{\sqrt{t + x}(t + y)}\,dt, \f] * where \e x ≥ 0 and \e y > 0. **********************************************************************/ static real RC(real x, real y); /** * Symmetric integral of the second kind RG. * * @param[in] x * @param[in] y * @param[in] z * @return RG(\e x, \e y, \e z). * * RG is defined in Carlson, eq 1.5 * \f[ R_G(x, y, z) = \frac14 * \int_0^\infty[(t + x) (t + y) (t + z)]^{-1/2} * \biggl( * \frac x{t + x} + \frac y{t + y} + \frac z{t + z} * \biggr)t\,dt, \f] * where at most one of arguments, \e x, \e y, \e z, can be zero and those * arguments that are nonzero must be positive. See also * https://dlmf.nist.gov/19.23.E6_5. If one of the arguments is zero, it * is more efficient to call the two-argument version of this function with * the non-zero arguments. **********************************************************************/ static real RG(real x, real y, real z); /** * Complete symmetric integral of the second kind, * RG with one argument zero. * * @param[in] x * @param[in] y * @return RG(\e x, \e y, 0). * * The arguments \e x and \e y must be positive. **********************************************************************/ static real RG(real x, real y); /** * Symmetric integral of the third kind RJ. * * @param[in] x * @param[in] y * @param[in] z * @param[in] p * @return RJ(\e x, \e y, \e z, \e p). * * RJ is defined in https://dlmf.nist.gov/19.16.E2 * \f[ R_J(x, y, z, p) = \frac32 * \int_0^\infty * [(t + x) (t + y) (t + z)]^{-1/2} (t + p)^{-1}\, dt, \f] * where \e p > 0, and \e x, \e y, \e z are nonnegative with at most one of * them being 0. **********************************************************************/ static real RJ(real x, real y, real z, real p); /** * Degenerate symmetric integral of the third kind * RD. * * @param[in] x * @param[in] y * @param[in] z * @return RD(\e x, \e y, \e z) = * RJ(\e x, \e y, \e z, \e z). * * RD is defined in https://dlmf.nist.gov/19.16.E5 * \f[ R_D(x, y, z) = \frac32 * \int_0^\infty[(t + x) (t + y)]^{-1/2} (t + z)^{-3/2}\, dt, \f] * where \e x, \e y, \e z are positive except that at most one of \e x and * \e y can be 0. **********************************************************************/ static real RD(real x, real y, real z); ///@} }; } // namespace GeographicLib #endif // GEOGRAPHICLIB_ELLIPTICFUNCTION_HPP geosphere/src/GeodesicLineExact.h0000644000176200001440000007277314323377037016543 0ustar liggesusers/** * \file GeodesicLineExact.hpp * \brief Header for GeographicLib::GeodesicLineExact class * * Copyright (c) Charles Karney (2012-2022) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ **********************************************************************/ #if !defined(GEOGRAPHICLIB_GEODESICLINEEXACT_HPP) #define GEOGRAPHICLIB_GEODESICLINEEXACT_HPP 1 #include "Constants.h" #include "GeodesicExact.h" #include "EllipticFunction.h" #include namespace GeographicLib { /** * \brief An exact geodesic line * * GeodesicLineExact facilitates the determination of a series of points on a * single geodesic. This is a companion to the GeodesicExact class. For * additional information on this class see the documentation on the * GeodesicLine class. * * Example of use: * \include example-GeodesicLineExact.cpp * * GeodSolve is a command-line utility * providing access to the functionality of GeodesicExact and * GeodesicLineExact (via the -E option). **********************************************************************/ class GeodesicLineExact { private: typedef Math::real real; friend class GeodesicExact; int _nC4; real tiny_; real _lat1, _lon1, _azi1; real _a, _f, _b, _c2, _f1, _e2, _salp0, _calp0, _k2, _salp1, _calp1, _ssig1, _csig1, _dn1, _stau1, _ctau1, _somg1, _comg1, _cchi1, _aA4, _eE0, _dD0, _hH0, _eE1, _dD1, _hH1; real _a13, _s13; real _bB41; std::vector _cC4a; EllipticFunction _eE; unsigned _caps; void LineInit(const GeodesicExact& g, real lat1, real lon1, real azi1, real salp1, real calp1, unsigned caps); GeodesicLineExact(const GeodesicExact& g, real lat1, real lon1, real azi1, real salp1, real calp1, unsigned caps, bool arcmode, real s13_a13); enum captype { CAP_NONE = GeodesicExact::CAP_NONE, CAP_E = GeodesicExact::CAP_E, CAP_D = GeodesicExact::CAP_D, CAP_H = GeodesicExact::CAP_H, CAP_C4 = GeodesicExact::CAP_C4, CAP_ALL = GeodesicExact::CAP_ALL, CAP_MASK = GeodesicExact::CAP_MASK, OUT_ALL = GeodesicExact::OUT_ALL, OUT_MASK = GeodesicExact::OUT_MASK, }; public: /** * Bit masks for what calculations to do. They signify to the * GeodesicLineExact::GeodesicLineExact constructor and to * GeodesicExact::Line what capabilities should be included in the * GeodesicLineExact object. This is merely a duplication of * GeodesicExact::mask. **********************************************************************/ enum mask { /** * No capabilities, no output. * @hideinitializer **********************************************************************/ NONE = GeodesicExact::NONE, /** * Calculate latitude \e lat2. (It's not necessary to include this as a * capability to GeodesicLineExact because this is included by default.) * @hideinitializer **********************************************************************/ LATITUDE = GeodesicExact::LATITUDE, /** * Calculate longitude \e lon2. * @hideinitializer **********************************************************************/ LONGITUDE = GeodesicExact::LONGITUDE, /** * Calculate azimuths \e azi1 and \e azi2. (It's not necessary to * include this as a capability to GeodesicLineExact because this is * included by default.) * @hideinitializer **********************************************************************/ AZIMUTH = GeodesicExact::AZIMUTH, /** * Calculate distance \e s12. * @hideinitializer **********************************************************************/ DISTANCE = GeodesicExact::DISTANCE, /** * A combination of the common capabilities: GeodesicLineExact::LATITUDE, * GeodesicLineExact::LONGITUDE, GeodesicLineExact::AZIMUTH, * GeodesicLineExact::DISTANCE. * @hideinitializer **********************************************************************/ STANDARD = GeodesicExact::STANDARD, /** * Allow distance \e s12 to be used as input in the direct geodesic * problem. * @hideinitializer **********************************************************************/ DISTANCE_IN = GeodesicExact::DISTANCE_IN, /** * Calculate reduced length \e m12. * @hideinitializer **********************************************************************/ REDUCEDLENGTH = GeodesicExact::REDUCEDLENGTH, /** * Calculate geodesic scales \e M12 and \e M21. * @hideinitializer **********************************************************************/ GEODESICSCALE = GeodesicExact::GEODESICSCALE, /** * Calculate area \e S12. * @hideinitializer **********************************************************************/ AREA = GeodesicExact::AREA, /** * Unroll \e lon2 in the direct calculation. * @hideinitializer **********************************************************************/ LONG_UNROLL = GeodesicExact::LONG_UNROLL, /** * All capabilities, calculate everything. (LONG_UNROLL is not * included in this mask.) * @hideinitializer **********************************************************************/ ALL = GeodesicExact::ALL, }; /** \name Constructors **********************************************************************/ ///@{ /** * Constructor for a geodesic line staring at latitude \e lat1, longitude * \e lon1, and azimuth \e azi1 (all in degrees). * * @param[in] g A GeodesicExact object used to compute the necessary * information about the GeodesicLineExact. * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] azi1 azimuth at point 1 (degrees). * @param[in] caps bitor'ed combination of GeodesicLineExact::mask values * specifying the capabilities the GeodesicLineExact object should * possess, i.e., which quantities can be returned in calls to * GeodesicLine::Position. * * \e lat1 should be in the range [−90°, 90°]. * * The GeodesicLineExact::mask values are * - \e caps |= GeodesicLineExact::LATITUDE for the latitude \e lat2; this * is added automatically; * - \e caps |= GeodesicLineExact::LONGITUDE for the latitude \e lon2; * - \e caps |= GeodesicLineExact::AZIMUTH for the latitude \e azi2; this * is added automatically; * - \e caps |= GeodesicLineExact::DISTANCE for the distance \e s12; * - \e caps |= GeodesicLineExact::REDUCEDLENGTH for the reduced length \e * m12; * - \e caps |= GeodesicLineExact::GEODESICSCALE for the geodesic scales \e * M12 and \e M21; * - \e caps |= GeodesicLineExact::AREA for the area \e S12; * - \e caps |= GeodesicLineExact::DISTANCE_IN permits the length of the * geodesic to be given in terms of \e s12; without this capability the * length can only be specified in terms of arc length; * - \e caps |= GeodesicLineExact::ALL for all of the above. * . * The default value of \e caps is GeodesicLineExact::ALL. * * If the point is at a pole, the azimuth is defined by keeping \e lon1 * fixed, writing \e lat1 = ±(90° − ε), and taking * the limit ε → 0+. **********************************************************************/ GEOGRAPHICLIB_EXPORT GeodesicLineExact(const GeodesicExact& g, real lat1, real lon1, real azi1, unsigned caps = ALL); /** * A default constructor. If GeodesicLineExact::Position is called on the * resulting object, it returns immediately (without doing any * calculations). The object can be set with a call to * GeodesicExact::Line. Use Init() to test whether object is still in this * uninitialized state. **********************************************************************/ GEOGRAPHICLIB_EXPORT GeodesicLineExact() : _caps(0U) {} ///@} /** \name Position in terms of distance **********************************************************************/ ///@{ /** * Compute the position of point 2 which is a distance \e s12 (meters) * from point 1. * * @param[in] s12 distance from point 1 to point 2 (meters); it can be * signed. * @param[out] lat2 latitude of point 2 (degrees). * @param[out] lon2 longitude of point 2 (degrees); requires that the * GeodesicLineExact object was constructed with \e caps |= * GeodesicLineExact::LONGITUDE. * @param[out] azi2 (forward) azimuth at point 2 (degrees). * @param[out] m12 reduced length of geodesic (meters); requires that the * GeodesicLineExact object was constructed with \e caps |= * GeodesicLineExact::REDUCEDLENGTH. * @param[out] M12 geodesic scale of point 2 relative to point 1 * (dimensionless); requires that the GeodesicLineExact object was * constructed with \e caps |= GeodesicLineExact::GEODESICSCALE. * @param[out] M21 geodesic scale of point 1 relative to point 2 * (dimensionless); requires that the GeodesicLineExact object was * constructed with \e caps |= GeodesicLineExact::GEODESICSCALE. * @param[out] S12 area under the geodesic (meters2); requires * that the GeodesicLineExact object was constructed with \e caps |= * GeodesicLineExact::AREA. * @return \e a12 arc length from point 1 to point 2 (degrees). * * The values of \e lon2 and \e azi2 returned are in the range * [−180°, 180°]. * * The GeodesicLineExact object \e must have been constructed with \e caps * |= GeodesicLineExact::DISTANCE_IN; otherwise Math::NaN() is returned and * no parameters are set. Requesting a value which the GeodesicLineExact * object is not capable of computing is not an error; the corresponding * argument will not be altered. * * The following functions are overloaded versions of * GeodesicLineExact::Position which omit some of the output parameters. * Note, however, that the arc length is always computed and returned as * the function value. **********************************************************************/ Math::real GEOGRAPHICLIB_EXPORT Position(real s12, real& lat2, real& lon2, real& azi2, real& m12, real& M12, real& M21, real& S12) const { real t; return GenPosition(false, s12, LATITUDE | LONGITUDE | AZIMUTH | REDUCEDLENGTH | GEODESICSCALE | AREA, lat2, lon2, azi2, t, m12, M12, M21, S12); } /** * See the documentation for GeodesicLineExact::Position. **********************************************************************/ Math::real GEOGRAPHICLIB_EXPORT Position(real s12, real& lat2, real& lon2) const { real t; return GenPosition(false, s12, LATITUDE | LONGITUDE, lat2, lon2, t, t, t, t, t, t); } /** * See the documentation for GeodesicLineExact::Position. **********************************************************************/ Math::real GEOGRAPHICLIB_EXPORT Position(real s12, real& lat2, real& lon2, real& azi2) const { real t; return GenPosition(false, s12, LATITUDE | LONGITUDE | AZIMUTH, lat2, lon2, azi2, t, t, t, t, t); } /** * See the documentation for GeodesicLineExact::Position. **********************************************************************/ Math::real GEOGRAPHICLIB_EXPORT Position(real s12, real& lat2, real& lon2, real& azi2, real& m12) const { real t; return GenPosition(false, s12, LATITUDE | LONGITUDE | AZIMUTH | REDUCEDLENGTH, lat2, lon2, azi2, t, m12, t, t, t); } /** * See the documentation for GeodesicLineExact::Position. **********************************************************************/ Math::real GEOGRAPHICLIB_EXPORT Position(real s12, real& lat2, real& lon2, real& azi2, real& M12, real& M21) const { real t; return GenPosition(false, s12, LATITUDE | LONGITUDE | AZIMUTH | GEODESICSCALE, lat2, lon2, azi2, t, t, M12, M21, t); } /** * See the documentation for GeodesicLineExact::Position. **********************************************************************/ Math::real GEOGRAPHICLIB_EXPORT Position(real s12, real& lat2, real& lon2, real& azi2, real& m12, real& M12, real& M21) const { real t; return GenPosition(false, s12, LATITUDE | LONGITUDE | AZIMUTH | REDUCEDLENGTH | GEODESICSCALE, lat2, lon2, azi2, t, m12, M12, M21, t); } ///@} /** \name Position in terms of arc length **********************************************************************/ ///@{ /** * Compute the position of point 2 which is an arc length \e a12 (degrees) * from point 1. * * @param[in] a12 arc length from point 1 to point 2 (degrees); it can * be signed. * @param[out] lat2 latitude of point 2 (degrees). * @param[out] lon2 longitude of point 2 (degrees); requires that the * GeodesicLineExact object was constructed with \e caps |= * GeodesicLineExact::LONGITUDE. * @param[out] azi2 (forward) azimuth at point 2 (degrees). * @param[out] s12 distance from point 1 to point 2 (meters); requires * that the GeodesicLineExact object was constructed with \e caps |= * GeodesicLineExact::DISTANCE. * @param[out] m12 reduced length of geodesic (meters); requires that the * GeodesicLineExact object was constructed with \e caps |= * GeodesicLineExact::REDUCEDLENGTH. * @param[out] M12 geodesic scale of point 2 relative to point 1 * (dimensionless); requires that the GeodesicLineExact object was * constructed with \e caps |= GeodesicLineExact::GEODESICSCALE. * @param[out] M21 geodesic scale of point 1 relative to point 2 * (dimensionless); requires that the GeodesicLineExact object was * constructed with \e caps |= GeodesicLineExact::GEODESICSCALE. * @param[out] S12 area under the geodesic (meters2); requires * that the GeodesicLineExact object was constructed with \e caps |= * GeodesicLineExact::AREA. * * The values of \e lon2 and \e azi2 returned are in the range * [−180°, 180°]. * * Requesting a value which the GeodesicLineExact object is not capable of * computing is not an error; the corresponding argument will not be * altered. * * The following functions are overloaded versions of * GeodesicLineExact::ArcPosition which omit some of the output parameters. **********************************************************************/ void GEOGRAPHICLIB_EXPORT ArcPosition(real a12, real& lat2, real& lon2, real& azi2, real& s12, real& m12, real& M12, real& M21, real& S12) const { GenPosition(true, a12, LATITUDE | LONGITUDE | AZIMUTH | DISTANCE | REDUCEDLENGTH | GEODESICSCALE | AREA, lat2, lon2, azi2, s12, m12, M12, M21, S12); } /** * See the documentation for GeodesicLineExact::ArcPosition. **********************************************************************/ void GEOGRAPHICLIB_EXPORT ArcPosition(real a12, real& lat2, real& lon2) const { real t; GenPosition(true, a12, LATITUDE | LONGITUDE, lat2, lon2, t, t, t, t, t, t); } /** * See the documentation for GeodesicLineExact::ArcPosition. **********************************************************************/ void GEOGRAPHICLIB_EXPORT ArcPosition(real a12, real& lat2, real& lon2, real& azi2) const { real t; GenPosition(true, a12, LATITUDE | LONGITUDE | AZIMUTH, lat2, lon2, azi2, t, t, t, t, t); } /** * See the documentation for GeodesicLineExact::ArcPosition. **********************************************************************/ void GEOGRAPHICLIB_EXPORT ArcPosition(real a12, real& lat2, real& lon2, real& azi2, real& s12) const { real t; GenPosition(true, a12, LATITUDE | LONGITUDE | AZIMUTH | DISTANCE, lat2, lon2, azi2, s12, t, t, t, t); } /** * See the documentation for GeodesicLineExact::ArcPosition. **********************************************************************/ void GEOGRAPHICLIB_EXPORT ArcPosition(real a12, real& lat2, real& lon2, real& azi2, real& s12, real& m12) const { real t; GenPosition(true, a12, LATITUDE | LONGITUDE | AZIMUTH | DISTANCE | REDUCEDLENGTH, lat2, lon2, azi2, s12, m12, t, t, t); } /** * See the documentation for GeodesicLineExact::ArcPosition. **********************************************************************/ void GEOGRAPHICLIB_EXPORT ArcPosition(real a12, real& lat2, real& lon2, real& azi2, real& s12, real& M12, real& M21) const { real t; GenPosition(true, a12, LATITUDE | LONGITUDE | AZIMUTH | DISTANCE | GEODESICSCALE, lat2, lon2, azi2, s12, t, M12, M21, t); } /** * See the documentation for GeodesicLineExact::ArcPosition. **********************************************************************/ void GEOGRAPHICLIB_EXPORT ArcPosition(real a12, real& lat2, real& lon2, real& azi2, real& s12, real& m12, real& M12, real& M21) const { real t; GenPosition(true, a12, LATITUDE | LONGITUDE | AZIMUTH | DISTANCE | REDUCEDLENGTH | GEODESICSCALE, lat2, lon2, azi2, s12, m12, M12, M21, t); } ///@} /** \name The general position function. **********************************************************************/ ///@{ /** * The general position function. GeodesicLineExact::Position and * GeodesicLineExact::ArcPosition are defined in terms of this function. * * @param[in] arcmode boolean flag determining the meaning of the second * parameter; if \e arcmode is false, then the GeodesicLineExact object * must have been constructed with \e caps |= * GeodesicLineExact::DISTANCE_IN. * @param[in] s12_a12 if \e arcmode is false, this is the distance between * point 1 and point 2 (meters); otherwise it is the arc length between * point 1 and point 2 (degrees); it can be signed. * @param[in] outmask a bitor'ed combination of GeodesicLineExact::mask * values specifying which of the following parameters should be set. * @param[out] lat2 latitude of point 2 (degrees). * @param[out] lon2 longitude of point 2 (degrees); requires that the * GeodesicLineExact object was constructed with \e caps |= * GeodesicLineExact::LONGITUDE. * @param[out] azi2 (forward) azimuth at point 2 (degrees). * @param[out] s12 distance from point 1 to point 2 (meters); requires * that the GeodesicLineExact object was constructed with \e caps |= * GeodesicLineExact::DISTANCE. * @param[out] m12 reduced length of geodesic (meters); requires that the * GeodesicLineExact object was constructed with \e caps |= * GeodesicLineExact::REDUCEDLENGTH. * @param[out] M12 geodesic scale of point 2 relative to point 1 * (dimensionless); requires that the GeodesicLineExact object was * constructed with \e caps |= GeodesicLineExact::GEODESICSCALE. * @param[out] M21 geodesic scale of point 1 relative to point 2 * (dimensionless); requires that the GeodesicLineExact object was * constructed with \e caps |= GeodesicLineExact::GEODESICSCALE. * @param[out] S12 area under the geodesic (meters2); requires * that the GeodesicLineExact object was constructed with \e caps |= * GeodesicLineExact::AREA. * @return \e a12 arc length from point 1 to point 2 (degrees). * * The GeodesicLineExact::mask values possible for \e outmask are * - \e outmask |= GeodesicLineExact::LATITUDE for the latitude \e lat2; * - \e outmask |= GeodesicLineExact::LONGITUDE for the latitude \e lon2; * - \e outmask |= GeodesicLineExact::AZIMUTH for the latitude \e azi2; * - \e outmask |= GeodesicLineExact::DISTANCE for the distance \e s12; * - \e outmask |= GeodesicLineExact::REDUCEDLENGTH for the reduced length * \e m12; * - \e outmask |= GeodesicLineExact::GEODESICSCALE for the geodesic scales * \e M12 and \e M21; * - \e outmask |= GeodesicLineExact::AREA for the area \e S12; * - \e outmask |= GeodesicLineExact::ALL for all of the above; * - \e outmask |= GeodesicLineExact::LONG_UNROLL to unroll \e lon2 instead * of wrapping it into the range [−180°, 180°]. * . * Requesting a value which the GeodesicLineExact object is not capable of * computing is not an error; the corresponding argument will not be * altered. Note, however, that the arc length is always computed and * returned as the function value. * * With the GeodesicLineExact::LONG_UNROLL bit set, the quantity \e lon2 * − \e lon1 indicates how many times and in what sense the geodesic * encircles the ellipsoid. **********************************************************************/ Math::real GEOGRAPHICLIB_EXPORT GenPosition(bool arcmode, real s12_a12, unsigned outmask, real& lat2, real& lon2, real& azi2, real& s12, real& m12, real& M12, real& M21, real& S12) const; ///@} /** \name Setting point 3 **********************************************************************/ ///@{ /** * Specify position of point 3 in terms of distance. * * @param[in] s13 the distance from point 1 to point 3 (meters); it * can be negative. * * This is only useful if the GeodesicLineExact object has been constructed * with \e caps |= GeodesicLineExact::DISTANCE_IN. **********************************************************************/ void GEOGRAPHICLIB_EXPORT SetDistance(real s13); /** * Specify position of point 3 in terms of arc length. * * @param[in] a13 the arc length from point 1 to point 3 (degrees); it * can be negative. * * The distance \e s13 is only set if the GeodesicLineExact object has been * constructed with \e caps |= GeodesicLineExact::DISTANCE. **********************************************************************/ void GEOGRAPHICLIB_EXPORT SetArc(real a13); /** * Specify position of point 3 in terms of either distance or arc length. * * @param[in] arcmode boolean flag determining the meaning of the second * parameter; if \e arcmode is false, then the GeodesicLineExact object * must have been constructed with \e caps |= * GeodesicLineExact::DISTANCE_IN. * @param[in] s13_a13 if \e arcmode is false, this is the distance from * point 1 to point 3 (meters); otherwise it is the arc length from * point 1 to point 3 (degrees); it can be negative. **********************************************************************/ void GEOGRAPHICLIB_EXPORT GenSetDistance(bool arcmode, real s13_a13); /** \name Inspector functions **********************************************************************/ ///@{ /** * @return true if the object has been initialized. **********************************************************************/ bool GEOGRAPHICLIB_EXPORT Init() const { return _caps != 0U; } /** * @return \e lat1 the latitude of point 1 (degrees). **********************************************************************/ Math::real GEOGRAPHICLIB_EXPORT Latitude() const { return Init() ? _lat1 : Math::NaN(); } /** * @return \e lon1 the longitude of point 1 (degrees). **********************************************************************/ Math::real GEOGRAPHICLIB_EXPORT Longitude() const { return Init() ? _lon1 : Math::NaN(); } /** * @return \e azi1 the azimuth (degrees) of the geodesic line at point 1. **********************************************************************/ Math::real GEOGRAPHICLIB_EXPORT Azimuth() const { return Init() ? _azi1 : Math::NaN(); } /** * The sine and cosine of \e azi1. * * @param[out] sazi1 the sine of \e azi1. * @param[out] cazi1 the cosine of \e azi1. **********************************************************************/ void GEOGRAPHICLIB_EXPORT Azimuth(real& sazi1, real& cazi1) const { if (Init()) { sazi1 = _salp1; cazi1 = _calp1; } } /** * @return \e azi0 the azimuth (degrees) of the geodesic line as it crosses * the equator in a northward direction. * * The result lies in [−90°, 90°]. **********************************************************************/ Math::real GEOGRAPHICLIB_EXPORT EquatorialAzimuth() const { return Init() ? Math::atan2d(_salp0, _calp0) : Math::NaN(); } /** * The sine and cosine of \e azi0. * * @param[out] sazi0 the sine of \e azi0. * @param[out] cazi0 the cosine of \e azi0. **********************************************************************/ void GEOGRAPHICLIB_EXPORT EquatorialAzimuth(real& sazi0, real& cazi0) const { if (Init()) { sazi0 = _salp0; cazi0 = _calp0; } } /** * @return \e a1 the arc length (degrees) between the northward equatorial * crossing and point 1. * * The result lies in [−180°, 180°]. **********************************************************************/ Math::real GEOGRAPHICLIB_EXPORT EquatorialArc() const { using std::atan2; return Init() ? atan2(_ssig1, _csig1) / Math::degree() : Math::NaN(); } /** * @return \e a the equatorial radius of the ellipsoid (meters). This is * the value inherited from the GeodesicExact object used in the * constructor. **********************************************************************/ Math::real GEOGRAPHICLIB_EXPORT EquatorialRadius() const { return Init() ? _a : Math::NaN(); } /** * @return \e f the flattening of the ellipsoid. This is the value * inherited from the GeodesicExact object used in the constructor. **********************************************************************/ Math::real GEOGRAPHICLIB_EXPORT Flattening() const { return Init() ? _f : Math::NaN(); } /** * @return \e caps the computational capabilities that this object was * constructed with. LATITUDE and AZIMUTH are always included. **********************************************************************/ unsigned GEOGRAPHICLIB_EXPORT Capabilities() const { return _caps; } /** * Test what capabilities are available. * * @param[in] testcaps a set of bitor'ed GeodesicLineExact::mask values. * @return true if the GeodesicLineExact object has all these capabilities. **********************************************************************/ bool GEOGRAPHICLIB_EXPORT Capabilities(unsigned testcaps) const { testcaps &= OUT_ALL; return (_caps & testcaps) == testcaps; } /** * The distance or arc length to point 3. * * @param[in] arcmode boolean flag determining the meaning of returned * value. * @return \e s13 if \e arcmode is false; \e a13 if \e arcmode is true. **********************************************************************/ Math::real GEOGRAPHICLIB_EXPORT GenDistance(bool arcmode) const { return Init() ? (arcmode ? _a13 : _s13) : Math::NaN(); } /** * @return \e s13, the distance to point 3 (meters). **********************************************************************/ Math::real GEOGRAPHICLIB_EXPORT Distance() const { return GenDistance(false); } /** * @return \e a13, the arc length to point 3 (degrees). **********************************************************************/ Math::real GEOGRAPHICLIB_EXPORT Arc() const { return GenDistance(true); } ///@} }; } // namespace GeographicLib #endif // GEOGRAPHICLIB_GEODESICLINEEXACT_HPP geosphere/src/AlbersEqualArea.cpp0000644000176200001440000005310614323376011016524 0ustar liggesusers/** * \file AlbersEqualArea.cpp * \brief Implementation for GeographicLib::AlbersEqualArea class * * Copyright (c) Charles Karney (2010-2022) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ **********************************************************************/ #include "AlbersEqualArea.h" #if defined(_MSC_VER) // Squelch warnings about constant conditional and enum-float expressions # pragma warning (disable: 4127 5055) #endif namespace GeographicLib { using namespace std; AlbersEqualArea::AlbersEqualArea(real a, real f, real stdlat, real k0) : eps_(numeric_limits::epsilon()) , epsx_(Math::sq(eps_)) , epsx2_(Math::sq(epsx_)) , tol_(sqrt(eps_)) , tol0_(tol_ * sqrt(sqrt(eps_))) , _a(a) , _f(f) , _fm(1 - _f) , _e2(_f * (2 - _f)) , _e(sqrt(fabs(_e2))) , _e2m(1 - _e2) , _qZ(1 + _e2m * atanhee(real(1))) , _qx(_qZ / ( 2 * _e2m )) { if (!(isfinite(_a) && _a > 0)) throw GeographicErr("Equatorial radius is not positive"); if (!(isfinite(_f) && _f < 1)) throw GeographicErr("Polar semi-axis is not positive"); if (!(isfinite(k0) && k0 > 0)) throw GeographicErr("Scale is not positive"); if (!(fabs(stdlat) <= Math::qd)) throw GeographicErr("Standard latitude not in [-" + to_string(Math::qd) + "d, " + to_string(Math::qd) + "d]"); real sphi, cphi; Math::sincosd(stdlat, sphi, cphi); Init(sphi, cphi, sphi, cphi, k0); } AlbersEqualArea::AlbersEqualArea(real a, real f, real stdlat1, real stdlat2, real k1) : eps_(numeric_limits::epsilon()) , epsx_(Math::sq(eps_)) , epsx2_(Math::sq(epsx_)) , tol_(sqrt(eps_)) , tol0_(tol_ * sqrt(sqrt(eps_))) , _a(a) , _f(f) , _fm(1 - _f) , _e2(_f * (2 - _f)) , _e(sqrt(fabs(_e2))) , _e2m(1 - _e2) , _qZ(1 + _e2m * atanhee(real(1))) , _qx(_qZ / ( 2 * _e2m )) { if (!(isfinite(_a) && _a > 0)) throw GeographicErr("Equatorial radius is not positive"); if (!(isfinite(_f) && _f < 1)) throw GeographicErr("Polar semi-axis is not positive"); if (!(isfinite(k1) && k1 > 0)) throw GeographicErr("Scale is not positive"); if (!(fabs(stdlat1) <= Math::qd)) throw GeographicErr("Standard latitude 1 not in [-" + to_string(Math::qd) + "d, " + to_string(Math::qd) + "d]"); if (!(fabs(stdlat2) <= Math::qd)) throw GeographicErr("Standard latitude 2 not in [-" + to_string(Math::qd) + "d, " + to_string(Math::qd) + "d]"); real sphi1, cphi1, sphi2, cphi2; Math::sincosd(stdlat1, sphi1, cphi1); Math::sincosd(stdlat2, sphi2, cphi2); Init(sphi1, cphi1, sphi2, cphi2, k1); } AlbersEqualArea::AlbersEqualArea(real a, real f, real sinlat1, real coslat1, real sinlat2, real coslat2, real k1) : eps_(numeric_limits::epsilon()) , epsx_(Math::sq(eps_)) , epsx2_(Math::sq(epsx_)) , tol_(sqrt(eps_)) , tol0_(tol_ * sqrt(sqrt(eps_))) , _a(a) , _f(f) , _fm(1 - _f) , _e2(_f * (2 - _f)) , _e(sqrt(fabs(_e2))) , _e2m(1 - _e2) , _qZ(1 + _e2m * atanhee(real(1))) , _qx(_qZ / ( 2 * _e2m )) { if (!(isfinite(_a) && _a > 0)) throw GeographicErr("Equatorial radius is not positive"); if (!(isfinite(_f) && _f < 1)) throw GeographicErr("Polar semi-axis is not positive"); if (!(isfinite(k1) && k1 > 0)) throw GeographicErr("Scale is not positive"); if (signbit(coslat1)) throw GeographicErr("Standard latitude 1 not in [-" + to_string(Math::qd) + "d, " + to_string(Math::qd) + "d]"); if (signbit(coslat2)) throw GeographicErr("Standard latitude 2 not in [-" + to_string(Math::qd) + "d, " + to_string(Math::qd) + "d]"); if (!(fabs(sinlat1) <= 1 && coslat1 <= 1) || (coslat1 == 0 && sinlat1 == 0)) throw GeographicErr("Bad sine/cosine of standard latitude 1"); if (!(fabs(sinlat2) <= 1 && coslat2 <= 1) || (coslat2 == 0 && sinlat2 == 0)) throw GeographicErr("Bad sine/cosine of standard latitude 2"); if (coslat1 == 0 && coslat2 == 0 && sinlat1 * sinlat2 <= 0) throw GeographicErr ("Standard latitudes cannot be opposite poles"); Init(sinlat1, coslat1, sinlat2, coslat2, k1); } void AlbersEqualArea::Init(real sphi1, real cphi1, real sphi2, real cphi2, real k1) { { real r; r = hypot(sphi1, cphi1); sphi1 /= r; cphi1 /= r; r = hypot(sphi2, cphi2); sphi2 /= r; cphi2 /= r; } bool polar = (cphi1 == 0); cphi1 = fmax(epsx_, cphi1); // Avoid singularities at poles cphi2 = fmax(epsx_, cphi2); // Determine hemisphere of tangent latitude _sign = sphi1 + sphi2 >= 0 ? 1 : -1; // Internally work with tangent latitude positive sphi1 *= _sign; sphi2 *= _sign; if (sphi1 > sphi2) { swap(sphi1, sphi2); swap(cphi1, cphi2); // Make phi1 < phi2 } real tphi1 = sphi1/cphi1, tphi2 = sphi2/cphi2; // q = (1-e^2)*(sphi/(1-e^2*sphi^2) - atanhee(sphi)) // qZ = q(pi/2) = (1 + (1-e^2)*atanhee(1)) // atanhee(x) = atanh(e*x)/e // q = sxi * qZ // dq/dphi = 2*(1-e^2)*cphi/(1-e^2*sphi^2)^2 // // n = (m1^2-m2^2)/(q2-q1) -> sin(phi0) for phi1, phi2 -> phi0 // C = m1^2 + n*q1 = (m1^2*q2-m2^2*q1)/(q2-q1) // let // rho(pi/2)/rho(-pi/2) = (1-s)/(1+s) // s = n*qZ/C // = qZ * (m1^2-m2^2)/(m1^2*q2-m2^2*q1) // = qZ * (scbet2^2 - scbet1^2)/(scbet2^2*q2 - scbet1^2*q1) // = (scbet2^2 - scbet1^2)/(scbet2^2*sxi2 - scbet1^2*sxi1) // = (tbet2^2 - tbet1^2)/(scbet2^2*sxi2 - scbet1^2*sxi1) // 1-s = -((1-sxi2)*scbet2^2 - (1-sxi1)*scbet1^2)/ // (scbet2^2*sxi2 - scbet1^2*sxi1) // // Define phi0 to give same value of s, i.e., // s = sphi0 * qZ / (m0^2 + sphi0*q0) // = sphi0 * scbet0^2 / (1/qZ + sphi0 * scbet0^2 * sxi0) real tphi0, C; if (polar || tphi1 == tphi2) { tphi0 = tphi2; C = 1; // ignored } else { real tbet1 = _fm * tphi1, scbet12 = 1 + Math::sq(tbet1), tbet2 = _fm * tphi2, scbet22 = 1 + Math::sq(tbet2), txi1 = txif(tphi1), cxi1 = 1/hyp(txi1), sxi1 = txi1 * cxi1, txi2 = txif(tphi2), cxi2 = 1/hyp(txi2), sxi2 = txi2 * cxi2, dtbet2 = _fm * (tbet1 + tbet2), es1 = 1 - _e2 * Math::sq(sphi1), es2 = 1 - _e2 * Math::sq(sphi2), /* dsxi = ( (_e2 * sq(sphi2 + sphi1) + es2 + es1) / (2 * es2 * es1) + Datanhee(sphi2, sphi1) ) * Dsn(tphi2, tphi1, sphi2, sphi1) / ( 2 * _qx ), */ dsxi = ( (1 + _e2 * sphi1 * sphi2) / (es2 * es1) + Datanhee(sphi2, sphi1) ) * Dsn(tphi2, tphi1, sphi2, sphi1) / ( 2 * _qx ), den = (sxi2 + sxi1) * dtbet2 + (scbet22 + scbet12) * dsxi, // s = (sq(tbet2) - sq(tbet1)) / (scbet22*sxi2 - scbet12*sxi1) s = 2 * dtbet2 / den, // 1-s = -(sq(scbet2)*(1-sxi2) - sq(scbet1)*(1-sxi1)) / // (scbet22*sxi2 - scbet12*sxi1) // Write // sq(scbet)*(1-sxi) = sq(scbet)*(1-sphi) * (1-sxi)/(1-sphi) sm1 = -Dsn(tphi2, tphi1, sphi2, sphi1) * ( -( ((sphi2 <= 0 ? (1 - sxi2) / (1 - sphi2) : Math::sq(cxi2/cphi2) * (1 + sphi2) / (1 + sxi2)) + (sphi1 <= 0 ? (1 - sxi1) / (1 - sphi1) : Math::sq(cxi1/cphi1) * (1 + sphi1) / (1 + sxi1))) ) * (1 + _e2 * (sphi1 + sphi2 + sphi1 * sphi2)) / (1 + (sphi1 + sphi2 + sphi1 * sphi2)) + (scbet22 * (sphi2 <= 0 ? 1 - sphi2 : Math::sq(cphi2) / ( 1 + sphi2)) + scbet12 * (sphi1 <= 0 ? 1 - sphi1 : Math::sq(cphi1) / ( 1 + sphi1))) * (_e2 * (1 + sphi1 + sphi2 + _e2 * sphi1 * sphi2)/(es1 * es2) +_e2m * DDatanhee(sphi1, sphi2) ) / _qZ ) / den; // C = (scbet22*sxi2 - scbet12*sxi1) / (scbet22 * scbet12 * (sx2 - sx1)) C = den / (2 * scbet12 * scbet22 * dsxi); tphi0 = (tphi2 + tphi1)/2; real stol = tol0_ * fmax(real(1), fabs(tphi0)); for (int i = 0; i < 2*numit0_ || GEOGRAPHICLIB_PANIC; ++i) { // Solve (scbet0^2 * sphi0) / (1/qZ + scbet0^2 * sphi0 * sxi0) = s // for tphi0 by Newton's method on // v(tphi0) = (scbet0^2 * sphi0) - s * (1/qZ + scbet0^2 * sphi0 * sxi0) // = 0 // Alt: // (scbet0^2 * sphi0) / (1/qZ - scbet0^2 * sphi0 * (1-sxi0)) // = s / (1-s) // w(tphi0) = (1-s) * (scbet0^2 * sphi0) // - s * (1/qZ - scbet0^2 * sphi0 * (1-sxi0)) // = (1-s) * (scbet0^2 * sphi0) // - S/qZ * (1 - scbet0^2 * sphi0 * (qZ-q0)) // Now // qZ-q0 = (1+e2*sphi0)*(1-sphi0)/(1-e2*sphi0^2) + // (1-e2)*atanhee((1-sphi0)/(1-e2*sphi0)) // In limit sphi0 -> 1, qZ-q0 -> 2*(1-sphi0)/(1-e2), so wrte // qZ-q0 = 2*(1-sphi0)/(1-e2) + A + B // A = (1-sphi0)*( (1+e2*sphi0)/(1-e2*sphi0^2) - (1+e2)/(1-e2) ) // = -e2 *(1-sphi0)^2 * (2+(1+e2)*sphi0) / ((1-e2)*(1-e2*sphi0^2)) // B = (1-e2)*atanhee((1-sphi0)/(1-e2*sphi0)) - (1-sphi0) // = (1-sphi0)*(1-e2)/(1-e2*sphi0)* // ((atanhee(x)/x-1) - e2*(1-sphi0)/(1-e2)) // x = (1-sphi0)/(1-e2*sphi0), atanhee(x)/x = atanh(e*x)/(e*x) // // 1 - scbet0^2 * sphi0 * (qZ-q0) // = 1 - scbet0^2 * sphi0 * (2*(1-sphi0)/(1-e2) + A + B) // = D - scbet0^2 * sphi0 * (A + B) // D = 1 - scbet0^2 * sphi0 * 2*(1-sphi0)/(1-e2) // = (1-sphi0)*(1-e2*(1+2*sphi0*(1+sphi0)))/((1-e2)*(1+sphi0)) // dD/dsphi0 = -2*(1-e2*sphi0^2*(2*sphi0+3))/((1-e2)*(1+sphi0)^2) // d(A+B)/dsphi0 = 2*(1-sphi0^2)*e2*(2-e2*(1+sphi0^2))/ // ((1-e2)*(1-e2*sphi0^2)^2) real scphi02 = 1 + Math::sq(tphi0), scphi0 = sqrt(scphi02), // sphi0m = 1-sin(phi0) = 1/( sec(phi0) * (tan(phi0) + sec(phi0)) ) sphi0 = tphi0 / scphi0, sphi0m = 1/(scphi0 * (tphi0 + scphi0)), // scbet0^2 * sphi0 g = (1 + Math::sq( _fm * tphi0 )) * sphi0, // dg/dsphi0 = dg/dtphi0 * scphi0^3 dg = _e2m * scphi02 * (1 + 2 * Math::sq(tphi0)) + _e2, D = sphi0m * (1 - _e2*(1 + 2*sphi0*(1+sphi0))) / (_e2m * (1+sphi0)), // dD/dsphi0 dD = -2 * (1 - _e2*Math::sq(sphi0) * (2*sphi0+3)) / (_e2m * Math::sq(1+sphi0)), A = -_e2 * Math::sq(sphi0m) * (2+(1+_e2)*sphi0) / (_e2m*(1-_e2*Math::sq(sphi0))), B = (sphi0m * _e2m / (1 - _e2*sphi0) * (atanhxm1(_e2 * Math::sq(sphi0m / (1-_e2*sphi0))) - _e2*sphi0m/_e2m)), // d(A+B)/dsphi0 dAB = (2 * _e2 * (2 - _e2 * (1 + Math::sq(sphi0))) / (_e2m * Math::sq(1 - _e2*Math::sq(sphi0)) * scphi02)), u = sm1 * g - s/_qZ * ( D - g * (A + B) ), // du/dsphi0 du = sm1 * dg - s/_qZ * (dD - dg * (A + B) - g * dAB), dtu = -u/du * (scphi0 * scphi02); tphi0 += dtu; if (!(fabs(dtu) >= stol)) break; } } _txi0 = txif(tphi0); _scxi0 = hyp(_txi0); _sxi0 = _txi0 / _scxi0; _n0 = tphi0/hyp(tphi0); _m02 = 1 / (1 + Math::sq(_fm * tphi0)); _nrho0 = polar ? 0 : _a * sqrt(_m02); _k0 = sqrt(tphi1 == tphi2 ? 1 : C / (_m02 + _n0 * _qZ * _sxi0)) * k1; _k2 = Math::sq(_k0); _lat0 = _sign * atan(tphi0)/Math::degree(); } const AlbersEqualArea& AlbersEqualArea::CylindricalEqualArea() { static const AlbersEqualArea cylindricalequalarea(Constants::WGS84_a(), Constants::WGS84_f(), real(0), real(1), real(0), real(1), real(1)); return cylindricalequalarea; } const AlbersEqualArea& AlbersEqualArea::AzimuthalEqualAreaNorth() { static const AlbersEqualArea azimuthalequalareanorth(Constants::WGS84_a(), Constants::WGS84_f(), real(1), real(0), real(1), real(0), real(1)); return azimuthalequalareanorth; } const AlbersEqualArea& AlbersEqualArea::AzimuthalEqualAreaSouth() { static const AlbersEqualArea azimuthalequalareasouth(Constants::WGS84_a(), Constants::WGS84_f(), real(-1), real(0), real(-1), real(0), real(1)); return azimuthalequalareasouth; } Math::real AlbersEqualArea::txif(real tphi) const { // sxi = ( sphi/(1-e2*sphi^2) + atanhee(sphi) ) / // ( 1/(1-e2) + atanhee(1) ) // // txi = ( sphi/(1-e2*sphi^2) + atanhee(sphi) ) / // sqrt( ( (1+e2*sphi)*(1-sphi)/( (1-e2*sphi^2) * (1-e2) ) + // atanhee((1-sphi)/(1-e2*sphi)) ) * // ( (1-e2*sphi)*(1+sphi)/( (1-e2*sphi^2) * (1-e2) ) + // atanhee((1+sphi)/(1+e2*sphi)) ) ) // = ( tphi/(1-e2*sphi^2) + atanhee(sphi, e2)/cphi ) / // sqrt( // ( (1+e2*sphi)/( (1-e2*sphi^2) * (1-e2) ) + Datanhee(1, sphi) ) * // ( (1-e2*sphi)/( (1-e2*sphi^2) * (1-e2) ) + Datanhee(1, -sphi) ) ) // // This function maintains odd parity real cphi = 1 / sqrt(1 + Math::sq(tphi)), sphi = tphi * cphi, es1 = _e2 * sphi, es2m1 = 1 - es1 * sphi, // 1 - e2 * sphi^2 es2m1a = _e2m * es2m1; // (1 - e2 * sphi^2) * (1 - e2) return ( tphi / es2m1 + atanhee(sphi) / cphi ) / sqrt( ( (1 + es1) / es2m1a + Datanhee(1, sphi) ) * ( (1 - es1) / es2m1a + Datanhee(1, -sphi) ) ); } Math::real AlbersEqualArea::tphif(real txi) const { real tphi = txi, stol = tol_ * fmax(real(1), fabs(txi)); // CHECK: min iterations = 1, max iterations = 2; mean = 1.99 for (int i = 0; i < numit_ || GEOGRAPHICLIB_PANIC; ++i) { // dtxi/dtphi = (scxi/scphi)^3 * 2*(1-e^2)/(qZ*(1-e^2*sphi^2)^2) real txia = txif(tphi), tphi2 = Math::sq(tphi), scphi2 = 1 + tphi2, scterm = scphi2/(1 + Math::sq(txia)), dtphi = (txi - txia) * scterm * sqrt(scterm) * _qx * Math::sq(1 - _e2 * tphi2 / scphi2); tphi += dtphi; if (!(fabs(dtphi) >= stol)) break; } return tphi; } // return atanh(sqrt(x))/sqrt(x) - 1 = x/3 + x^2/5 + x^3/7 + ... // typical x < e^2 = 2*f Math::real AlbersEqualArea::atanhxm1(real x) { real s = 0; if (fabs(x) < real(0.5)) { static const real lg2eps_ = -log2(numeric_limits::epsilon() / 2); int e; frexp(x, &e); e = -e; // x = [0.5,1) * 2^(-e) // estimate n s.t. x^n/(2*n+1) < x/3 * epsilon/2 // a stronger condition is x^(n-1) < epsilon/2 // taking log2 of both sides, a stronger condition is // (n-1)*(-e) < -lg2eps or (n-1)*e > lg2eps or n > ceiling(lg2eps/e)+1 int n = x == 0 ? 1 : int(ceil(lg2eps_ / e)) + 1; while (n--) // iterating from n-1 down to 0 s = x * s + (n ? 1 : 0)/Math::real(2*n + 1); } else { real xs = sqrt(fabs(x)); s = (x > 0 ? atanh(xs) : atan(xs)) / xs - 1; } return s; } // return (Datanhee(1,y) - Datanhee(1,x))/(y-x) Math::real AlbersEqualArea::DDatanhee(real x, real y) const { // This function is called with x = sphi1, y = sphi2, phi1 <= phi2, sphi2 // >= 0, abs(sphi1) <= phi2. However for safety's sake we enforce x <= y. if (y < x) swap(x, y); // ensure that x <= y real q1 = fabs(_e2), q2 = fabs(2 * _e / _e2m * (1 - x)); return x <= 0 || !(fmin(q1, q2) < real(0.75)) ? DDatanhee0(x, y) : (q1 < q2 ? DDatanhee1(x, y) : DDatanhee2(x, y)); } // Rearrange difference so that 1 - x is in the denominator, then do a // straight divided difference. Math::real AlbersEqualArea::DDatanhee0(real x, real y) const { return (Datanhee(1, y) - Datanhee(x, y))/(1 - x); } // The expansion for e2 small Math::real AlbersEqualArea::DDatanhee1(real x, real y) const { // The series in e2 is // sum( c[l] * e2^l, l, 1, N) // where // c[l] = sum( x^i * y^j; i >= 0, j >= 0, i+j < 2*l) / (2*l + 1) // = ( (x-y) - (1-y) * x^(2*l+1) + (1-x) * y^(2*l+1) ) / // ( (2*l+1) * (x-y) * (1-y) * (1-x) ) // For x = y = 1, c[l] = l // // In the limit x,y -> 1, // // DDatanhee -> e2/(1-e2)^2 = sum(l * e2^l, l, 1, inf) // // Use if e2 is sufficiently small. real s = 0; real z = 1, k = 1, t = 0, c = 0, en = 1; while (true) { t = y * t + z; c += t; z *= x; t = y * t + z; c += t; z *= x; k += 2; en *= _e2; // Here en[l] = e2^l, k[l] = 2*l + 1, // c[l] = sum( x^i * y^j; i >= 0, j >= 0, i+j < 2*l) / (2*l + 1) // Taylor expansion is // s = sum( c[l] * e2^l, l, 1, N) real ds = en * c / k; s += ds; if (!(fabs(ds) > fabs(s) * eps_/2)) break; // Iterate until the added term is sufficiently small } return s; } // The expansion for x (and y) close to 1 Math::real AlbersEqualArea::DDatanhee2(real x, real y) const { // If x and y are both close to 1, expand in Taylor series in dx = 1-x and // dy = 1-y: // // DDatanhee = sum(C_m * (dx^(m+1) - dy^(m+1)) / (dx - dy), m, 0, inf) // // where // // C_m = sum( (m+2)!! / (m+2-2*k)!! * // ((m+1)/2)! / ((m+1)/2-k)! / // (k! * (2*k-1)!!) * // e2^((m+1)/2+k), // k, 0, (m+1)/2) * (-1)^m / ((m+2) * (1-e2)^(m+2)) // for m odd, and // // C_m = sum( 2 * (m+1)!! / (m+1-2*k)!! * // (m/2+1)! / (m/2-k)! / // (k! * (2*k+1)!!) * // e2^(m/2+1+k), // k, 0, m/2)) * (-1)^m / ((m+2) * (1-e2)^(m+2)) // for m even. // // Here i!! is the double factorial extended to negative i with // i!! = (i+2)!!/(i+2). // // Note that // (dx^(m+1) - dy^(m+1)) / (dx - dy) = // dx^m + dx^(m-1)*dy ... + dx*dy^(m-1) + dy^m // // Leading (m = 0) term is e2 / (1 - e2)^2 // // Magnitude of mth term relative to the leading term scales as // // 2*(2*e/(1-e2)*dx)^m // // So use series if (2*e/(1-e2)*dx) is sufficiently small real s, dx = 1 - x, dy = 1 - y, xy = 1, yy = 1, ee = _e2 / Math::sq(_e2m); s = ee; for (int m = 1; ; ++m) { real c = m + 2, t = c; yy *= dy; // yy = dy^m xy = dx * xy + yy; // Now xy = dx^m + dx^(m-1)*dy ... + dx*dy^(m-1) + dy^m // = (dx^(m+1) - dy^(m+1)) / (dx - dy) // max value = (m+1) * max(dx,dy)^m ee /= -_e2m; if (m % 2 == 0) ee *= _e2; // Now ee = (-1)^m * e2^(floor(m/2)+1) / (1-e2)^(m+2) int kmax = (m+1)/2; for (int k = kmax - 1; k >= 0; --k) { // max coeff is less than 2^(m+1) c *= (k + 1) * (2 * (k + m - 2*kmax) + 3); c /= (kmax - k) * (2 * (kmax - k) + 1); // Horner sum for inner _e2 series t = _e2 * t + c; } // Straight sum for outer m series real ds = t * ee * xy / (m + 2); s = s + ds; if (!(fabs(ds) > fabs(s) * eps_/2)) break; // Iterate until the added term is sufficiently small } return s; } void AlbersEqualArea::Forward(real lon0, real lat, real lon, real& x, real& y, real& gamma, real& k) const { lon = Math::AngDiff(lon0, lon); lat *= _sign; real sphi, cphi; Math::sincosd(Math::LatFix(lat) * _sign, sphi, cphi); cphi = fmax(epsx_, cphi); real lam = lon * Math::degree(), tphi = sphi/cphi, txi = txif(tphi), sxi = txi/hyp(txi), dq = _qZ * Dsn(txi, _txi0, sxi, _sxi0) * (txi - _txi0), drho = - _a * dq / (sqrt(_m02 - _n0 * dq) + _nrho0 / _a), theta = _k2 * _n0 * lam, stheta = sin(theta), ctheta = cos(theta), t = _nrho0 + _n0 * drho; x = t * (_n0 != 0 ? stheta / _n0 : _k2 * lam) / _k0; y = (_nrho0 * (_n0 != 0 ? (ctheta < 0 ? 1 - ctheta : Math::sq(stheta)/(1 + ctheta)) / _n0 : 0) - drho * ctheta) / _k0; k = _k0 * (t != 0 ? t * hyp(_fm * tphi) / _a : 1); y *= _sign; gamma = _sign * theta / Math::degree(); } void AlbersEqualArea::Reverse(real lon0, real x, real y, real& lat, real& lon, real& gamma, real& k) const { y *= _sign; real nx = _k0 * _n0 * x, ny = _k0 * _n0 * y, y1 = _nrho0 - ny, den = hypot(nx, y1) + _nrho0, // 0 implies origin with polar aspect drho = den != 0 ? (_k0*x*nx - 2*_k0*y*_nrho0 + _k0*y*ny) / den : 0, // dsxia = scxi0 * dsxi dsxia = - _scxi0 * (2 * _nrho0 + _n0 * drho) * drho / (Math::sq(_a) * _qZ), txi = (_txi0 + dsxia) / sqrt(fmax(1 - dsxia * (2*_txi0 + dsxia), epsx2_)), tphi = tphif(txi), theta = atan2(nx, y1), lam = _n0 != 0 ? theta / (_k2 * _n0) : x / (y1 * _k0); gamma = _sign * theta / Math::degree(); lat = Math::atand(_sign * tphi); lon = lam / Math::degree(); lon = Math::AngNormalize(lon + Math::AngNormalize(lon0)); k = _k0 * (den != 0 ? (_nrho0 + _n0 * drho) * hyp(_fm * tphi) / _a : 1); } void AlbersEqualArea::SetScale(real lat, real k) { if (!(isfinite(k) && k > 0)) throw GeographicErr("Scale is not positive"); if (!(fabs(lat) < Math::qd)) throw GeographicErr("Latitude for SetScale not in (-" + to_string(Math::qd) + "d, " + to_string(Math::qd) + "d)"); real x, y, gamma, kold; Forward(0, lat, 0, x, y, gamma, kold); k /= kold; _k0 *= k; _k2 = Math::sq(_k0); } } // namespace GeographicLib geosphere/src/AlbersEqualArea.h0000644000176200001440000003347314323377037016207 0ustar liggesusers/** * \file AlbersEqualArea.hpp * \brief Header for GeographicLib::AlbersEqualArea class * * Copyright (c) Charles Karney (2010-2022) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ **********************************************************************/ #if !defined(GEOGRAPHICLIB_ALBERSEQUALAREA_HPP) #define GEOGRAPHICLIB_ALBERSEQUALAREA_HPP 1 #include "Constants.h" namespace GeographicLib { /** * \brief Albers equal area conic projection * * Implementation taken from the report, * - J. P. Snyder, * Map Projections: A * Working Manual, USGS Professional Paper 1395 (1987), * pp. 101--102. * * This is a implementation of the equations in Snyder except that divided * differences will be [have been] used to transform the expressions into * ones which may be evaluated accurately. [In this implementation, the * projection correctly becomes the cylindrical equal area or the azimuthal * equal area projection when the standard latitude is the equator or a * pole.] * * The ellipsoid parameters, the standard parallels, and the scale on the * standard parallels are set in the constructor. Internally, the case with * two standard parallels is converted into a single standard parallel, the * latitude of minimum azimuthal scale, with an azimuthal scale specified on * this parallel. This latitude is also used as the latitude of origin which * is returned by AlbersEqualArea::OriginLatitude. The azimuthal scale on * the latitude of origin is given by AlbersEqualArea::CentralScale. The * case with two standard parallels at opposite poles is singular and is * disallowed. The central meridian (which is a trivial shift of the * longitude) is specified as the \e lon0 argument of the * AlbersEqualArea::Forward and AlbersEqualArea::Reverse functions. * AlbersEqualArea::Forward and AlbersEqualArea::Reverse also return the * meridian convergence, γ, and azimuthal scale, \e k. A small square * aligned with the cardinal directions is projected to a rectangle with * dimensions \e k (in the E-W direction) and 1/\e k (in the N-S direction). * The E-W sides of the rectangle are oriented γ degrees * counter-clockwise from the \e x axis. There is no provision in this class * for specifying a false easting or false northing or a different latitude * of origin. * * Example of use: * \include example-AlbersEqualArea.cpp * * ConicProj is a command-line utility * providing access to the functionality of LambertConformalConic and * AlbersEqualArea. **********************************************************************/ class GEOGRAPHICLIB_EXPORT AlbersEqualArea { private: typedef Math::real real; real eps_, epsx_, epsx2_, tol_, tol0_; real _a, _f, _fm, _e2, _e, _e2m, _qZ, _qx; real _sign, _lat0, _k0; real _n0, _m02, _nrho0, _k2, _txi0, _scxi0, _sxi0; static const int numit_ = 5; // Newton iterations in Reverse static const int numit0_ = 20; // Newton iterations in Init static real hyp(real x) { using std::hypot; return hypot(real(1), x); } // atanh( e * x)/ e if f > 0 // atan (sqrt(-e2) * x)/sqrt(-e2) if f < 0 // x if f = 0 real atanhee(real x) const { using std::atan; using std::atanh; return _f > 0 ? atanh(_e * x)/_e : (_f < 0 ? (atan(_e * x)/_e) : x); } // return atanh(sqrt(x))/sqrt(x) - 1, accurate for small x static real atanhxm1(real x); // Divided differences // Definition: Df(x,y) = (f(x)-f(y))/(x-y) // See: // W. M. Kahan and R. J. Fateman, // Symbolic computation of divided differences, // SIGSAM Bull. 33(2), 7-28 (1999) // https://doi.org/10.1145/334714.334716 // http://www.cs.berkeley.edu/~fateman/papers/divdiff.pdf // // General rules // h(x) = f(g(x)): Dh(x,y) = Df(g(x),g(y))*Dg(x,y) // h(x) = f(x)*g(x): // Dh(x,y) = Df(x,y)*g(x) + Dg(x,y)*f(y) // = Df(x,y)*g(y) + Dg(x,y)*f(x) // = Df(x,y)*(g(x)+g(y))/2 + Dg(x,y)*(f(x)+f(y))/2 // // sn(x) = x/sqrt(1+x^2): Dsn(x,y) = (x+y)/((sn(x)+sn(y))*(1+x^2)*(1+y^2)) static real Dsn(real x, real y, real sx, real sy) { // sx = x/hyp(x) real t = x * y; return t > 0 ? (x + y) * Math::sq( (sx * sy)/t ) / (sx + sy) : (x - y != 0 ? (sx - sy) / (x - y) : 1); } // Datanhee(x,y) = (atanee(x)-atanee(y))/(x-y) // = atanhee((x-y)/(1-e^2*x*y))/(x-y) real Datanhee(real x, real y) const { real t = x - y, d = 1 - _e2 * x * y; return t == 0 ? 1 / d : (x*y < 0 ? atanhee(x) - atanhee(y) : atanhee(t / d)) / t; } // DDatanhee(x,y) = (Datanhee(1,y) - Datanhee(1,x))/(y-x) real DDatanhee(real x, real y) const; real DDatanhee0(real x, real y) const; real DDatanhee1(real x, real y) const; real DDatanhee2(real x, real y) const; void Init(real sphi1, real cphi1, real sphi2, real cphi2, real k1); real txif(real tphi) const; real tphif(real txi) const; friend class Ellipsoid; // For access to txif, tphif, etc. public: /** * Constructor with a single standard parallel. * * @param[in] a equatorial radius of ellipsoid (meters). * @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere. * Negative \e f gives a prolate ellipsoid. * @param[in] stdlat standard parallel (degrees), the circle of tangency. * @param[in] k0 azimuthal scale on the standard parallel. * @exception GeographicErr if \e a, (1 − \e f) \e a, or \e k0 is * not positive. * @exception GeographicErr if \e stdlat is not in [−90°, * 90°]. **********************************************************************/ AlbersEqualArea(real a, real f, real stdlat, real k0); /** * Constructor with two standard parallels. * * @param[in] a equatorial radius of ellipsoid (meters). * @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere. * Negative \e f gives a prolate ellipsoid. * @param[in] stdlat1 first standard parallel (degrees). * @param[in] stdlat2 second standard parallel (degrees). * @param[in] k1 azimuthal scale on the standard parallels. * @exception GeographicErr if \e a, (1 − \e f) \e a, or \e k1 is * not positive. * @exception GeographicErr if \e stdlat1 or \e stdlat2 is not in * [−90°, 90°], or if \e stdlat1 and \e stdlat2 are * opposite poles. **********************************************************************/ AlbersEqualArea(real a, real f, real stdlat1, real stdlat2, real k1); /** * Constructor with two standard parallels specified by sines and cosines. * * @param[in] a equatorial radius of ellipsoid (meters). * @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere. * Negative \e f gives a prolate ellipsoid. * @param[in] sinlat1 sine of first standard parallel. * @param[in] coslat1 cosine of first standard parallel. * @param[in] sinlat2 sine of second standard parallel. * @param[in] coslat2 cosine of second standard parallel. * @param[in] k1 azimuthal scale on the standard parallels. * @exception GeographicErr if \e a, (1 − \e f) \e a, or \e k1 is * not positive. * @exception GeographicErr if \e stdlat1 or \e stdlat2 is not in * [−90°, 90°], or if \e stdlat1 and \e stdlat2 are * opposite poles. * * This allows parallels close to the poles to be specified accurately. * This routine computes the latitude of origin and the azimuthal scale at * this latitude. If \e dlat = abs(\e lat2 − \e lat1) ≤ 160°, * then the error in the latitude of origin is less than 4.5 × * 10−14d;. **********************************************************************/ AlbersEqualArea(real a, real f, real sinlat1, real coslat1, real sinlat2, real coslat2, real k1); /** * Set the azimuthal scale for the projection. * * @param[in] lat (degrees). * @param[in] k azimuthal scale at latitude \e lat (default 1). * @exception GeographicErr \e k is not positive. * @exception GeographicErr if \e lat is not in (−90°, * 90°). * * This allows a "latitude of conformality" to be specified. **********************************************************************/ void SetScale(real lat, real k = real(1)); /** * Forward projection, from geographic to Lambert conformal conic. * * @param[in] lon0 central meridian longitude (degrees). * @param[in] lat latitude of point (degrees). * @param[in] lon longitude of point (degrees). * @param[out] x easting of point (meters). * @param[out] y northing of point (meters). * @param[out] gamma meridian convergence at point (degrees). * @param[out] k azimuthal scale of projection at point; the radial * scale is the 1/\e k. * * The latitude origin is given by AlbersEqualArea::LatitudeOrigin(). No * false easting or northing is added and \e lat should be in the range * [−90°, 90°]. The values of \e x and \e y returned for * points which project to infinity (i.e., one or both of the poles) will * be large but finite. **********************************************************************/ void Forward(real lon0, real lat, real lon, real& x, real& y, real& gamma, real& k) const; /** * Reverse projection, from Lambert conformal conic to geographic. * * @param[in] lon0 central meridian longitude (degrees). * @param[in] x easting of point (meters). * @param[in] y northing of point (meters). * @param[out] lat latitude of point (degrees). * @param[out] lon longitude of point (degrees). * @param[out] gamma meridian convergence at point (degrees). * @param[out] k azimuthal scale of projection at point; the radial * scale is the 1/\e k. * * The latitude origin is given by AlbersEqualArea::LatitudeOrigin(). No * false easting or northing is added. The value of \e lon returned is in * the range [−180°, 180°]. The value of \e lat returned is * in the range [−90°, 90°]. If the input point is outside * the legal projected space the nearest pole is returned. **********************************************************************/ void Reverse(real lon0, real x, real y, real& lat, real& lon, real& gamma, real& k) const; /** * AlbersEqualArea::Forward without returning the convergence and * scale. **********************************************************************/ void Forward(real lon0, real lat, real lon, real& x, real& y) const { real gamma, k; Forward(lon0, lat, lon, x, y, gamma, k); } /** * AlbersEqualArea::Reverse without returning the convergence and * scale. **********************************************************************/ void Reverse(real lon0, real x, real y, real& lat, real& lon) const { real gamma, k; Reverse(lon0, x, y, lat, lon, gamma, k); } /** \name Inspector functions **********************************************************************/ ///@{ /** * @return \e a the equatorial radius of the ellipsoid (meters). This is * the value used in the constructor. **********************************************************************/ Math::real EquatorialRadius() const { return _a; } /** * @return \e f the flattening of the ellipsoid. This is the value used in * the constructor. **********************************************************************/ Math::real Flattening() const { return _f; } /** * @return latitude of the origin for the projection (degrees). * * This is the latitude of minimum azimuthal scale and equals the \e stdlat * in the 1-parallel constructor and lies between \e stdlat1 and \e stdlat2 * in the 2-parallel constructors. **********************************************************************/ Math::real OriginLatitude() const { return _lat0; } /** * @return central scale for the projection. This is the azimuthal scale * on the latitude of origin. **********************************************************************/ Math::real CentralScale() const { return _k0; } ///@} /** * A global instantiation of AlbersEqualArea with the WGS84 ellipsoid, \e * stdlat = 0, and \e k0 = 1. This degenerates to the cylindrical equal * area projection. **********************************************************************/ static const AlbersEqualArea& CylindricalEqualArea(); /** * A global instantiation of AlbersEqualArea with the WGS84 ellipsoid, \e * stdlat = 90°, and \e k0 = 1. This degenerates to the * Lambert azimuthal equal area projection. **********************************************************************/ static const AlbersEqualArea& AzimuthalEqualAreaNorth(); /** * A global instantiation of AlbersEqualArea with the WGS84 ellipsoid, \e * stdlat = −90°, and \e k0 = 1. This degenerates to the * Lambert azimuthal equal area projection. **********************************************************************/ static const AlbersEqualArea& AzimuthalEqualAreaSouth(); }; } // namespace GeographicLib #endif // GEOGRAPHICLIB_ALBERSEQUALAREA_HPP geosphere/src/Accumulator.cpp0000644000176200001440000000113214323375571016014 0ustar liggesusers/** * \file Accumulator.cpp * \brief Implementation for GeographicLib::Accumulator class * * Copyright (c) Charles Karney (2013-2022) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ **********************************************************************/ #include "Accumulator.h" namespace GeographicLib { /// \cond SKIP // Need to instantiate Accumulator to get the code into the shared library. template class GEOGRAPHICLIB_EXPORT Accumulator; /// \endcond } // namespace GeographicLib geosphere/src/Math.h0000644000176200001440000005021614323400305014062 0ustar liggesusers/** * \file Math.hpp * \brief Header for GeographicLib::Math class * * Copyright (c) Charles Karney (2008-2022) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ **********************************************************************/ // Constants.hpp includes Math.hpp. Place this include outside Math.hpp's // include guard to enforce this ordering. #include "Constants.h" #if !defined(GEOGRAPHICLIB_MATH_HPP) #define GEOGRAPHICLIB_MATH_HPP 1 #if !defined(GEOGRAPHICLIB_WORDS_BIGENDIAN) # define GEOGRAPHICLIB_WORDS_BIGENDIAN 0 #endif #if !defined(GEOGRAPHICLIB_HAVE_LONG_DOUBLE) # define GEOGRAPHICLIB_HAVE_LONG_DOUBLE 0 #endif #if !defined(GEOGRAPHICLIB_PRECISION) /** * The precision of floating point numbers used in %GeographicLib. 1 means * float (single precision); 2 (the default) means double; 3 means long double; * 4 is reserved for quadruple precision. Nearly all the testing has been * carried out with doubles and that's the recommended configuration. In order * for long double to be used, GEOGRAPHICLIB_HAVE_LONG_DOUBLE needs to be * defined. Note that with Microsoft Visual Studio, long double is the same as * double. **********************************************************************/ # define GEOGRAPHICLIB_PRECISION 2 #endif #include #include #include #if GEOGRAPHICLIB_PRECISION == 4 #include #include #include #elif GEOGRAPHICLIB_PRECISION == 5 #include #endif #if GEOGRAPHICLIB_PRECISION > 3 // volatile keyword makes no sense for multiprec types #define GEOGRAPHICLIB_VOLATILE // Signal a convergence failure with multiprec types by throwing an exception // at loop exit. #define GEOGRAPHICLIB_PANIC \ (throw GeographicLib::GeographicErr("Convergence failure"), false) #else #define GEOGRAPHICLIB_VOLATILE volatile // Ignore convergence failures with standard floating points types by allowing // loop to exit cleanly. #define GEOGRAPHICLIB_PANIC false #endif namespace GeographicLib { /** * \brief Mathematical functions needed by %GeographicLib * * Define mathematical functions in order to localize system dependencies and * to provide generic versions of the functions. In addition define a real * type to be used by %GeographicLib. * * Example of use: * \include example-Math.cpp **********************************************************************/ class GEOGRAPHICLIB_EXPORT Math { private: void dummy(); // Static check for GEOGRAPHICLIB_PRECISION Math() = delete; // Disable constructor public: #if GEOGRAPHICLIB_HAVE_LONG_DOUBLE /** * The extended precision type for real numbers, used for some testing. * This is long double on computers with this type; otherwise it is double. **********************************************************************/ typedef long double extended; #else typedef double extended; #endif #if GEOGRAPHICLIB_PRECISION == 2 /** * The real type for %GeographicLib. Nearly all the testing has been done * with \e real = double. However, the algorithms should also work with * float and long double (where available). (CAUTION: reasonable * accuracy typically cannot be obtained using floats.) **********************************************************************/ typedef double real; #elif GEOGRAPHICLIB_PRECISION == 1 typedef float real; #elif GEOGRAPHICLIB_PRECISION == 3 typedef extended real; #elif GEOGRAPHICLIB_PRECISION == 4 typedef boost::multiprecision::float128 real; #elif GEOGRAPHICLIB_PRECISION == 5 typedef mpfr::mpreal real; #else typedef double real; #endif /** * The constants defining the standard (Babylonian) meanings of degrees, * minutes, and seconds, for angles. Read the constants as follows (for * example): \e ms = 60 is the ratio 1 minute / 1 second. The * abbreviations are * - \e t a whole turn (360°) * - \e h a half turn (180°) * - \e q a quarter turn (a right angle = 90°) * - \e d a degree * - \e m a minute * - \e s a second * . * Note that degree() is ratio 1 degree / 1 radian, thus, for example, * Math::degree() * Math::qd is the ratio 1 quarter turn / 1 radian = * π/2. * * Defining all these in one place would mean that it's simple to convert * to the centesimal system for measuring angles. The DMS class assumes * that Math::dm and Math::ms are less than or equal to 100 (so that two * digits suffice for the integer parts of the minutes and degrees * components of an angle). Switching to the centesimal convention will * break most of the tests. Also the normal definition of degree is baked * into some classes, e.g., UTMUPS, MGRS, Georef, Geohash, etc. **********************************************************************/ #if GEOGRAPHICLIB_PRECISION == 4 static const int #else enum dms { #endif qd = 90, ///< degrees per quarter turn dm = 60, ///< minutes per degree ms = 60, ///< seconds per minute hd = 2 * qd, ///< degrees per half turn td = 2 * hd, ///< degrees per turn ds = dm * ms ///< seconds per degree #if GEOGRAPHICLIB_PRECISION == 4 ; #else }; #endif /** * @return the number of bits of precision in a real number. **********************************************************************/ static int digits(); /** * Set the binary precision of a real number. * * @param[in] ndigits the number of bits of precision. * @return the resulting number of bits of precision. * * This only has an effect when GEOGRAPHICLIB_PRECISION = 5. See also * Utility::set_digits for caveats about when this routine should be * called. **********************************************************************/ static int set_digits(int ndigits); /** * @return the number of decimal digits of precision in a real number. **********************************************************************/ static int digits10(); /** * Number of additional decimal digits of precision for real relative to * double (0 for float). **********************************************************************/ static int extra_digits(); /** * true if the machine is big-endian. **********************************************************************/ static const bool bigendian = GEOGRAPHICLIB_WORDS_BIGENDIAN; /** * @tparam T the type of the returned value. * @return π. **********************************************************************/ template static T pi() { using std::atan2; static const T pi = atan2(T(0), T(-1)); return pi; } /** * @tparam T the type of the returned value. * @return the number of radians in a degree. **********************************************************************/ template static T degree() { static const T degree = pi() / T(hd); return degree; } /** * Square a number. * * @tparam T the type of the argument and the returned value. * @param[in] x * @return x2. **********************************************************************/ template static T sq(T x) { return x * x; } /** * Normalize a two-vector. * * @tparam T the type of the argument and the returned value. * @param[in,out] x on output set to x/hypot(x, y). * @param[in,out] y on output set to y/hypot(x, y). **********************************************************************/ template static void norm(T& x, T& y) { #if defined(_MSC_VER) && defined(_M_IX86) // hypot for Visual Studio (A=win32) fails monotonicity, e.g., with // x = 0.6102683302836215 // y1 = 0.7906090004346522 // y2 = y1 + 1e-16 // the test // hypot(x, y2) >= hypot(x, y1) // fails. Reported 2021-03-14: // https://developercommunity.visualstudio.com/t/1369259 // See also: // https://bugs.python.org/issue43088 using std::sqrt; T h = sqrt(x * x + y * y); #else using std::hypot; T h = hypot(x, y); #endif x /= h; y /= h; } /** * The error-free sum of two numbers. * * @tparam T the type of the argument and the returned value. * @param[in] u * @param[in] v * @param[out] t the exact error given by (\e u + \e v) - \e s. * @return \e s = round(\e u + \e v). * * See D. E. Knuth, TAOCP, Vol 2, 4.2.2, Theorem B. * * \note \e t can be the same as one of the first two arguments. **********************************************************************/ template static T sum(T u, T v, T& t); /** * Evaluate a polynomial. * * @tparam T the type of the arguments and returned value. * @param[in] N the order of the polynomial. * @param[in] p the coefficient array (of size \e N + 1) with * p0 being coefficient of xN. * @param[in] x the variable. * @return the value of the polynomial. * * Evaluate ∑n=0..N * pn xN−n. * Return 0 if \e N < 0. Return p0, if \e N = 0 (even * if \e x is infinite or a nan). The evaluation uses Horner's method. **********************************************************************/ template static T polyval(int N, const T p[], T x) { // This used to employ Math::fma; but that's too slow and it seemed not // to improve the accuracy noticeably. This might change when there's // direct hardware support for fma. T y = N < 0 ? 0 : *p++; while (--N >= 0) y = y * x + *p++; return y; } /** * Normalize an angle. * * @tparam T the type of the argument and returned value. * @param[in] x the angle in degrees. * @return the angle reduced to the range [−180°, 180°]. * * The range of \e x is unrestricted. If the result is ±0° or * ±180° then the sign is the sign of \e x. **********************************************************************/ template static T AngNormalize(T x); /** * Normalize a latitude. * * @tparam T the type of the argument and returned value. * @param[in] x the angle in degrees. * @return x if it is in the range [−90°, 90°], otherwise * return NaN. **********************************************************************/ template static T LatFix(T x) { using std::fabs; return fabs(x) > T(qd) ? NaN() : x; } /** * The exact difference of two angles reduced to * [−180°, 180°]. * * @tparam T the type of the arguments and returned value. * @param[in] x the first angle in degrees. * @param[in] y the second angle in degrees. * @param[out] e the error term in degrees. * @return \e d, the truncated value of \e y − \e x. * * This computes \e z = \e y − \e x exactly, reduced to * [−180°, 180°]; and then sets \e z = \e d + \e e where \e d * is the nearest representable number to \e z and \e e is the truncation * error. If \e z = ±0° or ±180°, then the sign of * \e d is given by the sign of \e y − \e x. The maximum absolute * value of \e e is 2−26 (for doubles). **********************************************************************/ template static T AngDiff(T x, T y, T& e); /** * Difference of two angles reduced to [−180°, 180°] * * @tparam T the type of the arguments and returned value. * @param[in] x the first angle in degrees. * @param[in] y the second angle in degrees. * @return \e y − \e x, reduced to the range [−180°, * 180°]. * * The result is equivalent to computing the difference exactly, reducing * it to [−180°, 180°] and rounding the result. **********************************************************************/ template static T AngDiff(T x, T y) { T e; return AngDiff(x, y, e); } /** * Coarsen a value close to zero. * * @tparam T the type of the argument and returned value. * @param[in] x * @return the coarsened value. * * The makes the smallest gap in \e x = 1/16 − nextafter(1/16, 0) = * 1/257 for doubles = 0.8 pm on the earth if \e x is an angle * in degrees. (This is about 2000 times more resolution than we get with * angles around 90°.) We use this to avoid having to deal with near * singular cases when \e x is non-zero but tiny (e.g., * 10−200). This sign of ±0 is preserved. **********************************************************************/ template static T AngRound(T x); /** * Evaluate the sine and cosine function with the argument in degrees * * @tparam T the type of the arguments. * @param[in] x in degrees. * @param[out] sinx sin(x). * @param[out] cosx cos(x). * * The results obey exactly the elementary properties of the trigonometric * functions, e.g., sin 9° = cos 81° = − sin 123456789°. * If x = −0 or a negative multiple of 180°, then \e sinx = * −0; this is the only case where −0 is returned. **********************************************************************/ template static void sincosd(T x, T& sinx, T& cosx); /** * Evaluate the sine and cosine with reduced argument plus correction * * @tparam T the type of the arguments. * @param[in] x reduced angle in degrees. * @param[in] t correction in degrees. * @param[out] sinx sin(x + t). * @param[out] cosx cos(x + t). * * This is a variant of Math::sincosd allowing a correction to the angle to * be supplied. \e x must be in [−180°, 180°] and \e t is * assumed to be a small correction. Math::AngRound is applied to * the reduced angle to prevent problems with \e x + \e t being extremely * close but not exactly equal to one of the four cardinal directions. **********************************************************************/ template static void sincosde(T x, T t, T& sinx, T& cosx); /** * Evaluate the sine function with the argument in degrees * * @tparam T the type of the argument and the returned value. * @param[in] x in degrees. * @return sin(x). * * The result is +0 for \e x = +0 and positive multiples of 180°. The * result is −0 for \e x = -0 and negative multiples of 180°. **********************************************************************/ template static T sind(T x); /** * Evaluate the cosine function with the argument in degrees * * @tparam T the type of the argument and the returned value. * @param[in] x in degrees. * @return cos(x). * * The result is +0 for \e x an odd multiple of 90°. **********************************************************************/ template static T cosd(T x); /** * Evaluate the tangent function with the argument in degrees * * @tparam T the type of the argument and the returned value. * @param[in] x in degrees. * @return tan(x). * * If \e x is an odd multiple of 90°, then a suitably large (but * finite) value is returned. **********************************************************************/ template static T tand(T x); /** * Evaluate the atan2 function with the result in degrees * * @tparam T the type of the arguments and the returned value. * @param[in] y * @param[in] x * @return atan2(y, x) in degrees. * * The result is in the range [−180° 180°]. N.B., * atan2d(±0, −1) = ±180°. **********************************************************************/ template static T atan2d(T y, T x); /** * Evaluate the atan function with the result in degrees * * @tparam T the type of the argument and the returned value. * @param[in] x * @return atan(x) in degrees. **********************************************************************/ template static T atand(T x); /** * Evaluate e atanh(e x) * * @tparam T the type of the argument and the returned value. * @param[in] x * @param[in] es the signed eccentricity = sign(e2) * sqrt(|e2|) * @return e atanh(e x) * * If e2 is negative (e is imaginary), the * expression is evaluated in terms of atan. **********************************************************************/ template static T eatanhe(T x, T es); /** * tanχ in terms of tanφ * * @tparam T the type of the argument and the returned value. * @param[in] tau τ = tanφ * @param[in] es the signed eccentricity = sign(e2) * sqrt(|e2|) * @return τ′ = tanχ * * See Eqs. (7--9) of * C. F. F. Karney, * * Transverse Mercator with an accuracy of a few nanometers, * J. Geodesy 85(8), 475--485 (Aug. 2011) * (preprint * arXiv:1002.1417). **********************************************************************/ template static T taupf(T tau, T es); /** * tanφ in terms of tanχ * * @tparam T the type of the argument and the returned value. * @param[in] taup τ′ = tanχ * @param[in] es the signed eccentricity = sign(e2) * sqrt(|e2|) * @return τ = tanφ * * See Eqs. (19--21) of * C. F. F. Karney, * * Transverse Mercator with an accuracy of a few nanometers, * J. Geodesy 85(8), 475--485 (Aug. 2011) * (preprint * arXiv:1002.1417). **********************************************************************/ template static T tauf(T taup, T es); /** * The NaN (not a number) * * @tparam T the type of the returned value. * @return NaN if available, otherwise return the max real of type T. **********************************************************************/ template static T NaN(); /** * Infinity * * @tparam T the type of the returned value. * @return infinity if available, otherwise return the max real. **********************************************************************/ template static T infinity(); /** * Swap the bytes of a quantity * * @tparam T the type of the argument and the returned value. * @param[in] x * @return x with its bytes swapped. **********************************************************************/ template static T swab(T x) { union { T r; unsigned char c[sizeof(T)]; } b; b.r = x; for (int i = sizeof(T)/2; i--; ) std::swap(b.c[i], b.c[sizeof(T) - 1 - i]); return b.r; } }; } // namespace GeographicLib #endif // GEOGRAPHICLIB_MATH_HPP geosphere/src/GeodesicLineExact.cpp0000644000176200001440000002622214323376011017051 0ustar liggesusers/** * \file GeodesicLineExact.cpp * \brief Implementation for GeographicLib::GeodesicLineExact class * * Copyright (c) Charles Karney (2012-2022) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ * * This is a reformulation of the geodesic problem. The notation is as * follows: * - at a general point (no suffix or 1 or 2 as suffix) * - phi = latitude * - beta = latitude on auxiliary sphere * - omega = longitude on auxiliary sphere * - lambda = longitude * - alpha = azimuth of great circle * - sigma = arc length along great circle * - s = distance * - tau = scaled distance (= sigma at multiples of pi/2) * - at northwards equator crossing * - beta = phi = 0 * - omega = lambda = 0 * - alpha = alpha0 * - sigma = s = 0 * - a 12 suffix means a difference, e.g., s12 = s2 - s1. * - s and c prefixes mean sin and cos **********************************************************************/ #include "GeodesicLineExact.h" #if defined(_MSC_VER) // Squelch warnings about mixing enums # pragma warning (disable: 5054) #endif namespace GeographicLib { using namespace std; void GeodesicLineExact::LineInit(const GeodesicExact& g, real lat1, real lon1, real azi1, real salp1, real calp1, unsigned caps) { tiny_ = g.tiny_; _lat1 = Math::LatFix(lat1); _lon1 = lon1; _azi1 = azi1; _salp1 = salp1; _calp1 = calp1; _a = g._a; _f = g._f; _b = g._b; _c2 = g._c2; _f1 = g._f1; _e2 = g._e2; _nC4 = g._nC4; // Always allow latitude and azimuth and unrolling of longitude _caps = caps | LATITUDE | AZIMUTH | LONG_UNROLL; real cbet1, sbet1; Math::sincosd(Math::AngRound(_lat1), sbet1, cbet1); sbet1 *= _f1; // Ensure cbet1 = +epsilon at poles Math::norm(sbet1, cbet1); cbet1 = fmax(tiny_, cbet1); _dn1 = (_f >= 0 ? sqrt(1 + g._ep2 * Math::sq(sbet1)) : sqrt(1 - _e2 * Math::sq(cbet1)) / _f1); // Evaluate alp0 from sin(alp1) * cos(bet1) = sin(alp0), _salp0 = _salp1 * cbet1; // alp0 in [0, pi/2 - |bet1|] // Alt: calp0 = hypot(sbet1, calp1 * cbet1). The following // is slightly better (consider the case salp1 = 0). _calp0 = hypot(_calp1, _salp1 * sbet1); // Evaluate sig with tan(bet1) = tan(sig1) * cos(alp1). // sig = 0 is nearest northward crossing of equator. // With bet1 = 0, alp1 = pi/2, we have sig1 = 0 (equatorial line). // With bet1 = pi/2, alp1 = -pi, sig1 = pi/2 // With bet1 = -pi/2, alp1 = 0 , sig1 = -pi/2 // Evaluate omg1 with tan(omg1) = sin(alp0) * tan(sig1). // With alp0 in (0, pi/2], quadrants for sig and omg coincide. // No atan2(0,0) ambiguity at poles since cbet1 = +epsilon. // With alp0 = 0, omg1 = 0 for alp1 = 0, omg1 = pi for alp1 = pi. _ssig1 = sbet1; _somg1 = _salp0 * sbet1; _csig1 = _comg1 = sbet1 != 0 || _calp1 != 0 ? cbet1 * _calp1 : 1; // Without normalization we have schi1 = somg1. _cchi1 = _f1 * _dn1 * _comg1; Math::norm(_ssig1, _csig1); // sig1 in (-pi, pi] // Math::norm(_somg1, _comg1); -- don't need to normalize! // Math::norm(_schi1, _cchi1); -- don't need to normalize! _k2 = Math::sq(_calp0) * g._ep2; _eE.Reset(-_k2, -g._ep2, 1 + _k2, 1 + g._ep2); if (_caps & CAP_E) { _eE0 = _eE.E() / (Math::pi() / 2); _eE1 = _eE.deltaE(_ssig1, _csig1, _dn1); real s = sin(_eE1), c = cos(_eE1); // tau1 = sig1 + B11 _stau1 = _ssig1 * c + _csig1 * s; _ctau1 = _csig1 * c - _ssig1 * s; // Not necessary because Einv inverts E // _eE1 = -_eE.deltaEinv(_stau1, _ctau1); } if (_caps & CAP_D) { _dD0 = _eE.D() / (Math::pi() / 2); _dD1 = _eE.deltaD(_ssig1, _csig1, _dn1); } if (_caps & CAP_H) { _hH0 = _eE.H() / (Math::pi() / 2); _hH1 = _eE.deltaH(_ssig1, _csig1, _dn1); } if (_caps & CAP_C4) { // Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0) _aA4 = Math::sq(_a) * _calp0 * _salp0 * _e2; if (_aA4 == 0) _bB41 = 0; else { GeodesicExact::I4Integrand i4(g._ep2, _k2); _cC4a.resize(_nC4); g._fft.transform(i4, _cC4a.data()); _bB41 = DST::integral(_ssig1, _csig1, _cC4a.data(), _nC4); } } _a13 = _s13 = Math::NaN(); } GeodesicLineExact::GeodesicLineExact(const GeodesicExact& g, real lat1, real lon1, real azi1, unsigned caps) { azi1 = Math::AngNormalize(azi1); real salp1, calp1; // Guard against underflow in salp0. Also -0 is converted to +0. Math::sincosd(Math::AngRound(azi1), salp1, calp1); LineInit(g, lat1, lon1, azi1, salp1, calp1, caps); } GeodesicLineExact::GeodesicLineExact(const GeodesicExact& g, real lat1, real lon1, real azi1, real salp1, real calp1, unsigned caps, bool arcmode, real s13_a13) { LineInit(g, lat1, lon1, azi1, salp1, calp1, caps); GenSetDistance(arcmode, s13_a13); } Math::real GeodesicLineExact::GenPosition(bool arcmode, real s12_a12, unsigned outmask, real& lat2, real& lon2, real& azi2, real& s12, real& m12, real& M12, real& M21, real& S12) const { outmask &= _caps & OUT_MASK; if (!( Init() && (arcmode || (_caps & (OUT_MASK & DISTANCE_IN))) )) // Uninitialized or impossible distance calculation requested return Math::NaN(); // Avoid warning about uninitialized B12. real sig12, ssig12, csig12, E2 = 0, AB1 = 0; if (arcmode) { // Interpret s12_a12 as spherical arc length sig12 = s12_a12 * Math::degree(); Math::sincosd(s12_a12, ssig12, csig12); } else { // Interpret s12_a12 as distance real tau12 = s12_a12 / (_b * _eE0), s = sin(tau12), c = cos(tau12); // tau2 = tau1 + tau12 E2 = - _eE.deltaEinv(_stau1 * c + _ctau1 * s, _ctau1 * c - _stau1 * s); sig12 = tau12 - (E2 - _eE1); ssig12 = sin(sig12); csig12 = cos(sig12); } real ssig2, csig2, sbet2, cbet2, salp2, calp2; // sig2 = sig1 + sig12 ssig2 = _ssig1 * csig12 + _csig1 * ssig12; csig2 = _csig1 * csig12 - _ssig1 * ssig12; real dn2 = _eE.Delta(ssig2, csig2); if (outmask & (DISTANCE | REDUCEDLENGTH | GEODESICSCALE)) { if (arcmode) { E2 = _eE.deltaE(ssig2, csig2, dn2); } AB1 = _eE0 * (E2 - _eE1); } // sin(bet2) = cos(alp0) * sin(sig2) sbet2 = _calp0 * ssig2; // Alt: cbet2 = hypot(csig2, salp0 * ssig2); cbet2 = hypot(_salp0, _calp0 * csig2); if (cbet2 == 0) // I.e., salp0 = 0, csig2 = 0. Break the degeneracy in this case cbet2 = csig2 = tiny_; // tan(alp0) = cos(sig2)*tan(alp2) salp2 = _salp0; calp2 = _calp0 * csig2; // No need to normalize if (outmask & DISTANCE) s12 = arcmode ? _b * (_eE0 * sig12 + AB1) : s12_a12; if (outmask & LONGITUDE) { real somg2 = _salp0 * ssig2, comg2 = csig2, // No need to normalize E = copysign(real(1), _salp0); // east-going? // Without normalization we have schi2 = somg2. real cchi2 = _f1 * dn2 * comg2; real chi12 = outmask & LONG_UNROLL ? E * (sig12 - (atan2( ssig2, csig2) - atan2( _ssig1, _csig1)) + (atan2(E * somg2, cchi2) - atan2(E * _somg1, _cchi1))) : atan2(somg2 * _cchi1 - cchi2 * _somg1, cchi2 * _cchi1 + somg2 * _somg1); real lam12 = chi12 - _e2/_f1 * _salp0 * _hH0 * (sig12 + (_eE.deltaH(ssig2, csig2, dn2) - _hH1)); real lon12 = lam12 / Math::degree(); lon2 = outmask & LONG_UNROLL ? _lon1 + lon12 : Math::AngNormalize(Math::AngNormalize(_lon1) + Math::AngNormalize(lon12)); } if (outmask & LATITUDE) lat2 = Math::atan2d(sbet2, _f1 * cbet2); if (outmask & AZIMUTH) azi2 = Math::atan2d(salp2, calp2); if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) { real J12 = _k2 * _dD0 * (sig12 + (_eE.deltaD(ssig2, csig2, dn2) - _dD1)); if (outmask & REDUCEDLENGTH) // Add parens around (_csig1 * ssig2) and (_ssig1 * csig2) to ensure // accurate cancellation in the case of coincident points. m12 = _b * ((dn2 * (_csig1 * ssig2) - _dn1 * (_ssig1 * csig2)) - _csig1 * csig2 * J12); if (outmask & GEODESICSCALE) { real t = _k2 * (ssig2 - _ssig1) * (ssig2 + _ssig1) / (_dn1 + dn2); M12 = csig12 + (t * ssig2 - csig2 * J12) * _ssig1 / _dn1; M21 = csig12 - (t * _ssig1 - _csig1 * J12) * ssig2 / dn2; } } if (outmask & AREA) { real B42 = _aA4 == 0 ? 0 : DST::integral(ssig2, csig2, _cC4a.data(), _nC4); real salp12, calp12; if (_calp0 == 0 || _salp0 == 0) { // alp12 = alp2 - alp1, used in atan2 so no need to normalize salp12 = salp2 * _calp1 - calp2 * _salp1; calp12 = calp2 * _calp1 + salp2 * _salp1; // We used to include here some patch up code that purported to deal // with nearly meridional geodesics properly. However, this turned out // to be wrong once _salp1 = -0 was allowed (via // GeodesicExact::InverseLine). In fact, the calculation of {s,c}alp12 // was already correct (following the IEEE rules for handling signed // zeros). So the patch up code was unnecessary (as well as // dangerous). } else { // tan(alp) = tan(alp0) * sec(sig) // tan(alp2-alp1) = (tan(alp2) -tan(alp1)) / (tan(alp2)*tan(alp1)+1) // = calp0 * salp0 * (csig1-csig2) / (salp0^2 + calp0^2 * csig1*csig2) // If csig12 > 0, write // csig1 - csig2 = ssig12 * (csig1 * ssig12 / (1 + csig12) + ssig1) // else // csig1 - csig2 = csig1 * (1 - csig12) + ssig12 * ssig1 // No need to normalize salp12 = _calp0 * _salp0 * (csig12 <= 0 ? _csig1 * (1 - csig12) + ssig12 * _ssig1 : ssig12 * (_csig1 * ssig12 / (1 + csig12) + _ssig1)); calp12 = Math::sq(_salp0) + Math::sq(_calp0) * _csig1 * csig2; } S12 = _c2 * atan2(salp12, calp12) + _aA4 * (B42 - _bB41); } return arcmode ? s12_a12 : sig12 / Math::degree(); } void GeodesicLineExact::SetDistance(real s13) { _s13 = s13; real t; // This will set _a13 to NaN if the GeodesicLineExact doesn't have the // DISTANCE_IN capability. _a13 = GenPosition(false, _s13, 0u, t, t, t, t, t, t, t, t); } void GeodesicLineExact::SetArc(real a13) { _a13 = a13; // In case the GeodesicLineExact doesn't have the DISTANCE capability. _s13 = Math::NaN(); real t; GenPosition(true, _a13, DISTANCE, t, t, t, _s13, t, t, t, t); } void GeodesicLineExact::GenSetDistance(bool arcmode, real s13_a13) { arcmode ? SetArc(s13_a13) : SetDistance(s13_a13); } } // namespace GeographicLib geosphere/src/EllipticFunction.cpp0000644000176200001440000004505214323376011017007 0ustar liggesusers/** * \file EllipticFunction.cpp * \brief Implementation for GeographicLib::EllipticFunction class * * Copyright (c) Charles Karney (2008-2022) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ **********************************************************************/ #include "EllipticFunction.h" #if defined(_MSC_VER) // Squelch warnings about constant conditional and enum-float expressions # pragma warning (disable: 4127 5055) #endif namespace GeographicLib { using namespace std; /* * Implementation of methods given in * * B. C. Carlson * Computation of elliptic integrals * Numerical Algorithms 10, 13-26 (1995) */ Math::real EllipticFunction::RF(real x, real y, real z) { // Carlson, eqs 2.2 - 2.7 static const real tolRF = pow(3 * numeric_limits::epsilon() * real(0.01), 1/real(8)); real A0 = (x + y + z)/3, An = A0, Q = fmax(fmax(fabs(A0-x), fabs(A0-y)), fabs(A0-z)) / tolRF, x0 = x, y0 = y, z0 = z, mul = 1; while (Q >= mul * fabs(An)) { // Max 6 trips real lam = sqrt(x0)*sqrt(y0) + sqrt(y0)*sqrt(z0) + sqrt(z0)*sqrt(x0); An = (An + lam)/4; x0 = (x0 + lam)/4; y0 = (y0 + lam)/4; z0 = (z0 + lam)/4; mul *= 4; } real X = (A0 - x) / (mul * An), Y = (A0 - y) / (mul * An), Z = - (X + Y), E2 = X*Y - Z*Z, E3 = X*Y*Z; // https://dlmf.nist.gov/19.36.E1 // Polynomial is // (1 - E2/10 + E3/14 + E2^2/24 - 3*E2*E3/44 // - 5*E2^3/208 + 3*E3^2/104 + E2^2*E3/16) // convert to Horner form... return (E3 * (6930 * E3 + E2 * (15015 * E2 - 16380) + 17160) + E2 * ((10010 - 5775 * E2) * E2 - 24024) + 240240) / (240240 * sqrt(An)); } Math::real EllipticFunction::RF(real x, real y) { // Carlson, eqs 2.36 - 2.38 static const real tolRG0 = real(2.7) * sqrt((numeric_limits::epsilon() * real(0.01))); real xn = sqrt(x), yn = sqrt(y); if (xn < yn) swap(xn, yn); while (fabs(xn-yn) > tolRG0 * xn) { // Max 4 trips real t = (xn + yn) /2; yn = sqrt(xn * yn); xn = t; } return Math::pi() / (xn + yn); } Math::real EllipticFunction::RC(real x, real y) { // Defined only for y != 0 and x >= 0. return ( !(x >= y) ? // x < y and catch nans // https://dlmf.nist.gov/19.2.E18 atan(sqrt((y - x) / x)) / sqrt(y - x) : ( x == y ? 1 / sqrt(y) : asinh( y > 0 ? // https://dlmf.nist.gov/19.2.E19 // atanh(sqrt((x - y) / x)) sqrt((x - y) / y) : // https://dlmf.nist.gov/19.2.E20 // atanh(sqrt(x / (x - y))) sqrt(-x / y) ) / sqrt(x - y) ) ); } Math::real EllipticFunction::RG(real x, real y, real z) { if (z == 0) swap(y, z); // Carlson, eq 1.7 return (z * RF(x, y, z) - (x-z) * (y-z) * RD(x, y, z) / 3 + sqrt(x * y / z)) / 2; } Math::real EllipticFunction::RG(real x, real y) { // Carlson, eqs 2.36 - 2.39 static const real tolRG0 = real(2.7) * sqrt((numeric_limits::epsilon() * real(0.01))); real x0 = sqrt(fmax(x, y)), y0 = sqrt(fmin(x, y)), xn = x0, yn = y0, s = 0, mul = real(0.25); while (fabs(xn-yn) > tolRG0 * xn) { // Max 4 trips real t = (xn + yn) /2; yn = sqrt(xn * yn); xn = t; mul *= 2; t = xn - yn; s += mul * t * t; } return (Math::sq( (x0 + y0)/2 ) - s) * Math::pi() / (2 * (xn + yn)); } Math::real EllipticFunction::RJ(real x, real y, real z, real p) { // Carlson, eqs 2.17 - 2.25 static const real tolRD = pow(real(0.2) * (numeric_limits::epsilon() * real(0.01)), 1/real(8)); real A0 = (x + y + z + 2*p)/5, An = A0, delta = (p-x) * (p-y) * (p-z), Q = fmax(fmax(fabs(A0-x), fabs(A0-y)), fmax(fabs(A0-z), fabs(A0-p))) / tolRD, x0 = x, y0 = y, z0 = z, p0 = p, mul = 1, mul3 = 1, s = 0; while (Q >= mul * fabs(An)) { // Max 7 trips real lam = sqrt(x0)*sqrt(y0) + sqrt(y0)*sqrt(z0) + sqrt(z0)*sqrt(x0), d0 = (sqrt(p0)+sqrt(x0)) * (sqrt(p0)+sqrt(y0)) * (sqrt(p0)+sqrt(z0)), e0 = delta/(mul3 * Math::sq(d0)); s += RC(1, 1 + e0)/(mul * d0); An = (An + lam)/4; x0 = (x0 + lam)/4; y0 = (y0 + lam)/4; z0 = (z0 + lam)/4; p0 = (p0 + lam)/4; mul *= 4; mul3 *= 64; } real X = (A0 - x) / (mul * An), Y = (A0 - y) / (mul * An), Z = (A0 - z) / (mul * An), P = -(X + Y + Z) / 2, E2 = X*Y + X*Z + Y*Z - 3*P*P, E3 = X*Y*Z + 2*P * (E2 + 2*P*P), E4 = (2*X*Y*Z + P * (E2 + 3*P*P)) * P, E5 = X*Y*Z*P*P; // https://dlmf.nist.gov/19.36.E2 // Polynomial is // (1 - 3*E2/14 + E3/6 + 9*E2^2/88 - 3*E4/22 - 9*E2*E3/52 + 3*E5/26 // - E2^3/16 + 3*E3^2/40 + 3*E2*E4/20 + 45*E2^2*E3/272 // - 9*(E3*E4+E2*E5)/68) return ((471240 - 540540 * E2) * E5 + (612612 * E2 - 540540 * E3 - 556920) * E4 + E3 * (306306 * E3 + E2 * (675675 * E2 - 706860) + 680680) + E2 * ((417690 - 255255 * E2) * E2 - 875160) + 4084080) / (4084080 * mul * An * sqrt(An)) + 6 * s; } Math::real EllipticFunction::RD(real x, real y, real z) { // Carlson, eqs 2.28 - 2.34 static const real tolRD = pow(real(0.2) * (numeric_limits::epsilon() * real(0.01)), 1/real(8)); real A0 = (x + y + 3*z)/5, An = A0, Q = fmax(fmax(fabs(A0-x), fabs(A0-y)), fabs(A0-z)) / tolRD, x0 = x, y0 = y, z0 = z, mul = 1, s = 0; while (Q >= mul * fabs(An)) { // Max 7 trips real lam = sqrt(x0)*sqrt(y0) + sqrt(y0)*sqrt(z0) + sqrt(z0)*sqrt(x0); s += 1/(mul * sqrt(z0) * (z0 + lam)); An = (An + lam)/4; x0 = (x0 + lam)/4; y0 = (y0 + lam)/4; z0 = (z0 + lam)/4; mul *= 4; } real X = (A0 - x) / (mul * An), Y = (A0 - y) / (mul * An), Z = -(X + Y) / 3, E2 = X*Y - 6*Z*Z, E3 = (3*X*Y - 8*Z*Z)*Z, E4 = 3 * (X*Y - Z*Z) * Z*Z, E5 = X*Y*Z*Z*Z; // https://dlmf.nist.gov/19.36.E2 // Polynomial is // (1 - 3*E2/14 + E3/6 + 9*E2^2/88 - 3*E4/22 - 9*E2*E3/52 + 3*E5/26 // - E2^3/16 + 3*E3^2/40 + 3*E2*E4/20 + 45*E2^2*E3/272 // - 9*(E3*E4+E2*E5)/68) return ((471240 - 540540 * E2) * E5 + (612612 * E2 - 540540 * E3 - 556920) * E4 + E3 * (306306 * E3 + E2 * (675675 * E2 - 706860) + 680680) + E2 * ((417690 - 255255 * E2) * E2 - 875160) + 4084080) / (4084080 * mul * An * sqrt(An)) + 3 * s; } void EllipticFunction::Reset(real k2, real alpha2, real kp2, real alphap2) { // Accept nans here (needed for GeodesicExact) if (k2 > 1) throw GeographicErr("Parameter k2 is not in (-inf, 1]"); if (alpha2 > 1) throw GeographicErr("Parameter alpha2 is not in (-inf, 1]"); if (kp2 < 0) throw GeographicErr("Parameter kp2 is not in [0, inf)"); if (alphap2 < 0) throw GeographicErr("Parameter alphap2 is not in [0, inf)"); _k2 = k2; _kp2 = kp2; _alpha2 = alpha2; _alphap2 = alphap2; _eps = _k2/Math::sq(sqrt(_kp2) + 1); // Values of complete elliptic integrals for k = 0,1 and alpha = 0,1 // K E D // k = 0: pi/2 pi/2 pi/4 // k = 1: inf 1 inf // Pi G H // k = 0, alpha = 0: pi/2 pi/2 pi/4 // k = 1, alpha = 0: inf 1 1 // k = 0, alpha = 1: inf inf pi/2 // k = 1, alpha = 1: inf inf inf // // Pi(0, k) = K(k) // G(0, k) = E(k) // H(0, k) = K(k) - D(k) // Pi(0, k) = K(k) // G(0, k) = E(k) // H(0, k) = K(k) - D(k) // Pi(alpha2, 0) = pi/(2*sqrt(1-alpha2)) // G(alpha2, 0) = pi/(2*sqrt(1-alpha2)) // H(alpha2, 0) = pi/(2*(1 + sqrt(1-alpha2))) // Pi(alpha2, 1) = inf // H(1, k) = K(k) // G(alpha2, 1) = H(alpha2, 1) = RC(1, alphap2) if (_k2 != 0) { // Complete elliptic integral K(k), Carlson eq. 4.1 // https://dlmf.nist.gov/19.25.E1 _kKc = _kp2 != 0 ? RF(_kp2, 1) : Math::infinity(); // Complete elliptic integral E(k), Carlson eq. 4.2 // https://dlmf.nist.gov/19.25.E1 _eEc = _kp2 != 0 ? 2 * RG(_kp2, 1) : 1; // D(k) = (K(k) - E(k))/k^2, Carlson eq.4.3 // https://dlmf.nist.gov/19.25.E1 _dDc = _kp2 != 0 ? RD(0, _kp2, 1) / 3 : Math::infinity(); } else { _kKc = _eEc = Math::pi()/2; _dDc = _kKc/2; } if (_alpha2 != 0) { // https://dlmf.nist.gov/19.25.E2 real rj = (_kp2 != 0 && _alphap2 != 0) ? RJ(0, _kp2, 1, _alphap2) : Math::infinity(), // Only use rc if _kp2 = 0. rc = _kp2 != 0 ? 0 : (_alphap2 != 0 ? RC(1, _alphap2) : Math::infinity()); // Pi(alpha^2, k) _pPic = _kp2 != 0 ? _kKc + _alpha2 * rj / 3 : Math::infinity(); // G(alpha^2, k) _gGc = _kp2 != 0 ? _kKc + (_alpha2 - _k2) * rj / 3 : rc; // H(alpha^2, k) _hHc = _kp2 != 0 ? _kKc - (_alphap2 != 0 ? _alphap2 * rj : 0) / 3 : rc; } else { _pPic = _kKc; _gGc = _eEc; // Hc = Kc - Dc but this involves large cancellations if k2 is close to // 1. So write (for alpha2 = 0) // Hc = int(cos(phi)^2/sqrt(1-k2*sin(phi)^2),phi,0,pi/2) // = 1/sqrt(1-k2) * int(sin(phi)^2/sqrt(1-k2/kp2*sin(phi)^2,...) // = 1/kp * D(i*k/kp) // and use D(k) = RD(0, kp2, 1) / 3 // so Hc = 1/kp * RD(0, 1/kp2, 1) / 3 // = kp2 * RD(0, 1, kp2) / 3 // using https://dlmf.nist.gov/19.20.E18 // Equivalently // RF(x, 1) - RD(0, x, 1)/3 = x * RD(0, 1, x)/3 for x > 0 // For k2 = 1 and alpha2 = 0, we have // Hc = int(cos(phi),...) = 1 _hHc = _kp2 != 0 ? _kp2 * RD(0, 1, _kp2) / 3 : 1; } } /* * Implementation of methods given in * * R. Bulirsch * Numerical Calculation of Elliptic Integrals and Elliptic Functions * Numericshe Mathematik 7, 78-90 (1965) */ void EllipticFunction::sncndn(real x, real& sn, real& cn, real& dn) const { // Bulirsch's sncndn routine, p 89. static const real tolJAC = sqrt(numeric_limits::epsilon() * real(0.01)); if (_kp2 != 0) { real mc = _kp2, d = 0; if (signbit(_kp2)) { d = 1 - mc; mc /= -d; d = sqrt(d); x *= d; } real c = 0; // To suppress warning about uninitialized variable real m[num_], n[num_]; unsigned l = 0; for (real a = 1; l < num_ || GEOGRAPHICLIB_PANIC; ++l) { // This converges quadratically. Max 5 trips m[l] = a; n[l] = mc = sqrt(mc); c = (a + mc) / 2; if (!(fabs(a - mc) > tolJAC * a)) { ++l; break; } mc *= a; a = c; } x *= c; sn = sin(x); cn = cos(x); dn = 1; if (sn != 0) { real a = cn / sn; c *= a; while (l--) { real b = m[l]; a *= c; c *= dn; dn = (n[l] + a) / (b + a); a = c / b; } a = 1 / sqrt(c*c + 1); sn = signbit(sn) ? -a : a; cn = c * sn; if (signbit(_kp2)) { swap(cn, dn); sn /= d; } } } else { sn = tanh(x); dn = cn = 1 / cosh(x); } } Math::real EllipticFunction::F(real sn, real cn, real dn) const { // Carlson, eq. 4.5 and // https://dlmf.nist.gov/19.25.E5 real cn2 = cn*cn, dn2 = dn*dn, fi = cn2 != 0 ? fabs(sn) * RF(cn2, dn2, 1) : K(); // Enforce usual trig-like symmetries if (signbit(cn)) fi = 2 * K() - fi; return copysign(fi, sn); } Math::real EllipticFunction::E(real sn, real cn, real dn) const { real cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn, ei = cn2 != 0 ? fabs(sn) * ( _k2 <= 0 ? // Carlson, eq. 4.6 and // https://dlmf.nist.gov/19.25.E9 RF(cn2, dn2, 1) - _k2 * sn2 * RD(cn2, dn2, 1) / 3 : ( _kp2 >= 0 ? // https://dlmf.nist.gov/19.25.E10 _kp2 * RF(cn2, dn2, 1) + _k2 * _kp2 * sn2 * RD(cn2, 1, dn2) / 3 + _k2 * fabs(cn) / dn : // https://dlmf.nist.gov/19.25.E11 - _kp2 * sn2 * RD(dn2, 1, cn2) / 3 + dn / fabs(cn) ) ) : E(); // Enforce usual trig-like symmetries if (signbit(cn)) ei = 2 * E() - ei; return copysign(ei, sn); } Math::real EllipticFunction::D(real sn, real cn, real dn) const { // Carlson, eq. 4.8 and // https://dlmf.nist.gov/19.25.E13 real cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn, di = cn2 != 0 ? fabs(sn) * sn2 * RD(cn2, dn2, 1) / 3 : D(); // Enforce usual trig-like symmetries if (signbit(cn)) di = 2 * D() - di; return copysign(di, sn); } Math::real EllipticFunction::Pi(real sn, real cn, real dn) const { // Carlson, eq. 4.7 and // https://dlmf.nist.gov/19.25.E14 real cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn, pii = cn2 != 0 ? fabs(sn) * (RF(cn2, dn2, 1) + _alpha2 * sn2 * RJ(cn2, dn2, 1, cn2 + _alphap2 * sn2) / 3) : Pi(); // Enforce usual trig-like symmetries if (signbit(cn)) pii = 2 * Pi() - pii; return copysign(pii, sn); } Math::real EllipticFunction::G(real sn, real cn, real dn) const { real cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn, gi = cn2 != 0 ? fabs(sn) * (RF(cn2, dn2, 1) + (_alpha2 - _k2) * sn2 * RJ(cn2, dn2, 1, cn2 + _alphap2 * sn2) / 3) : G(); // Enforce usual trig-like symmetries if (signbit(cn)) gi = 2 * G() - gi; return copysign(gi, sn); } Math::real EllipticFunction::H(real sn, real cn, real dn) const { real cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn, // WARNING: large cancellation if k2 = 1, alpha2 = 0, and phi near pi/2 hi = cn2 != 0 ? fabs(sn) * (RF(cn2, dn2, 1) - _alphap2 * sn2 * RJ(cn2, dn2, 1, cn2 + _alphap2 * sn2) / 3) : H(); // Enforce usual trig-like symmetries if (signbit(cn)) hi = 2 * H() - hi; return copysign(hi, sn); } Math::real EllipticFunction::deltaF(real sn, real cn, real dn) const { // Function is periodic with period pi if (signbit(cn)) { cn = -cn; sn = -sn; } return F(sn, cn, dn) * (Math::pi()/2) / K() - atan2(sn, cn); } Math::real EllipticFunction::deltaE(real sn, real cn, real dn) const { // Function is periodic with period pi if (signbit(cn)) { cn = -cn; sn = -sn; } return E(sn, cn, dn) * (Math::pi()/2) / E() - atan2(sn, cn); } Math::real EllipticFunction::deltaPi(real sn, real cn, real dn) const { // Function is periodic with period pi if (signbit(cn)) { cn = -cn; sn = -sn; } return Pi(sn, cn, dn) * (Math::pi()/2) / Pi() - atan2(sn, cn); } Math::real EllipticFunction::deltaD(real sn, real cn, real dn) const { // Function is periodic with period pi if (signbit(cn)) { cn = -cn; sn = -sn; } return D(sn, cn, dn) * (Math::pi()/2) / D() - atan2(sn, cn); } Math::real EllipticFunction::deltaG(real sn, real cn, real dn) const { // Function is periodic with period pi if (signbit(cn)) { cn = -cn; sn = -sn; } return G(sn, cn, dn) * (Math::pi()/2) / G() - atan2(sn, cn); } Math::real EllipticFunction::deltaH(real sn, real cn, real dn) const { // Function is periodic with period pi if (signbit(cn)) { cn = -cn; sn = -sn; } return H(sn, cn, dn) * (Math::pi()/2) / H() - atan2(sn, cn); } Math::real EllipticFunction::F(real phi) const { real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn); return fabs(phi) < Math::pi() ? F(sn, cn, dn) : (deltaF(sn, cn, dn) + phi) * K() / (Math::pi()/2); } Math::real EllipticFunction::E(real phi) const { real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn); return fabs(phi) < Math::pi() ? E(sn, cn, dn) : (deltaE(sn, cn, dn) + phi) * E() / (Math::pi()/2); } Math::real EllipticFunction::Ed(real ang) const { // ang - Math::AngNormalize(ang) is (nearly) an exact multiple of 360 real n = round((ang - Math::AngNormalize(ang))/Math::td); real sn, cn; Math::sincosd(ang, sn, cn); return E(sn, cn, Delta(sn, cn)) + 4 * E() * n; } Math::real EllipticFunction::Pi(real phi) const { real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn); return fabs(phi) < Math::pi() ? Pi(sn, cn, dn) : (deltaPi(sn, cn, dn) + phi) * Pi() / (Math::pi()/2); } Math::real EllipticFunction::D(real phi) const { real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn); return fabs(phi) < Math::pi() ? D(sn, cn, dn) : (deltaD(sn, cn, dn) + phi) * D() / (Math::pi()/2); } Math::real EllipticFunction::G(real phi) const { real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn); return fabs(phi) < Math::pi() ? G(sn, cn, dn) : (deltaG(sn, cn, dn) + phi) * G() / (Math::pi()/2); } Math::real EllipticFunction::H(real phi) const { real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn); return fabs(phi) < Math::pi() ? H(sn, cn, dn) : (deltaH(sn, cn, dn) + phi) * H() / (Math::pi()/2); } Math::real EllipticFunction::Einv(real x) const { static const real tolJAC = sqrt(numeric_limits::epsilon() * real(0.01)); real n = floor(x / (2 * _eEc) + real(0.5)); x -= 2 * _eEc * n; // x now in [-ec, ec) // Linear approximation real phi = Math::pi() * x / (2 * _eEc); // phi in [-pi/2, pi/2) // First order correction phi -= _eps * sin(2 * phi) / 2; // For kp2 close to zero use asin(x/_eEc) or // J. P. Boyd, Applied Math. and Computation 218, 7005-7013 (2012) // https://doi.org/10.1016/j.amc.2011.12.021 for (int i = 0; i < num_ || GEOGRAPHICLIB_PANIC; ++i) { real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn), err = (E(sn, cn, dn) - x)/dn; phi -= err; if (!(fabs(err) > tolJAC)) break; } return n * Math::pi() + phi; } Math::real EllipticFunction::deltaEinv(real stau, real ctau) const { // Function is periodic with period pi if (signbit(ctau)) { ctau = -ctau; stau = -stau; } real tau = atan2(stau, ctau); return Einv( tau * E() / (Math::pi()/2) ) - tau; } } // namespace GeographicLib geosphere/src/Ellipsoid.h0000644000176200001440000005653214323377037015143 0ustar liggesusers/** * \file Ellipsoid.hpp * \brief Header for GeographicLib::Ellipsoid class * * Copyright (c) Charles Karney (2012-2022) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ **********************************************************************/ #if !defined(GEOGRAPHICLIB_ELLIPSOID_HPP) #define GEOGRAPHICLIB_ELLIPSOID_HPP 1 #include "Constants.h" #include "TransverseMercator.h" #include "EllipticFunction.h" #include "AlbersEqualArea.h" namespace GeographicLib { /** * \brief Properties of an ellipsoid * * This class returns various properties of the ellipsoid and converts * between various types of latitudes. The latitude conversions are also * possible using the various projections supported by %GeographicLib; but * Ellipsoid provides more direct access (sometimes using private functions * of the projection classes). Ellipsoid::RectifyingLatitude, * Ellipsoid::InverseRectifyingLatitude, and Ellipsoid::MeridianDistance * provide functionality which can be provided by the Geodesic class. * However Geodesic uses a series approximation (valid for abs \e f < 1/150), * whereas Ellipsoid computes these quantities using EllipticFunction which * provides accurate results even when \e f is large. Use of this class * should be limited to −3 < \e f < 3/4 (i.e., 1/4 < b/a < 4). * * Example of use: * \include example-Ellipsoid.cpp **********************************************************************/ class GEOGRAPHICLIB_EXPORT Ellipsoid { private: typedef Math::real real; static const int numit_ = 10; real stol_; real _a, _f, _f1, _f12, _e2, _es, _e12, _n, _b; TransverseMercator _tm; EllipticFunction _ell; AlbersEqualArea _au; // These are the alpha and beta coefficients in the Krueger series from // TransverseMercator. Thy are used by RhumbSolve to compute // (psi2-psi1)/(mu2-mu1). const Math::real* ConformalToRectifyingCoeffs() const { return _tm._alp; } const Math::real* RectifyingToConformalCoeffs() const { return _tm._bet; } friend class Rhumb; friend class RhumbLine; public: /** \name Constructor **********************************************************************/ ///@{ /** * Constructor for an ellipsoid with * * @param[in] a equatorial radius (meters). * @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere. * Negative \e f gives a prolate ellipsoid. * @exception GeographicErr if \e a or (1 − \e f) \e a is not * positive. **********************************************************************/ Ellipsoid(real a, real f); ///@} /** \name %Ellipsoid dimensions. **********************************************************************/ ///@{ /** * @return \e a the equatorial radius of the ellipsoid (meters). This is * the value used in the constructor. **********************************************************************/ Math::real EquatorialRadius() const { return _a; } /** * @return \e b the polar semi-axis (meters). **********************************************************************/ Math::real PolarRadius() const { return _b; } /** * @return \e L the distance between the equator and a pole along a * meridian (meters). For a sphere \e L = (π/2) \e a. The radius * of a sphere with the same meridian length is \e L / (π/2). **********************************************************************/ Math::real QuarterMeridian() const; /** * @return \e A the total area of the ellipsoid (meters2). For * a sphere \e A = 4π a2. The radius of a sphere * with the same area is sqrt(\e A / (4π)). **********************************************************************/ Math::real Area() const; /** * @return \e V the total volume of the ellipsoid (meters3). * For a sphere \e V = (4π / 3) a3. The radius of * a sphere with the same volume is cbrt(\e V / (4π/3)). **********************************************************************/ Math::real Volume() const { return (4 * Math::pi()) * Math::sq(_a) * _b / 3; } ///@} /** \name %Ellipsoid shape **********************************************************************/ ///@{ /** * @return \e f = (\e a − \e b) / \e a, the flattening of the * ellipsoid. This is the value used in the constructor. This is zero, * positive, or negative for a sphere, oblate ellipsoid, or prolate * ellipsoid. **********************************************************************/ Math::real Flattening() const { return _f; } /** * @return \e f ' = (\e a − \e b) / \e b, the second flattening of * the ellipsoid. This is zero, positive, or negative for a sphere, * oblate ellipsoid, or prolate ellipsoid. **********************************************************************/ Math::real SecondFlattening() const { return _f / (1 - _f); } /** * @return \e n = (\e a − \e b) / (\e a + \e b), the third flattening * of the ellipsoid. This is zero, positive, or negative for a sphere, * oblate ellipsoid, or prolate ellipsoid. **********************************************************************/ Math::real ThirdFlattening() const { return _n; } /** * @return e2 = (a2 − * b2) / a2, the eccentricity squared * of the ellipsoid. This is zero, positive, or negative for a sphere, * oblate ellipsoid, or prolate ellipsoid. **********************************************************************/ Math::real EccentricitySq() const { return _e2; } /** * @return e' 2 = (a2 − * b2) / b2, the second eccentricity * squared of the ellipsoid. This is zero, positive, or negative for a * sphere, oblate ellipsoid, or prolate ellipsoid. **********************************************************************/ Math::real SecondEccentricitySq() const { return _e12; } /** * @return e'' 2 = (a2 − * b2) / (a2 + b2), * the third eccentricity squared of the ellipsoid. This is zero, * positive, or negative for a sphere, oblate ellipsoid, or prolate * ellipsoid. **********************************************************************/ Math::real ThirdEccentricitySq() const { return _e2 / (2 - _e2); } ///@} /** \name Latitude conversion. **********************************************************************/ ///@{ /** * @param[in] phi the geographic latitude (degrees). * @return β the parametric latitude (degrees). * * The geographic latitude, φ, is the angle between the equatorial * plane and a vector normal to the surface of the ellipsoid. * * The parametric latitude (also called the reduced latitude), β, * allows the cartesian coordinated of a meridian to be expressed * conveniently in parametric form as * - \e R = \e a cos β * - \e Z = \e b sin β * . * where \e a and \e b are the equatorial radius and the polar semi-axis. * For a sphere β = φ. * * φ must lie in the range [−90°, 90°]; the * result is undefined if this condition does not hold. The returned value * β lies in [−90°, 90°]. **********************************************************************/ Math::real ParametricLatitude(real phi) const; /** * @param[in] beta the parametric latitude (degrees). * @return φ the geographic latitude (degrees). * * β must lie in the range [−90°, 90°]; the * result is undefined if this condition does not hold. The returned value * φ lies in [−90°, 90°]. **********************************************************************/ Math::real InverseParametricLatitude(real beta) const; /** * @param[in] phi the geographic latitude (degrees). * @return θ the geocentric latitude (degrees). * * The geocentric latitude, θ, is the angle between the equatorial * plane and a line between the center of the ellipsoid and a point on the * ellipsoid. For a sphere θ = φ. * * φ must lie in the range [−90°, 90°]; the * result is undefined if this condition does not hold. The returned value * θ lies in [−90°, 90°]. **********************************************************************/ Math::real GeocentricLatitude(real phi) const; /** * @param[in] theta the geocentric latitude (degrees). * @return φ the geographic latitude (degrees). * * θ must lie in the range [−90°, 90°]; the * result is undefined if this condition does not hold. The returned value * φ lies in [−90°, 90°]. **********************************************************************/ Math::real InverseGeocentricLatitude(real theta) const; /** * @param[in] phi the geographic latitude (degrees). * @return μ the rectifying latitude (degrees). * * The rectifying latitude, μ, has the property that the distance along * a meridian of the ellipsoid between two points with rectifying latitudes * μ1 and μ2 is equal to * (μ2 - μ1) \e L / 90°, * where \e L = QuarterMeridian(). For a sphere μ = φ. * * φ must lie in the range [−90°, 90°]; the * result is undefined if this condition does not hold. The returned value * μ lies in [−90°, 90°]. **********************************************************************/ Math::real RectifyingLatitude(real phi) const; /** * @param[in] mu the rectifying latitude (degrees). * @return φ the geographic latitude (degrees). * * μ must lie in the range [−90°, 90°]; the * result is undefined if this condition does not hold. The returned value * φ lies in [−90°, 90°]. **********************************************************************/ Math::real InverseRectifyingLatitude(real mu) const; /** * @param[in] phi the geographic latitude (degrees). * @return ξ the authalic latitude (degrees). * * The authalic latitude, ξ, has the property that the area of the * ellipsoid between two circles with authalic latitudes * ξ1 and ξ2 is equal to (sin * ξ2 - sin ξ1) \e A / 2, where \e A * = Area(). For a sphere ξ = φ. * * φ must lie in the range [−90°, 90°]; the * result is undefined if this condition does not hold. The returned value * ξ lies in [−90°, 90°]. **********************************************************************/ Math::real AuthalicLatitude(real phi) const; /** * @param[in] xi the authalic latitude (degrees). * @return φ the geographic latitude (degrees). * * ξ must lie in the range [−90°, 90°]; the * result is undefined if this condition does not hold. The returned value * φ lies in [−90°, 90°]. **********************************************************************/ Math::real InverseAuthalicLatitude(real xi) const; /** * @param[in] phi the geographic latitude (degrees). * @return χ the conformal latitude (degrees). * * The conformal latitude, χ, gives the mapping of the ellipsoid to a * sphere which which is conformal (angles are preserved) and in which the * equator of the ellipsoid maps to the equator of the sphere. For a * sphere χ = φ. * * φ must lie in the range [−90°, 90°]; the * result is undefined if this condition does not hold. The returned value * χ lies in [−90°, 90°]. **********************************************************************/ Math::real ConformalLatitude(real phi) const; /** * @param[in] chi the conformal latitude (degrees). * @return φ the geographic latitude (degrees). * * χ must lie in the range [−90°, 90°]; the * result is undefined if this condition does not hold. The returned value * φ lies in [−90°, 90°]. **********************************************************************/ Math::real InverseConformalLatitude(real chi) const; /** * @param[in] phi the geographic latitude (degrees). * @return ψ the isometric latitude (degrees). * * The isometric latitude gives the mapping of the ellipsoid to a plane * which which is conformal (angles are preserved) and in which the equator * of the ellipsoid maps to a straight line of constant scale; this mapping * defines the Mercator projection. For a sphere ψ = * sinh−1 tan φ. * * φ must lie in the range [−90°, 90°]; the result is * undefined if this condition does not hold. The value returned for φ * = ±90° is some (positive or negative) large but finite value, * such that InverseIsometricLatitude returns the original value of φ. **********************************************************************/ Math::real IsometricLatitude(real phi) const; /** * @param[in] psi the isometric latitude (degrees). * @return φ the geographic latitude (degrees). * * The returned value φ lies in [−90°, 90°]. For a * sphere φ = tan−1 sinh ψ. **********************************************************************/ Math::real InverseIsometricLatitude(real psi) const; ///@} /** \name Other quantities. **********************************************************************/ ///@{ /** * @param[in] phi the geographic latitude (degrees). * @return \e R = \e a cos β the radius of a circle of latitude * φ (meters). \e R (π/180°) gives meters per degree * longitude measured along a circle of latitude. * * φ must lie in the range [−90°, 90°]; the * result is undefined if this condition does not hold. **********************************************************************/ Math::real CircleRadius(real phi) const; /** * @param[in] phi the geographic latitude (degrees). * @return \e Z = \e b sin β the distance of a circle of latitude * φ from the equator measured parallel to the ellipsoid axis * (meters). * * φ must lie in the range [−90°, 90°]; the * result is undefined if this condition does not hold. **********************************************************************/ Math::real CircleHeight(real phi) const; /** * @param[in] phi the geographic latitude (degrees). * @return \e s the distance along a meridian * between the equator and a point of latitude φ (meters). \e s is * given by \e s = μ \e L / 90°, where \e L = * QuarterMeridian()). * * φ must lie in the range [−90°, 90°]; the * result is undefined if this condition does not hold. **********************************************************************/ Math::real MeridianDistance(real phi) const; /** * @param[in] phi the geographic latitude (degrees). * @return ρ the meridional radius of curvature of the ellipsoid at * latitude φ (meters); this is the curvature of the meridian. \e * rho is given by ρ = (180°/π) d\e s / dφ, * where \e s = MeridianDistance(); thus ρ (π/180°) * gives meters per degree latitude measured along a meridian. * * φ must lie in the range [−90°, 90°]; the * result is undefined if this condition does not hold. **********************************************************************/ Math::real MeridionalCurvatureRadius(real phi) const; /** * @param[in] phi the geographic latitude (degrees). * @return ν the transverse radius of curvature of the ellipsoid at * latitude φ (meters); this is the curvature of a curve on the * ellipsoid which also lies in a plane perpendicular to the ellipsoid * and to the meridian. ν is related to \e R = CircleRadius() by \e * R = ν cos φ. * * φ must lie in the range [−90°, 90°]; the * result is undefined if this condition does not hold. **********************************************************************/ Math::real TransverseCurvatureRadius(real phi) const; /** * @param[in] phi the geographic latitude (degrees). * @param[in] azi the angle between the meridian and the normal section * (degrees). * @return the radius of curvature of the ellipsoid in the normal * section at latitude φ inclined at an angle \e azi to the * meridian (meters). * * φ must lie in the range [−90°, 90°]; the result is * undefined this condition does not hold. **********************************************************************/ Math::real NormalCurvatureRadius(real phi, real azi) const; ///@} /** \name Eccentricity conversions. **********************************************************************/ ///@{ /** * @param[in] fp = \e f ' = (\e a − \e b) / \e b, the second * flattening. * @return \e f = (\e a − \e b) / \e a, the flattening. * * \e f ' should lie in (−1, ∞). * The returned value \e f lies in (−∞, 1). **********************************************************************/ static Math::real SecondFlatteningToFlattening(real fp) { return fp / (1 + fp); } /** * @param[in] f = (\e a − \e b) / \e a, the flattening. * @return \e f ' = (\e a − \e b) / \e b, the second flattening. * * \e f should lie in (−∞, 1). * The returned value \e f ' lies in (−1, ∞). **********************************************************************/ static Math::real FlatteningToSecondFlattening(real f) { return f / (1 - f); } /** * @param[in] n = (\e a − \e b) / (\e a + \e b), the third * flattening. * @return \e f = (\e a − \e b) / \e a, the flattening. * * \e n should lie in (−1, 1). * The returned value \e f lies in (−∞, 1). **********************************************************************/ static Math::real ThirdFlatteningToFlattening(real n) { return 2 * n / (1 + n); } /** * @param[in] f = (\e a − \e b) / \e a, the flattening. * @return \e n = (\e a − \e b) / (\e a + \e b), the third * flattening. * * \e f should lie in (−∞, 1). * The returned value \e n lies in (−1, 1). **********************************************************************/ static Math::real FlatteningToThirdFlattening(real f) { return f / (2 - f); } /** * @param[in] e2 = e2 = (a2 − * b2) / a2, the eccentricity * squared. * @return \e f = (\e a − \e b) / \e a, the flattening. * * e2 should lie in (−∞, 1). * The returned value \e f lies in (−∞, 1). **********************************************************************/ static Math::real EccentricitySqToFlattening(real e2) { using std::sqrt; return e2 / (sqrt(1 - e2) + 1); } /** * @param[in] f = (\e a − \e b) / \e a, the flattening. * @return e2 = (a2 − * b2) / a2, the eccentricity * squared. * * \e f should lie in (−∞, 1). * The returned value e2 lies in (−∞, 1). **********************************************************************/ static Math::real FlatteningToEccentricitySq(real f) { return f * (2 - f); } /** * @param[in] ep2 = e' 2 = (a2 − * b2) / b2, the second eccentricity * squared. * @return \e f = (\e a − \e b) / \e a, the flattening. * * e' 2 should lie in (−1, ∞). * The returned value \e f lies in (−∞, 1). **********************************************************************/ static Math::real SecondEccentricitySqToFlattening(real ep2) { using std::sqrt; return ep2 / (sqrt(1 + ep2) + 1 + ep2); } /** * @param[in] f = (\e a − \e b) / \e a, the flattening. * @return e' 2 = (a2 − * b2) / b2, the second eccentricity * squared. * * \e f should lie in (−∞, 1). * The returned value e' 2 lies in (−1, ∞). **********************************************************************/ static Math::real FlatteningToSecondEccentricitySq(real f) { return f * (2 - f) / Math::sq(1 - f); } /** * @param[in] epp2 = e'' 2 = (a2 * − b2) / (a2 + * b2), the third eccentricity squared. * @return \e f = (\e a − \e b) / \e a, the flattening. * * e'' 2 should lie in (−1, 1). * The returned value \e f lies in (−∞, 1). **********************************************************************/ static Math::real ThirdEccentricitySqToFlattening(real epp2) { using std::sqrt; return 2 * epp2 / (sqrt((1 - epp2) * (1 + epp2)) + 1 + epp2); } /** * @param[in] f = (\e a − \e b) / \e a, the flattening. * @return e'' 2 = (a2 − * b2) / (a2 + b2), * the third eccentricity squared. * * \e f should lie in (−∞, 1). * The returned value e'' 2 lies in (−1, 1). **********************************************************************/ static Math::real FlatteningToThirdEccentricitySq(real f) { return f * (2 - f) / (1 + Math::sq(1 - f)); } ///@} /** * A global instantiation of Ellipsoid with the parameters for the WGS84 * ellipsoid. **********************************************************************/ static const Ellipsoid& WGS84(); }; } // namespace GeographicLib #endif // GEOGRAPHICLIB_ELLIPSOID_HPP geosphere/src/GeodesicExact.h0000644000176200001440000011311714323377037015717 0ustar liggesusers/** * \file GeodesicExact.hpp * \brief Header for GeographicLib::GeodesicExact class * * Copyright (c) Charles Karney (2012-2022) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ **********************************************************************/ #if !defined(GEOGRAPHICLIB_GEODESICEXACT_HPP) #define GEOGRAPHICLIB_GEODESICEXACT_HPP 1 #include "Constants.h" #include "EllipticFunction.h" #include "DST.h" #include namespace GeographicLib { class GEOGRAPHICLIB_EXPORT DST; class GeodesicLineExact; /** * \brief Exact geodesic calculations * * The equations for geodesics on an ellipsoid can be expressed in terms of * incomplete elliptic integrals. The Geodesic class expands these integrals * in a series in the flattening \e f and this provides an accurate solution * for \e f ∈ [-0.01, 0.01]. The GeodesicExact class computes the * ellitpic integrals directly and so provides a solution which is valid for * all \e f. However, in practice, its use should be limited to about * b/\e a ∈ [0.01, 100] or \e f ∈ [−99, 0.99]. * * For the WGS84 ellipsoid, these classes are 2--3 times \e slower than the * series solution and 2--3 times \e less \e accurate (because it's less easy * to control round-off errors with the elliptic integral formulation); i.e., * the error is about 40 nm (40 nanometers) instead of 15 nm. However the * error in the series solution scales as f7 while the * error in the elliptic integral solution depends weakly on \e f. If the * quarter meridian distance is 10000 km and the ratio b/\e a = 1 * − \e f is varied then the approximate maximum error (expressed as a * distance) is 
   *       1 - f  error (nm)
   *       1/128     387
   *       1/64      345
   *       1/32      269
   *       1/16      210
   *       1/8       115
   *       1/4        69
   *       1/2        36
   *         1        15
   *         2        25
   *         4        96
   *         8       318
   *        16       985
   *        32      2352
   *        64      6008
   *       128     19024
   *
* * The area in this classes is computing by finding an accurate approximation * to the area integrand using a discrete sine transform fitting \e N equally * spaced points in σ. \e N chosen to ensure full accuracy for * b/\e a ∈ [0.01, 100] or \e f ∈ [−99, 0.99]. * * See \ref geodellip for the formulation. See the documentation on the * Geodesic class for additional information on the geodesic problems. * * Example of use: * \include example-GeodesicExact.cpp * * GeodSolve is a command-line utility * providing access to the functionality of GeodesicExact and * GeodesicLineExact (via the -E option). **********************************************************************/ class GEOGRAPHICLIB_EXPORT GeodesicExact { private: typedef Math::real real; friend class GeodesicLineExact; static const unsigned maxit1_ = 20; unsigned maxit2_; real tiny_, tol0_, tol1_, tol2_, tolb_, xthresh_; enum captype { CAP_NONE = 0U, CAP_E = 1U<<0, // Skip 1U<<1 for compatibility with Geodesic (not required) CAP_D = 1U<<2, CAP_H = 1U<<3, CAP_C4 = 1U<<4, CAP_ALL = 0x1FU, CAP_MASK = CAP_ALL, OUT_ALL = 0x7F80U, OUT_MASK = 0xFF80U, // Includes LONG_UNROLL }; static real Astroid(real x, real y); real _a, _f, _f1, _e2, _ep2, _n, _b, _c2, _etol2; int _nC4; DST _fft; void Lengths(const EllipticFunction& E, real sig12, real ssig1, real csig1, real dn1, real ssig2, real csig2, real dn2, real cbet1, real cbet2, unsigned outmask, real& s12s, real& m12a, real& m0, real& M12, real& M21) const; real InverseStart(EllipticFunction& E, real sbet1, real cbet1, real dn1, real sbet2, real cbet2, real dn2, real lam12, real slam12, real clam12, real& salp1, real& calp1, real& salp2, real& calp2, real& dnm) const; real Lambda12(real sbet1, real cbet1, real dn1, real sbet2, real cbet2, real dn2, real salp1, real calp1, real slam120, real clam120, real& salp2, real& calp2, real& sig12, real& ssig1, real& csig1, real& ssig2, real& csig2, EllipticFunction& E, real& domg12, bool diffp, real& dlam12) const; real GenInverse(real lat1, real lon1, real lat2, real lon2, unsigned outmask, real& s12, real& salp1, real& calp1, real& salp2, real& calp2, real& m12, real& M12, real& M21, real& S12) const; class I4Integrand { private: real X, tX, tdX, sX, sX1, sXX1, asinhsX, _k2; static real asinhsqrt(real x); static real t(real x); static real td(real x); // static real Dt(real x, real y); real DtX(real y) const; public: I4Integrand(real ep2, real k2); real operator()(real sig) const; }; public: /** * Bit masks for what calculations to do. These masks do double duty. * They signify to the GeodesicLineExact::GeodesicLineExact constructor and * to GeodesicExact::Line what capabilities should be included in the * GeodesicLineExact object. They also specify which results to return in * the general routines GeodesicExact::GenDirect and * GeodesicExact::GenInverse routines. GeodesicLineExact::mask is a * duplication of this enum. **********************************************************************/ enum mask { /** * No capabilities, no output. * @hideinitializer **********************************************************************/ NONE = 0U, /** * Calculate latitude \e lat2. (It's not necessary to include this as a * capability to GeodesicLineExact because this is included by default.) * @hideinitializer **********************************************************************/ LATITUDE = 1U<<7 | CAP_NONE, /** * Calculate longitude \e lon2. * @hideinitializer **********************************************************************/ LONGITUDE = 1U<<8 | CAP_H, /** * Calculate azimuths \e azi1 and \e azi2. (It's not necessary to * include this as a capability to GeodesicLineExact because this is * included by default.) * @hideinitializer **********************************************************************/ AZIMUTH = 1U<<9 | CAP_NONE, /** * Calculate distance \e s12. * @hideinitializer **********************************************************************/ DISTANCE = 1U<<10 | CAP_E, /** * A combination of the common capabilities: GeodesicExact::LATITUDE, * GeodesicExact::LONGITUDE, GeodesicExact::AZIMUTH, * GeodesicExact::DISTANCE. * @hideinitializer **********************************************************************/ STANDARD = LATITUDE | LONGITUDE | AZIMUTH | DISTANCE, /** * Allow distance \e s12 to be used as input in the direct geodesic * problem. * @hideinitializer **********************************************************************/ DISTANCE_IN = 1U<<11 | CAP_E, /** * Calculate reduced length \e m12. * @hideinitializer **********************************************************************/ REDUCEDLENGTH = 1U<<12 | CAP_D, /** * Calculate geodesic scales \e M12 and \e M21. * @hideinitializer **********************************************************************/ GEODESICSCALE = 1U<<13 | CAP_D, /** * Calculate area \e S12. * @hideinitializer **********************************************************************/ AREA = 1U<<14 | CAP_C4, /** * Unroll \e lon2 in the direct calculation. * @hideinitializer **********************************************************************/ LONG_UNROLL = 1U<<15, /** * All capabilities, calculate everything. (GeodesicExact::LONG_UNROLL * is not included in this mask.) * @hideinitializer **********************************************************************/ ALL = OUT_ALL| CAP_ALL, }; /** \name Constructor **********************************************************************/ ///@{ /** * Constructor for an ellipsoid with * * @param[in] a equatorial radius (meters). * @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere. * Negative \e f gives a prolate ellipsoid. * @exception GeographicErr if \e a or (1 − \e f) \e a is not * positive. **********************************************************************/ GeodesicExact(real a, real f); ///@} /** \name Direct geodesic problem specified in terms of distance. **********************************************************************/ ///@{ /** * Perform the direct geodesic calculation where the length of the geodesic * is specified in terms of distance. * * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] azi1 azimuth at point 1 (degrees). * @param[in] s12 distance between point 1 and point 2 (meters); it can be * signed. * @param[out] lat2 latitude of point 2 (degrees). * @param[out] lon2 longitude of point 2 (degrees). * @param[out] azi2 (forward) azimuth at point 2 (degrees). * @param[out] m12 reduced length of geodesic (meters). * @param[out] M12 geodesic scale of point 2 relative to point 1 * (dimensionless). * @param[out] M21 geodesic scale of point 1 relative to point 2 * (dimensionless). * @param[out] S12 area under the geodesic (meters2). * @return \e a12 arc length of between point 1 and point 2 (degrees). * * \e lat1 should be in the range [−90°, 90°]. The values of * \e lon2 and \e azi2 returned are in the range [−180°, * 180°]. * * If either point is at a pole, the azimuth is defined by keeping the * longitude fixed, writing \e lat = ±(90° − ε), * and taking the limit ε → 0+. An arc length greater that * 180° signifies a geodesic which is not a shortest path. (For a * prolate ellipsoid, an additional condition is necessary for a shortest * path: the longitudinal extent must not exceed of 180°.) * * The following functions are overloaded versions of GeodesicExact::Direct * which omit some of the output parameters. Note, however, that the arc * length is always computed and returned as the function value. **********************************************************************/ Math::real Direct(real lat1, real lon1, real azi1, real s12, real& lat2, real& lon2, real& azi2, real& m12, real& M12, real& M21, real& S12) const { real t; return GenDirect(lat1, lon1, azi1, false, s12, LATITUDE | LONGITUDE | AZIMUTH | REDUCEDLENGTH | GEODESICSCALE | AREA, lat2, lon2, azi2, t, m12, M12, M21, S12); } /** * See the documentation for GeodesicExact::Direct. **********************************************************************/ Math::real Direct(real lat1, real lon1, real azi1, real s12, real& lat2, real& lon2) const { real t; return GenDirect(lat1, lon1, azi1, false, s12, LATITUDE | LONGITUDE, lat2, lon2, t, t, t, t, t, t); } /** * See the documentation for GeodesicExact::Direct. **********************************************************************/ Math::real Direct(real lat1, real lon1, real azi1, real s12, real& lat2, real& lon2, real& azi2) const { real t; return GenDirect(lat1, lon1, azi1, false, s12, LATITUDE | LONGITUDE | AZIMUTH, lat2, lon2, azi2, t, t, t, t, t); } /** * See the documentation for GeodesicExact::Direct. **********************************************************************/ Math::real Direct(real lat1, real lon1, real azi1, real s12, real& lat2, real& lon2, real& azi2, real& m12) const { real t; return GenDirect(lat1, lon1, azi1, false, s12, LATITUDE | LONGITUDE | AZIMUTH | REDUCEDLENGTH, lat2, lon2, azi2, t, m12, t, t, t); } /** * See the documentation for GeodesicExact::Direct. **********************************************************************/ Math::real Direct(real lat1, real lon1, real azi1, real s12, real& lat2, real& lon2, real& azi2, real& M12, real& M21) const { real t; return GenDirect(lat1, lon1, azi1, false, s12, LATITUDE | LONGITUDE | AZIMUTH | GEODESICSCALE, lat2, lon2, azi2, t, t, M12, M21, t); } /** * See the documentation for GeodesicExact::Direct. **********************************************************************/ Math::real Direct(real lat1, real lon1, real azi1, real s12, real& lat2, real& lon2, real& azi2, real& m12, real& M12, real& M21) const { real t; return GenDirect(lat1, lon1, azi1, false, s12, LATITUDE | LONGITUDE | AZIMUTH | REDUCEDLENGTH | GEODESICSCALE, lat2, lon2, azi2, t, m12, M12, M21, t); } ///@} /** \name Direct geodesic problem specified in terms of arc length. **********************************************************************/ ///@{ /** * Perform the direct geodesic calculation where the length of the geodesic * is specified in terms of arc length. * * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] azi1 azimuth at point 1 (degrees). * @param[in] a12 arc length between point 1 and point 2 (degrees); it can * be signed. * @param[out] lat2 latitude of point 2 (degrees). * @param[out] lon2 longitude of point 2 (degrees). * @param[out] azi2 (forward) azimuth at point 2 (degrees). * @param[out] s12 distance between point 1 and point 2 (meters). * @param[out] m12 reduced length of geodesic (meters). * @param[out] M12 geodesic scale of point 2 relative to point 1 * (dimensionless). * @param[out] M21 geodesic scale of point 1 relative to point 2 * (dimensionless). * @param[out] S12 area under the geodesic (meters2). * * \e lat1 should be in the range [−90°, 90°]. The values of * \e lon2 and \e azi2 returned are in the range [−180°, * 180°]. * * If either point is at a pole, the azimuth is defined by keeping the * longitude fixed, writing \e lat = ±(90° − ε), * and taking the limit ε → 0+. An arc length greater that * 180° signifies a geodesic which is not a shortest path. (For a * prolate ellipsoid, an additional condition is necessary for a shortest * path: the longitudinal extent must not exceed of 180°.) * * The following functions are overloaded versions of GeodesicExact::Direct * which omit some of the output parameters. **********************************************************************/ void ArcDirect(real lat1, real lon1, real azi1, real a12, real& lat2, real& lon2, real& azi2, real& s12, real& m12, real& M12, real& M21, real& S12) const { GenDirect(lat1, lon1, azi1, true, a12, LATITUDE | LONGITUDE | AZIMUTH | DISTANCE | REDUCEDLENGTH | GEODESICSCALE | AREA, lat2, lon2, azi2, s12, m12, M12, M21, S12); } /** * See the documentation for GeodesicExact::ArcDirect. **********************************************************************/ void ArcDirect(real lat1, real lon1, real azi1, real a12, real& lat2, real& lon2) const { real t; GenDirect(lat1, lon1, azi1, true, a12, LATITUDE | LONGITUDE, lat2, lon2, t, t, t, t, t, t); } /** * See the documentation for GeodesicExact::ArcDirect. **********************************************************************/ void ArcDirect(real lat1, real lon1, real azi1, real a12, real& lat2, real& lon2, real& azi2) const { real t; GenDirect(lat1, lon1, azi1, true, a12, LATITUDE | LONGITUDE | AZIMUTH, lat2, lon2, azi2, t, t, t, t, t); } /** * See the documentation for GeodesicExact::ArcDirect. **********************************************************************/ void ArcDirect(real lat1, real lon1, real azi1, real a12, real& lat2, real& lon2, real& azi2, real& s12) const { real t; GenDirect(lat1, lon1, azi1, true, a12, LATITUDE | LONGITUDE | AZIMUTH | DISTANCE, lat2, lon2, azi2, s12, t, t, t, t); } /** * See the documentation for GeodesicExact::ArcDirect. **********************************************************************/ void ArcDirect(real lat1, real lon1, real azi1, real a12, real& lat2, real& lon2, real& azi2, real& s12, real& m12) const { real t; GenDirect(lat1, lon1, azi1, true, a12, LATITUDE | LONGITUDE | AZIMUTH | DISTANCE | REDUCEDLENGTH, lat2, lon2, azi2, s12, m12, t, t, t); } /** * See the documentation for GeodesicExact::ArcDirect. **********************************************************************/ void ArcDirect(real lat1, real lon1, real azi1, real a12, real& lat2, real& lon2, real& azi2, real& s12, real& M12, real& M21) const { real t; GenDirect(lat1, lon1, azi1, true, a12, LATITUDE | LONGITUDE | AZIMUTH | DISTANCE | GEODESICSCALE, lat2, lon2, azi2, s12, t, M12, M21, t); } /** * See the documentation for GeodesicExact::ArcDirect. **********************************************************************/ void ArcDirect(real lat1, real lon1, real azi1, real a12, real& lat2, real& lon2, real& azi2, real& s12, real& m12, real& M12, real& M21) const { real t; GenDirect(lat1, lon1, azi1, true, a12, LATITUDE | LONGITUDE | AZIMUTH | DISTANCE | REDUCEDLENGTH | GEODESICSCALE, lat2, lon2, azi2, s12, m12, M12, M21, t); } ///@} /** \name General version of the direct geodesic solution. **********************************************************************/ ///@{ /** * The general direct geodesic calculation. GeodesicExact::Direct and * GeodesicExact::ArcDirect are defined in terms of this function. * * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] azi1 azimuth at point 1 (degrees). * @param[in] arcmode boolean flag determining the meaning of the second * parameter. * @param[in] s12_a12 if \e arcmode is false, this is the distance between * point 1 and point 2 (meters); otherwise it is the arc length between * point 1 and point 2 (degrees); it can be signed. * @param[in] outmask a bitor'ed combination of GeodesicExact::mask values * specifying which of the following parameters should be set. * @param[out] lat2 latitude of point 2 (degrees). * @param[out] lon2 longitude of point 2 (degrees). * @param[out] azi2 (forward) azimuth at point 2 (degrees). * @param[out] s12 distance between point 1 and point 2 (meters). * @param[out] m12 reduced length of geodesic (meters). * @param[out] M12 geodesic scale of point 2 relative to point 1 * (dimensionless). * @param[out] M21 geodesic scale of point 1 relative to point 2 * (dimensionless). * @param[out] S12 area under the geodesic (meters2). * @return \e a12 arc length of between point 1 and point 2 (degrees). * * The GeodesicExact::mask values possible for \e outmask are * - \e outmask |= GeodesicExact::LATITUDE for the latitude \e lat2; * - \e outmask |= GeodesicExact::LONGITUDE for the latitude \e lon2; * - \e outmask |= GeodesicExact::AZIMUTH for the latitude \e azi2; * - \e outmask |= GeodesicExact::DISTANCE for the distance \e s12; * - \e outmask |= GeodesicExact::REDUCEDLENGTH for the reduced length \e * m12; * - \e outmask |= GeodesicExact::GEODESICSCALE for the geodesic scales \e * M12 and \e M21; * - \e outmask |= GeodesicExact::AREA for the area \e S12; * - \e outmask |= GeodesicExact::ALL for all of the above; * - \e outmask |= GeodesicExact::LONG_UNROLL to unroll \e lon2 instead of * wrapping it into the range [−180°, 180°]. * . * The function value \e a12 is always computed and returned and this * equals \e s12_a12 is \e arcmode is true. If \e outmask includes * GeodesicExact::DISTANCE and \e arcmode is false, then \e s12 = \e * s12_a12. It is not necessary to include GeodesicExact::DISTANCE_IN in * \e outmask; this is automatically included is \e arcmode is false. * * With the GeodesicExact::LONG_UNROLL bit set, the quantity \e lon2 * − \e lon1 indicates how many times and in what sense the geodesic * encircles the ellipsoid. **********************************************************************/ Math::real GenDirect(real lat1, real lon1, real azi1, bool arcmode, real s12_a12, unsigned outmask, real& lat2, real& lon2, real& azi2, real& s12, real& m12, real& M12, real& M21, real& S12) const; ///@} /** \name Inverse geodesic problem. **********************************************************************/ ///@{ /** * Perform the inverse geodesic calculation. * * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] lat2 latitude of point 2 (degrees). * @param[in] lon2 longitude of point 2 (degrees). * @param[out] s12 distance between point 1 and point 2 (meters). * @param[out] azi1 azimuth at point 1 (degrees). * @param[out] azi2 (forward) azimuth at point 2 (degrees). * @param[out] m12 reduced length of geodesic (meters). * @param[out] M12 geodesic scale of point 2 relative to point 1 * (dimensionless). * @param[out] M21 geodesic scale of point 1 relative to point 2 * (dimensionless). * @param[out] S12 area under the geodesic (meters2). * @return \e a12 arc length of between point 1 and point 2 (degrees). * * \e lat1 and \e lat2 should be in the range [−90°, 90°]. * The values of \e azi1 and \e azi2 returned are in the range * [−180°, 180°]. * * If either point is at a pole, the azimuth is defined by keeping the * longitude fixed, writing \e lat = ±(90° − ε), * and taking the limit ε → 0+. * * The following functions are overloaded versions of * GeodesicExact::Inverse which omit some of the output parameters. Note, * however, that the arc length is always computed and returned as the * function value. **********************************************************************/ Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real& s12, real& azi1, real& azi2, real& m12, real& M12, real& M21, real& S12) const { return GenInverse(lat1, lon1, lat2, lon2, DISTANCE | AZIMUTH | REDUCEDLENGTH | GEODESICSCALE | AREA, s12, azi1, azi2, m12, M12, M21, S12); } /** * See the documentation for GeodesicExact::Inverse. **********************************************************************/ Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real& s12) const { real t; return GenInverse(lat1, lon1, lat2, lon2, DISTANCE, s12, t, t, t, t, t, t); } /** * See the documentation for GeodesicExact::Inverse. **********************************************************************/ Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real& azi1, real& azi2) const { real t; return GenInverse(lat1, lon1, lat2, lon2, AZIMUTH, t, azi1, azi2, t, t, t, t); } /** * See the documentation for GeodesicExact::Inverse. **********************************************************************/ Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real& s12, real& azi1, real& azi2) const { real t; return GenInverse(lat1, lon1, lat2, lon2, DISTANCE | AZIMUTH, s12, azi1, azi2, t, t, t, t); } /** * See the documentation for GeodesicExact::Inverse. **********************************************************************/ Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real& s12, real& azi1, real& azi2, real& m12) const { real t; return GenInverse(lat1, lon1, lat2, lon2, DISTANCE | AZIMUTH | REDUCEDLENGTH, s12, azi1, azi2, m12, t, t, t); } /** * See the documentation for GeodesicExact::Inverse. **********************************************************************/ Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real& s12, real& azi1, real& azi2, real& M12, real& M21) const { real t; return GenInverse(lat1, lon1, lat2, lon2, DISTANCE | AZIMUTH | GEODESICSCALE, s12, azi1, azi2, t, M12, M21, t); } /** * See the documentation for GeodesicExact::Inverse. **********************************************************************/ Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real& s12, real& azi1, real& azi2, real& m12, real& M12, real& M21) const { real t; return GenInverse(lat1, lon1, lat2, lon2, DISTANCE | AZIMUTH | REDUCEDLENGTH | GEODESICSCALE, s12, azi1, azi2, m12, M12, M21, t); } ///@} /** \name General version of inverse geodesic solution. **********************************************************************/ ///@{ /** * The general inverse geodesic calculation. GeodesicExact::Inverse is * defined in terms of this function. * * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] lat2 latitude of point 2 (degrees). * @param[in] lon2 longitude of point 2 (degrees). * @param[in] outmask a bitor'ed combination of GeodesicExact::mask values * specifying which of the following parameters should be set. * @param[out] s12 distance between point 1 and point 2 (meters). * @param[out] azi1 azimuth at point 1 (degrees). * @param[out] azi2 (forward) azimuth at point 2 (degrees). * @param[out] m12 reduced length of geodesic (meters). * @param[out] M12 geodesic scale of point 2 relative to point 1 * (dimensionless). * @param[out] M21 geodesic scale of point 1 relative to point 2 * (dimensionless). * @param[out] S12 area under the geodesic (meters2). * @return \e a12 arc length of between point 1 and point 2 (degrees). * * The GeodesicExact::mask values possible for \e outmask are * - \e outmask |= GeodesicExact::DISTANCE for the distance \e s12; * - \e outmask |= GeodesicExact::AZIMUTH for the latitude \e azi2; * - \e outmask |= GeodesicExact::REDUCEDLENGTH for the reduced length \e * m12; * - \e outmask |= GeodesicExact::GEODESICSCALE for the geodesic scales \e * M12 and \e M21; * - \e outmask |= GeodesicExact::AREA for the area \e S12; * - \e outmask |= GeodesicExact::ALL for all of the above. * . * The arc length is always computed and returned as the function value. **********************************************************************/ Math::real GenInverse(real lat1, real lon1, real lat2, real lon2, unsigned outmask, real& s12, real& azi1, real& azi2, real& m12, real& M12, real& M21, real& S12) const; ///@} /** \name Interface to GeodesicLineExact. **********************************************************************/ ///@{ /** * Set up to compute several points on a single geodesic. * * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] azi1 azimuth at point 1 (degrees). * @param[in] caps bitor'ed combination of GeodesicExact::mask values * specifying the capabilities the GeodesicLineExact object should * possess, i.e., which quantities can be returned in calls to * GeodesicLineExact::Position. * @return a GeodesicLineExact object. * * \e lat1 should be in the range [−90°, 90°]. * * The GeodesicExact::mask values are * - \e caps |= GeodesicExact::LATITUDE for the latitude \e lat2; this is * added automatically; * - \e caps |= GeodesicExact::LONGITUDE for the latitude \e lon2; * - \e caps |= GeodesicExact::AZIMUTH for the azimuth \e azi2; this is * added automatically; * - \e caps |= GeodesicExact::DISTANCE for the distance \e s12; * - \e caps |= GeodesicExact::REDUCEDLENGTH for the reduced length \e m12; * - \e caps |= GeodesicExact::GEODESICSCALE for the geodesic scales \e M12 * and \e M21; * - \e caps |= GeodesicExact::AREA for the area \e S12; * - \e caps |= GeodesicExact::DISTANCE_IN permits the length of the * geodesic to be given in terms of \e s12; without this capability the * length can only be specified in terms of arc length; * - \e caps |= GeodesicExact::ALL for all of the above. * . * The default value of \e caps is GeodesicExact::ALL which turns on all * the capabilities. * * If the point is at a pole, the azimuth is defined by keeping \e lon1 * fixed, writing \e lat1 = ±(90 − ε), and taking the * limit ε → 0+. **********************************************************************/ GeodesicLineExact Line(real lat1, real lon1, real azi1, unsigned caps = ALL) const; /** * Define a GeodesicLineExact in terms of the inverse geodesic problem. * * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] lat2 latitude of point 2 (degrees). * @param[in] lon2 longitude of point 2 (degrees). * @param[in] caps bitor'ed combination of GeodesicExact::mask values * specifying the capabilities the GeodesicLineExact object should * possess, i.e., which quantities can be returned in calls to * GeodesicLineExact::Position. * @return a GeodesicLineExact object. * * This function sets point 3 of the GeodesicLineExact to correspond to * point 2 of the inverse geodesic problem. * * \e lat1 and \e lat2 should be in the range [−90°, 90°]. **********************************************************************/ GeodesicLineExact InverseLine(real lat1, real lon1, real lat2, real lon2, unsigned caps = ALL) const; /** * Define a GeodesicLineExact in terms of the direct geodesic problem * specified in terms of distance. * * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] azi1 azimuth at point 1 (degrees). * @param[in] s12 distance between point 1 and point 2 (meters); it can be * negative. * @param[in] caps bitor'ed combination of GeodesicExact::mask values * specifying the capabilities the GeodesicLineExact object should * possess, i.e., which quantities can be returned in calls to * GeodesicLineExact::Position. * @return a GeodesicLineExact object. * * This function sets point 3 of the GeodesicLineExact to correspond to * point 2 of the direct geodesic problem. * * \e lat1 should be in the range [−90°, 90°]. **********************************************************************/ GeodesicLineExact DirectLine(real lat1, real lon1, real azi1, real s12, unsigned caps = ALL) const; /** * Define a GeodesicLineExact in terms of the direct geodesic problem * specified in terms of arc length. * * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] azi1 azimuth at point 1 (degrees). * @param[in] a12 arc length between point 1 and point 2 (degrees); it can * be negative. * @param[in] caps bitor'ed combination of GeodesicExact::mask values * specifying the capabilities the GeodesicLineExact object should * possess, i.e., which quantities can be returned in calls to * GeodesicLineExact::Position. * @return a GeodesicLineExact object. * * This function sets point 3 of the GeodesicLineExact to correspond to * point 2 of the direct geodesic problem. * * \e lat1 should be in the range [−90°, 90°]. **********************************************************************/ GeodesicLineExact ArcDirectLine(real lat1, real lon1, real azi1, real a12, unsigned caps = ALL) const; /** * Define a GeodesicLineExact in terms of the direct geodesic problem * specified in terms of either distance or arc length. * * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] azi1 azimuth at point 1 (degrees). * @param[in] arcmode boolean flag determining the meaning of the \e * s12_a12. * @param[in] s12_a12 if \e arcmode is false, this is the distance between * point 1 and point 2 (meters); otherwise it is the arc length between * point 1 and point 2 (degrees); it can be negative. * @param[in] caps bitor'ed combination of GeodesicExact::mask values * specifying the capabilities the GeodesicLineExact object should * possess, i.e., which quantities can be returned in calls to * GeodesicLineExact::Position. * @return a GeodesicLineExact object. * * This function sets point 3 of the GeodesicLineExact to correspond to * point 2 of the direct geodesic problem. * * \e lat1 should be in the range [−90°, 90°]. **********************************************************************/ GeodesicLineExact GenDirectLine(real lat1, real lon1, real azi1, bool arcmode, real s12_a12, unsigned caps = ALL) const; ///@} /** \name Inspector functions. **********************************************************************/ ///@{ /** * @return \e a the equatorial radius of the ellipsoid (meters). This is * the value used in the constructor. **********************************************************************/ Math::real EquatorialRadius() const { return _a; } /** * @return \e f the flattening of the ellipsoid. This is the * value used in the constructor. **********************************************************************/ Math::real Flattening() const { return _f; } /** * @return total area of ellipsoid in meters2. The area of a * polygon encircling a pole can be found by adding * GeodesicExact::EllipsoidArea()/2 to the sum of \e S12 for each side of * the polygon. **********************************************************************/ Math::real EllipsoidArea() const { return 4 * Math::pi() * _c2; } ///@} /** * A global instantiation of GeodesicExact with the parameters for the * WGS84 ellipsoid. **********************************************************************/ static const GeodesicExact& WGS84(); }; } // namespace GeographicLib #endif // GEOGRAPHICLIB_GEODESICEXACT_HPP geosphere/src/Config.h0000644000176200001440000000063014064202407014376 0ustar liggesusers// This will be overwritten by ./configure #define GEOGRAPHICLIB_VERSION_STRING "1.52" #define GEOGRAPHICLIB_VERSION_MAJOR 1 #define GEOGRAPHICLIB_VERSION_MINOR 52 #define GEOGRAPHICLIB_VERSION_PATCH 0 // Undefine HAVE_LONG_DOUBLE if this type is unknown to the compiler #define GEOGRAPHICLIB_HAVE_LONG_DOUBLE 1 // Define WORDS_BIGENDIAN to be 1 if your machine is big endian /* #undef WORDS_BIGENDIAN */ geosphere/src/Constants.h0000644000176200001440000003171114323377171015162 0ustar liggesusers/** * \file Constants.hpp * \brief Header for GeographicLib::Constants class * * Copyright (c) Charles Karney (2008-2022) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ **********************************************************************/ #if !defined(GEOGRAPHICLIB_CONSTANTS_HPP) #define GEOGRAPHICLIB_CONSTANTS_HPP 1 #include "Config.h" /** * @relates GeographicLib::Constants * Pack the version components into a single integer. Users should not rely on * this particular packing of the components of the version number; see the * documentation for GEOGRAPHICLIB_VERSION, below. **********************************************************************/ #define GEOGRAPHICLIB_VERSION_NUM(a,b,c) ((((a) * 10000 + (b)) * 100) + (c)) /** * @relates GeographicLib::Constants * The version of GeographicLib as a single integer, packed as MMmmmmpp where * MM is the major version, mmmm is the minor version, and pp is the patch * level. Users should not rely on this particular packing of the components * of the version number. Instead they should use a test such as \code #if GEOGRAPHICLIB_VERSION >= GEOGRAPHICLIB_VERSION_NUM(1,37,0) ... #endif * \endcode **********************************************************************/ #define GEOGRAPHICLIB_VERSION \ GEOGRAPHICLIB_VERSION_NUM(GEOGRAPHICLIB_VERSION_MAJOR, \ GEOGRAPHICLIB_VERSION_MINOR, \ GEOGRAPHICLIB_VERSION_PATCH) // For reference, here is a table of Visual Studio and _MSC_VER // correspondences: // // _MSC_VER Visual Studio // 1100 vc5 // 1200 vc6 // 1300 vc7 // 1310 vc7.1 (2003) // 1400 vc8 (2005) // 1500 vc9 (2008) // 1600 vc10 (2010) // 1700 vc11 (2012) // 1800 vc12 (2013) // 1900 vc14 (2015) First version of VS to include enough C++11 support // 191[0-9] vc15 (2017) // 192[0-9] vc16 (2019) // 193[0-9] vc17 (2022) #if defined(_MSC_VER) && defined(GEOGRAPHICLIB_SHARED_LIB) && \ GEOGRAPHICLIB_SHARED_LIB # if GEOGRAPHICLIB_SHARED_LIB > 1 # error GEOGRAPHICLIB_SHARED_LIB must be 0 or 1 # elif defined(GeographicLib_SHARED_EXPORTS) # define GEOGRAPHICLIB_EXPORT __declspec(dllexport) # else # define GEOGRAPHICLIB_EXPORT __declspec(dllimport) # endif #else # define GEOGRAPHICLIB_EXPORT #endif // Use GEOGRAPHICLIB_DEPRECATED to mark functions, types or variables as // deprecated. Code inspired by Apache Subversion's svn_types.h file (via // MPFR). #if defined(__GNUC__) # if __GNUC__ > 4 # define GEOGRAPHICLIB_DEPRECATED(msg) __attribute__((deprecated(msg))) # else # define GEOGRAPHICLIB_DEPRECATED(msg) __attribute__((deprecated)) # endif #elif defined(_MSC_VER) && _MSC_VER >= 1300 # define GEOGRAPHICLIB_DEPRECATED(msg) __declspec(deprecated(msg)) #else # define GEOGRAPHICLIB_DEPRECATED(msg) #endif #include #include #include "Math.h" /** * \brief Namespace for %GeographicLib * * All of %GeographicLib is defined within the GeographicLib namespace. In * addition all the header files are included via %GeographicLib/Class.hpp. * This minimizes the likelihood of conflicts with other packages. **********************************************************************/ namespace GeographicLib { /** * \brief %Constants needed by %GeographicLib * * Define constants specifying the WGS84 ellipsoid, the UTM and UPS * projections, and various unit conversions. * * Example of use: * \include example-Constants.cpp **********************************************************************/ class GEOGRAPHICLIB_EXPORT Constants { private: typedef Math::real real; Constants() = delete; // Disable constructor public: /** * A synonym for Math::degree(). **********************************************************************/ static Math::real degree() { return Math::degree(); } /** * @return the number of radians in an arcminute. **********************************************************************/ static Math::real arcminute() { return Math::degree() / Math::real(Math::dm); } /** * @return the number of radians in an arcsecond. **********************************************************************/ static Math::real arcsecond() { return Math::degree() / Math::real(Math::ds); } /** \name Ellipsoid parameters **********************************************************************/ ///@{ /** * @tparam T the type of the returned value. * @return the equatorial radius of WGS84 ellipsoid (6378137 m). **********************************************************************/ template static T WGS84_a() { return 6378137 * meter(); } /** * @tparam T the type of the returned value. * @return the flattening of WGS84 ellipsoid (1/298.257223563). **********************************************************************/ template static T WGS84_f() { // Evaluating this as 1000000000 / T(298257223563LL) reduces the // round-off error by about 10%. However, expressing the flattening as // 1/298.257223563 is well ingrained. return 1 / ( T(298257223563LL) / 1000000000 ); } /** * @tparam T the type of the returned value. * @return the gravitational constant of the WGS84 ellipsoid, \e GM, in * m3 s−2. **********************************************************************/ template static T WGS84_GM() { return T(3986004) * 100000000 + 41800000; } /** * @tparam T the type of the returned value. * @return the angular velocity of the WGS84 ellipsoid, ω, in rad * s−1. **********************************************************************/ template static T WGS84_omega() { return 7292115 / (T(1000000) * 100000); } /** * @tparam T the type of the returned value. * @return the equatorial radius of GRS80 ellipsoid, \e a, in m. **********************************************************************/ template static T GRS80_a() { return 6378137 * meter(); } /** * @tparam T the type of the returned value. * @return the gravitational constant of the GRS80 ellipsoid, \e GM, in * m3 s−2. **********************************************************************/ template static T GRS80_GM() { return T(3986005) * 100000000; } /** * @tparam T the type of the returned value. * @return the angular velocity of the GRS80 ellipsoid, ω, in rad * s−1. * * This is about 2 π 366.25 / (365.25 × 24 × 3600) rad * s−1. 365.25 is the number of days in a Julian year and * 365.35/366.25 converts from solar days to sidereal days. Using the * number of days in a Gregorian year (365.2425) results in a worse * approximation (because the Gregorian year includes the precession of the * earth's axis). **********************************************************************/ template static T GRS80_omega() { return 7292115 / (T(1000000) * 100000); } /** * @tparam T the type of the returned value. * @return the dynamical form factor of the GRS80 ellipsoid, * J2. **********************************************************************/ template static T GRS80_J2() { return T(108263) / 100000000; } /** * @tparam T the type of the returned value. * @return the central scale factor for UTM (0.9996). **********************************************************************/ template static T UTM_k0() {return T(9996) / 10000; } /** * @tparam T the type of the returned value. * @return the central scale factor for UPS (0.994). **********************************************************************/ template static T UPS_k0() { return T(994) / 1000; } ///@} /** \name SI units **********************************************************************/ ///@{ /** * @tparam T the type of the returned value. * @return the number of meters in a meter. * * This is unity, but this lets the internal system of units be changed if * necessary. **********************************************************************/ template static T meter() { return T(1); } /** * @return the number of meters in a kilometer. **********************************************************************/ static Math::real kilometer() { return 1000 * meter(); } /** * @return the number of meters in a nautical mile (approximately 1 arc * minute) **********************************************************************/ static Math::real nauticalmile() { return 1852 * meter(); } /** * @tparam T the type of the returned value. * @return the number of square meters in a square meter. * * This is unity, but this lets the internal system of units be changed if * necessary. **********************************************************************/ template static T square_meter() { return meter() * meter(); } /** * @return the number of square meters in a hectare. **********************************************************************/ static Math::real hectare() { return 10000 * square_meter(); } /** * @return the number of square meters in a square kilometer. **********************************************************************/ static Math::real square_kilometer() { return kilometer() * kilometer(); } /** * @return the number of square meters in a square nautical mile. **********************************************************************/ static Math::real square_nauticalmile() { return nauticalmile() * nauticalmile(); } ///@} /** \name Anachronistic British units **********************************************************************/ ///@{ /** * @return the number of meters in an international foot. **********************************************************************/ static Math::real foot() { return real(254 * 12) / 10000 * meter(); } /** * @return the number of meters in a yard. **********************************************************************/ static Math::real yard() { return 3 * foot(); } /** * @return the number of meters in a fathom. **********************************************************************/ static Math::real fathom() { return 2 * yard(); } /** * @return the number of meters in a chain. **********************************************************************/ static Math::real chain() { return 22 * yard(); } /** * @return the number of meters in a furlong. **********************************************************************/ static Math::real furlong() { return 10 * chain(); } /** * @return the number of meters in a statute mile. **********************************************************************/ static Math::real mile() { return 8 * furlong(); } /** * @return the number of square meters in an acre. **********************************************************************/ static Math::real acre() { return chain() * furlong(); } /** * @return the number of square meters in a square statute mile. **********************************************************************/ static Math::real square_mile() { return mile() * mile(); } ///@} /** \name Anachronistic US units **********************************************************************/ ///@{ /** * @return the number of meters in a US survey foot. **********************************************************************/ static Math::real surveyfoot() { return real(1200) / 3937 * meter(); } ///@} }; /** * \brief Exception handling for %GeographicLib * * A class to handle exceptions. It's derived from std::runtime_error so it * can be caught by the usual catch clauses. * * Example of use: * \include example-GeographicErr.cpp **********************************************************************/ class GeographicErr : public std::runtime_error { public: /** * Constructor * * @param[in] msg a string message, which is accessible in the catch * clause via what(). **********************************************************************/ GeographicErr(const std::string& msg) : std::runtime_error(msg) {} }; } // namespace GeographicLib #endif // GEOGRAPHICLIB_CONSTANTS_HPP geosphere/src/Math.cpp0000644000176200001440000002625714323376012014434 0ustar liggesusers/** * \file Math.cpp * \brief Implementation for GeographicLib::Math class * * Copyright (c) Charles Karney (2015-2022) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ **********************************************************************/ #include "Math.h" #if defined(_MSC_VER) // Squelch warnings about constant conditional and enum-float expressions # pragma warning (disable: 4127 5055) #endif namespace GeographicLib { using namespace std; void Math::dummy() { static_assert(GEOGRAPHICLIB_PRECISION >= 1 && GEOGRAPHICLIB_PRECISION <= 5, "Bad value of precision"); } int Math::digits() { #if GEOGRAPHICLIB_PRECISION != 5 return numeric_limits::digits; #else return numeric_limits::digits(); #endif } int Math::set_digits(int ndigits) { #if GEOGRAPHICLIB_PRECISION != 5 (void)ndigits; #else mpfr::mpreal::set_default_prec(ndigits >= 2 ? ndigits : 2); #endif return digits(); } int Math::digits10() { #if GEOGRAPHICLIB_PRECISION != 5 return numeric_limits::digits10; #else return numeric_limits::digits10(); #endif } int Math::extra_digits() { return digits10() > numeric_limits::digits10 ? digits10() - numeric_limits::digits10 : 0; } template T Math::sum(T u, T v, T& t) { GEOGRAPHICLIB_VOLATILE T s = u + v; GEOGRAPHICLIB_VOLATILE T up = s - v; GEOGRAPHICLIB_VOLATILE T vpp = s - up; up -= u; vpp -= v; // if s = 0, then t = 0 and give t the same sign as s // mpreal needs T(0) here t = s != 0 ? T(0) - (up + vpp) : s; // u + v = s + t // = round(u + v) + t return s; } template T Math::AngNormalize(T x) { T y = remainder(x, T(td)); #if GEOGRAPHICLIB_PRECISION == 4 // boost-quadmath doesn't set the sign of 0 correctly, see // https://github.com/boostorg/multiprecision/issues/426 // Fixed by https://github.com/boostorg/multiprecision/pull/428 if (y == 0) y = copysign(y, x); #endif return fabs(y) == T(hd) ? copysign(T(hd), x) : y; } template T Math::AngDiff(T x, T y, T& e) { // Use remainder instead of AngNormalize, since we treat boundary cases // later taking account of the error T d = sum(remainder(-x, T(td)), remainder( y, T(td)), e); // This second sum can only change d if abs(d) < 128, so don't need to // apply remainder yet again. d = sum(remainder(d, T(td)), e, e); // Fix the sign if d = -180, 0, 180. if (d == 0 || fabs(d) == hd) // If e == 0, take sign from y - x // else (e != 0, implies d = +/-180), d and e must have opposite signs d = copysign(d, e == 0 ? y - x : -e); return d; } template T Math::AngRound(T x) { static const T z = T(1)/T(16); GEOGRAPHICLIB_VOLATILE T y = fabs(x); GEOGRAPHICLIB_VOLATILE T w = z - y; // The compiler mustn't "simplify" z - (z - y) to y y = w > 0 ? z - w : y; return copysign(y, x); } template void Math::sincosd(T x, T& sinx, T& cosx) { // In order to minimize round-off errors, this function exactly reduces // the argument to the range [-45, 45] before converting it to radians. T r; int q = 0; r = remquo(x, T(qd), &q); // now abs(r) <= 45 r *= degree(); // g++ -O turns these two function calls into a call to sincos T s = sin(r), c = cos(r); switch (unsigned(q) & 3U) { case 0U: sinx = s; cosx = c; break; case 1U: sinx = c; cosx = -s; break; case 2U: sinx = -s; cosx = -c; break; default: sinx = -c; cosx = s; break; // case 3U } // http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1950.pdf // mpreal needs T(0) here cosx += T(0); // special values from F.10.1.12 if (sinx == 0) sinx = copysign(sinx, x); // special values from F.10.1.13 } template void Math::sincosde(T x, T t, T& sinx, T& cosx) { // In order to minimize round-off errors, this function exactly reduces // the argument to the range [-45, 45] before converting it to radians. // This implementation allows x outside [-180, 180], but implementations in // other languages may not. T r; int q = 0; r = AngRound(remquo(x, T(qd), &q) + t); // now abs(r) <= 45 r *= degree(); // g++ -O turns these two function calls into a call to sincos T s = sin(r), c = cos(r); switch (unsigned(q) & 3U) { case 0U: sinx = s; cosx = c; break; case 1U: sinx = c; cosx = -s; break; case 2U: sinx = -s; cosx = -c; break; default: sinx = -c; cosx = s; break; // case 3U } // http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1950.pdf // mpreal needs T(0) here cosx += T(0); // special values from F.10.1.12 if (sinx == 0) sinx = copysign(sinx, x); // special values from F.10.1.13 } template T Math::sind(T x) { // See sincosd T r; int q = 0; r = remquo(x, T(qd), &q); // now abs(r) <= 45 r *= degree(); unsigned p = unsigned(q); r = p & 1U ? cos(r) : sin(r); if (p & 2U) r = -r; if (r == 0) r = copysign(r, x); return r; } template T Math::cosd(T x) { // See sincosd T r; int q = 0; r = remquo(x, T(qd), &q); // now abs(r) <= 45 r *= degree(); unsigned p = unsigned(q + 1); r = p & 1U ? cos(r) : sin(r); if (p & 2U) r = -r; // mpreal needs T(0) here return T(0) + r; } template T Math::tand(T x) { static const T overflow = 1 / sq(numeric_limits::epsilon()); T s, c; sincosd(x, s, c); // http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1950.pdf T r = s / c; // special values from F.10.1.14 // With C++17 this becomes clamp(s / c, -overflow, overflow); // Use max/min here (instead of fmax/fmin) to preserve NaN return min(max(r, -overflow), overflow); } template T Math::atan2d(T y, T x) { // In order to minimize round-off errors, this function rearranges the // arguments so that result of atan2 is in the range [-pi/4, pi/4] before // converting it to degrees and mapping the result to the correct // quadrant. int q = 0; if (fabs(y) > fabs(x)) { swap(x, y); q = 2; } if (signbit(x)) { x = -x; ++q; } // here x >= 0 and x >= abs(y), so angle is in [-pi/4, pi/4] T ang = atan2(y, x) / degree(); switch (q) { case 1: ang = copysign(T(hd), y) - ang; break; case 2: ang = qd - ang; break; case 3: ang = -qd + ang; break; default: break; } return ang; } template T Math::atand(T x) { return atan2d(x, T(1)); } template T Math::eatanhe(T x, T es) { return es > 0 ? es * atanh(es * x) : -es * atan(es * x); } template T Math::taupf(T tau, T es) { // Need this test, otherwise tau = +/-inf gives taup = nan. if (isfinite(tau)) { T tau1 = hypot(T(1), tau), sig = sinh( eatanhe(tau / tau1, es ) ); return hypot(T(1), sig) * tau - sig * tau1; } else return tau; } template T Math::tauf(T taup, T es) { static const int numit = 5; // min iterations = 1, max iterations = 2; mean = 1.95 static const T tol = sqrt(numeric_limits::epsilon()) / 10; static const T taumax = 2 / sqrt(numeric_limits::epsilon()); T e2m = 1 - sq(es), // To lowest order in e^2, taup = (1 - e^2) * tau = _e2m * tau; so use // tau = taup/e2m as a starting guess. Only 1 iteration is needed for // |lat| < 3.35 deg, otherwise 2 iterations are needed. If, instead, tau // = taup is used the mean number of iterations increases to 1.999 (2 // iterations are needed except near tau = 0). // // For large tau, taup = exp(-es*atanh(es)) * tau. Use this as for the // initial guess for |taup| > 70 (approx |phi| > 89deg). Then for // sufficiently large tau (such that sqrt(1+tau^2) = |tau|), we can exit // with the intial guess and avoid overflow problems. This also reduces // the mean number of iterations slightly from 1.963 to 1.954. tau = fabs(taup) > 70 ? taup * exp(eatanhe(T(1), es)) : taup/e2m, stol = tol * fmax(T(1), fabs(taup)); if (!(fabs(tau) < taumax)) return tau; // handles +/-inf and nan for (int i = 0; i < numit || GEOGRAPHICLIB_PANIC; ++i) { T taupa = taupf(tau, es), dtau = (taup - taupa) * (1 + e2m * sq(tau)) / ( e2m * hypot(T(1), tau) * hypot(T(1), taupa) ); tau += dtau; if (!(fabs(dtau) >= stol)) break; } return tau; } template T Math::NaN() { #if defined(_MSC_VER) return numeric_limits::has_quiet_NaN ? numeric_limits::quiet_NaN() : (numeric_limits::max)(); #else return numeric_limits::has_quiet_NaN ? numeric_limits::quiet_NaN() : numeric_limits::max(); #endif } template T Math::infinity() { #if defined(_MSC_VER) return numeric_limits::has_infinity ? numeric_limits::infinity() : (numeric_limits::max)(); #else return numeric_limits::has_infinity ? numeric_limits::infinity() : numeric_limits::max(); #endif } /// \cond SKIP // Instantiate #define GEOGRAPHICLIB_MATH_INSTANTIATE(T) \ template T GEOGRAPHICLIB_EXPORT Math::sum (T, T, T&); \ template T GEOGRAPHICLIB_EXPORT Math::AngNormalize (T); \ template T GEOGRAPHICLIB_EXPORT Math::AngDiff (T, T, T&); \ template T GEOGRAPHICLIB_EXPORT Math::AngRound (T); \ template void GEOGRAPHICLIB_EXPORT Math::sincosd (T, T&, T&); \ template void GEOGRAPHICLIB_EXPORT Math::sincosde (T, T, T&, T&); \ template T GEOGRAPHICLIB_EXPORT Math::sind (T); \ template T GEOGRAPHICLIB_EXPORT Math::cosd (T); \ template T GEOGRAPHICLIB_EXPORT Math::tand (T); \ template T GEOGRAPHICLIB_EXPORT Math::atan2d (T, T); \ template T GEOGRAPHICLIB_EXPORT Math::atand (T); \ template T GEOGRAPHICLIB_EXPORT Math::eatanhe (T, T); \ template T GEOGRAPHICLIB_EXPORT Math::taupf (T, T); \ template T GEOGRAPHICLIB_EXPORT Math::tauf (T, T); \ template T GEOGRAPHICLIB_EXPORT Math::NaN (); \ template T GEOGRAPHICLIB_EXPORT Math::infinity (); // Instantiate with the standard floating type GEOGRAPHICLIB_MATH_INSTANTIATE(float) GEOGRAPHICLIB_MATH_INSTANTIATE(double) #if GEOGRAPHICLIB_HAVE_LONG_DOUBLE // Instantiate if long double is distinct from double GEOGRAPHICLIB_MATH_INSTANTIATE(long double) #endif #if GEOGRAPHICLIB_PRECISION > 3 // Instantiate with the high precision type GEOGRAPHICLIB_MATH_INSTANTIATE(Math::real) #endif #undef GEOGRAPHICLIB_MATH_INSTANTIATE // Also we need int versions for Utility::nummatch template int GEOGRAPHICLIB_EXPORT Math::NaN (); template int GEOGRAPHICLIB_EXPORT Math::infinity(); /// \endcond } // namespace GeographicLib geosphere/src/RcppExports.cpp0000644000176200001440000001065514323401003016014 0ustar liggesusers// Generated by using Rcpp::compileAttributes() -> do not edit by hand // Generator token: 10BE3573-1514-4C36-9D1C-5A225CD40393 #include using namespace Rcpp; #ifdef RCPP_USE_GLOBAL_ROSTREAM Rcpp::Rostream& Rcpp::Rcout = Rcpp::Rcpp_cout_get(); Rcpp::Rostream& Rcpp::Rcerr = Rcpp::Rcpp_cerr_get(); #endif // geodesic std::vector geodesic(std::vector lon1, std::vector lat1, std::vector azi1, std::vector s12, double a, double f); RcppExport SEXP _geosphere_geodesic(SEXP lon1SEXP, SEXP lat1SEXP, SEXP azi1SEXP, SEXP s12SEXP, SEXP aSEXP, SEXP fSEXP) { BEGIN_RCPP Rcpp::RObject rcpp_result_gen; Rcpp::RNGScope rcpp_rngScope_gen; Rcpp::traits::input_parameter< std::vector >::type lon1(lon1SEXP); Rcpp::traits::input_parameter< std::vector >::type lat1(lat1SEXP); Rcpp::traits::input_parameter< std::vector >::type azi1(azi1SEXP); Rcpp::traits::input_parameter< std::vector >::type s12(s12SEXP); Rcpp::traits::input_parameter< double >::type a(aSEXP); Rcpp::traits::input_parameter< double >::type f(fSEXP); rcpp_result_gen = Rcpp::wrap(geodesic(lon1, lat1, azi1, s12, a, f)); return rcpp_result_gen; END_RCPP } // inversegeodesic std::vector inversegeodesic(std::vector lon1, std::vector lat1, std::vector lon2, std::vector lat2, double a, double f); RcppExport SEXP _geosphere_inversegeodesic(SEXP lon1SEXP, SEXP lat1SEXP, SEXP lon2SEXP, SEXP lat2SEXP, SEXP aSEXP, SEXP fSEXP) { BEGIN_RCPP Rcpp::RObject rcpp_result_gen; Rcpp::RNGScope rcpp_rngScope_gen; Rcpp::traits::input_parameter< std::vector >::type lon1(lon1SEXP); Rcpp::traits::input_parameter< std::vector >::type lat1(lat1SEXP); Rcpp::traits::input_parameter< std::vector >::type lon2(lon2SEXP); Rcpp::traits::input_parameter< std::vector >::type lat2(lat2SEXP); Rcpp::traits::input_parameter< double >::type a(aSEXP); Rcpp::traits::input_parameter< double >::type f(fSEXP); rcpp_result_gen = Rcpp::wrap(inversegeodesic(lon1, lat1, lon2, lat2, a, f)); return rcpp_result_gen; END_RCPP } // polygonarea std::vector polygonarea(std::vector lon, std::vector lat, double a, double f); RcppExport SEXP _geosphere_polygonarea(SEXP lonSEXP, SEXP latSEXP, SEXP aSEXP, SEXP fSEXP) { BEGIN_RCPP Rcpp::RObject rcpp_result_gen; Rcpp::RNGScope rcpp_rngScope_gen; Rcpp::traits::input_parameter< std::vector >::type lon(lonSEXP); Rcpp::traits::input_parameter< std::vector >::type lat(latSEXP); Rcpp::traits::input_parameter< double >::type a(aSEXP); Rcpp::traits::input_parameter< double >::type f(fSEXP); rcpp_result_gen = Rcpp::wrap(polygonarea(lon, lat, a, f)); return rcpp_result_gen; END_RCPP } // geodesic_nodes std::vector> geodesic_nodes(double lon1, double lat1, double lon2, double lat2, size_t n, double distance, bool arc, double a, double f); RcppExport SEXP _geosphere_geodesic_nodes(SEXP lon1SEXP, SEXP lat1SEXP, SEXP lon2SEXP, SEXP lat2SEXP, SEXP nSEXP, SEXP distanceSEXP, SEXP arcSEXP, SEXP aSEXP, SEXP fSEXP) { BEGIN_RCPP Rcpp::RObject rcpp_result_gen; Rcpp::RNGScope rcpp_rngScope_gen; Rcpp::traits::input_parameter< double >::type lon1(lon1SEXP); Rcpp::traits::input_parameter< double >::type lat1(lat1SEXP); Rcpp::traits::input_parameter< double >::type lon2(lon2SEXP); Rcpp::traits::input_parameter< double >::type lat2(lat2SEXP); Rcpp::traits::input_parameter< size_t >::type n(nSEXP); Rcpp::traits::input_parameter< double >::type distance(distanceSEXP); Rcpp::traits::input_parameter< bool >::type arc(arcSEXP); Rcpp::traits::input_parameter< double >::type a(aSEXP); Rcpp::traits::input_parameter< double >::type f(fSEXP); rcpp_result_gen = Rcpp::wrap(geodesic_nodes(lon1, lat1, lon2, lat2, n, distance, arc, a, f)); return rcpp_result_gen; END_RCPP } static const R_CallMethodDef CallEntries[] = { {"_geosphere_geodesic", (DL_FUNC) &_geosphere_geodesic, 6}, {"_geosphere_inversegeodesic", (DL_FUNC) &_geosphere_inversegeodesic, 6}, {"_geosphere_polygonarea", (DL_FUNC) &_geosphere_polygonarea, 4}, {"_geosphere_geodesic_nodes", (DL_FUNC) &_geosphere_geodesic_nodes, 9}, {NULL, NULL, 0} }; RcppExport void R_init_geosphere(DllInfo *dll) { R_registerRoutines(dll, NULL, CallEntries, NULL, NULL); R_useDynamicSymbols(dll, FALSE); } geosphere/src/GeodesicLine.cpp0000644000176200001440000003102614323376011016062 0ustar liggesusers/** * \file GeodesicLine.cpp * \brief Implementation for GeographicLib::GeodesicLine class * * Copyright (c) Charles Karney (2009-2022) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ * * This is a reformulation of the geodesic problem. The notation is as * follows: * - at a general point (no suffix or 1 or 2 as suffix) * - phi = latitude * - beta = latitude on auxiliary sphere * - omega = longitude on auxiliary sphere * - lambda = longitude * - alpha = azimuth of great circle * - sigma = arc length along great circle * - s = distance * - tau = scaled distance (= sigma at multiples of pi/2) * - at northwards equator crossing * - beta = phi = 0 * - omega = lambda = 0 * - alpha = alpha0 * - sigma = s = 0 * - a 12 suffix means a difference, e.g., s12 = s2 - s1. * - s and c prefixes mean sin and cos **********************************************************************/ #include "GeodesicLine.h" #if defined(_MSC_VER) // Squelch warnings about mixing enums # pragma warning (disable: 5054) #endif namespace GeographicLib { using namespace std; void GeodesicLine::LineInit(const Geodesic& g, real lat1, real lon1, real azi1, real salp1, real calp1, unsigned caps) { tiny_ = g.tiny_; _lat1 = Math::LatFix(lat1); _lon1 = lon1; _azi1 = azi1; _salp1 = salp1; _calp1 = calp1; _a = g._a; _f = g._f; _b = g._b; _c2 = g._c2; _f1 = g._f1; // Always allow latitude and azimuth and unrolling of longitude _caps = caps | LATITUDE | AZIMUTH | LONG_UNROLL; real cbet1, sbet1; Math::sincosd(Math::AngRound(_lat1), sbet1, cbet1); sbet1 *= _f1; // Ensure cbet1 = +epsilon at poles Math::norm(sbet1, cbet1); cbet1 = fmax(tiny_, cbet1); _dn1 = sqrt(1 + g._ep2 * Math::sq(sbet1)); // Evaluate alp0 from sin(alp1) * cos(bet1) = sin(alp0), _salp0 = _salp1 * cbet1; // alp0 in [0, pi/2 - |bet1|] // Alt: calp0 = hypot(sbet1, calp1 * cbet1). The following // is slightly better (consider the case salp1 = 0). _calp0 = hypot(_calp1, _salp1 * sbet1); // Evaluate sig with tan(bet1) = tan(sig1) * cos(alp1). // sig = 0 is nearest northward crossing of equator. // With bet1 = 0, alp1 = pi/2, we have sig1 = 0 (equatorial line). // With bet1 = pi/2, alp1 = -pi, sig1 = pi/2 // With bet1 = -pi/2, alp1 = 0 , sig1 = -pi/2 // Evaluate omg1 with tan(omg1) = sin(alp0) * tan(sig1). // With alp0 in (0, pi/2], quadrants for sig and omg coincide. // No atan2(0,0) ambiguity at poles since cbet1 = +epsilon. // With alp0 = 0, omg1 = 0 for alp1 = 0, omg1 = pi for alp1 = pi. _ssig1 = sbet1; _somg1 = _salp0 * sbet1; _csig1 = _comg1 = sbet1 != 0 || _calp1 != 0 ? cbet1 * _calp1 : 1; Math::norm(_ssig1, _csig1); // sig1 in (-pi, pi] // Math::norm(_somg1, _comg1); -- don't need to normalize! _k2 = Math::sq(_calp0) * g._ep2; real eps = _k2 / (2 * (1 + sqrt(1 + _k2)) + _k2); if (_caps & CAP_C1) { _aA1m1 = Geodesic::A1m1f(eps); Geodesic::C1f(eps, _cC1a); _bB11 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _cC1a, nC1_); real s = sin(_bB11), c = cos(_bB11); // tau1 = sig1 + B11 _stau1 = _ssig1 * c + _csig1 * s; _ctau1 = _csig1 * c - _ssig1 * s; // Not necessary because C1pa reverts C1a // _bB11 = -SinCosSeries(true, _stau1, _ctau1, _cC1pa, nC1p_); } if (_caps & CAP_C1p) Geodesic::C1pf(eps, _cC1pa); if (_caps & CAP_C2) { _aA2m1 = Geodesic::A2m1f(eps); Geodesic::C2f(eps, _cC2a); _bB21 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _cC2a, nC2_); } if (_caps & CAP_C3) { g.C3f(eps, _cC3a); _aA3c = -_f * _salp0 * g.A3f(eps); _bB31 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _cC3a, nC3_-1); } if (_caps & CAP_C4) { g.C4f(eps, _cC4a); // Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0) _aA4 = Math::sq(_a) * _calp0 * _salp0 * g._e2; _bB41 = Geodesic::SinCosSeries(false, _ssig1, _csig1, _cC4a, nC4_); } _a13 = _s13 = Math::NaN(); } GeodesicLine::GeodesicLine(const Geodesic& g, real lat1, real lon1, real azi1, unsigned caps) { azi1 = Math::AngNormalize(azi1); real salp1, calp1; // Guard against underflow in salp0. Also -0 is converted to +0. Math::sincosd(Math::AngRound(azi1), salp1, calp1); LineInit(g, lat1, lon1, azi1, salp1, calp1, caps); } GeodesicLine::GeodesicLine(const Geodesic& g, real lat1, real lon1, real azi1, real salp1, real calp1, unsigned caps, bool arcmode, real s13_a13) { LineInit(g, lat1, lon1, azi1, salp1, calp1, caps); GenSetDistance(arcmode, s13_a13); } Math::real GeodesicLine::GenPosition(bool arcmode, real s12_a12, unsigned outmask, real& lat2, real& lon2, real& azi2, real& s12, real& m12, real& M12, real& M21, real& S12) const { outmask &= _caps & OUT_MASK; if (!( Init() && (arcmode || (_caps & (OUT_MASK & DISTANCE_IN))) )) // Uninitialized or impossible distance calculation requested return Math::NaN(); // Avoid warning about uninitialized B12. real sig12, ssig12, csig12, B12 = 0, AB1 = 0; if (arcmode) { // Interpret s12_a12 as spherical arc length sig12 = s12_a12 * Math::degree(); Math::sincosd(s12_a12, ssig12, csig12); } else { // Interpret s12_a12 as distance real tau12 = s12_a12 / (_b * (1 + _aA1m1)), s = sin(tau12), c = cos(tau12); // tau2 = tau1 + tau12 B12 = - Geodesic::SinCosSeries(true, _stau1 * c + _ctau1 * s, _ctau1 * c - _stau1 * s, _cC1pa, nC1p_); sig12 = tau12 - (B12 - _bB11); ssig12 = sin(sig12); csig12 = cos(sig12); if (fabs(_f) > 0.01) { // Reverted distance series is inaccurate for |f| > 1/100, so correct // sig12 with 1 Newton iteration. The following table shows the // approximate maximum error for a = WGS_a() and various f relative to // GeodesicExact. // erri = the error in the inverse solution (nm) // errd = the error in the direct solution (series only) (nm) // errda = the error in the direct solution // (series + 1 Newton) (nm) // // f erri errd errda // -1/5 12e6 1.2e9 69e6 // -1/10 123e3 12e6 765e3 // -1/20 1110 108e3 7155 // -1/50 18.63 200.9 27.12 // -1/100 18.63 23.78 23.37 // -1/150 18.63 21.05 20.26 // 1/150 22.35 24.73 25.83 // 1/100 22.35 25.03 25.31 // 1/50 29.80 231.9 30.44 // 1/20 5376 146e3 10e3 // 1/10 829e3 22e6 1.5e6 // 1/5 157e6 3.8e9 280e6 real ssig2 = _ssig1 * csig12 + _csig1 * ssig12, csig2 = _csig1 * csig12 - _ssig1 * ssig12; B12 = Geodesic::SinCosSeries(true, ssig2, csig2, _cC1a, nC1_); real serr = (1 + _aA1m1) * (sig12 + (B12 - _bB11)) - s12_a12 / _b; sig12 = sig12 - serr / sqrt(1 + _k2 * Math::sq(ssig2)); ssig12 = sin(sig12); csig12 = cos(sig12); // Update B12 below } } real ssig2, csig2, sbet2, cbet2, salp2, calp2; // sig2 = sig1 + sig12 ssig2 = _ssig1 * csig12 + _csig1 * ssig12; csig2 = _csig1 * csig12 - _ssig1 * ssig12; real dn2 = sqrt(1 + _k2 * Math::sq(ssig2)); if (outmask & (DISTANCE | REDUCEDLENGTH | GEODESICSCALE)) { if (arcmode || fabs(_f) > 0.01) B12 = Geodesic::SinCosSeries(true, ssig2, csig2, _cC1a, nC1_); AB1 = (1 + _aA1m1) * (B12 - _bB11); } // sin(bet2) = cos(alp0) * sin(sig2) sbet2 = _calp0 * ssig2; // Alt: cbet2 = hypot(csig2, salp0 * ssig2); cbet2 = hypot(_salp0, _calp0 * csig2); if (cbet2 == 0) // I.e., salp0 = 0, csig2 = 0. Break the degeneracy in this case cbet2 = csig2 = tiny_; // tan(alp0) = cos(sig2)*tan(alp2) salp2 = _salp0; calp2 = _calp0 * csig2; // No need to normalize if (outmask & DISTANCE) s12 = arcmode ? _b * ((1 + _aA1m1) * sig12 + AB1) : s12_a12; if (outmask & LONGITUDE) { // tan(omg2) = sin(alp0) * tan(sig2) real somg2 = _salp0 * ssig2, comg2 = csig2, // No need to normalize E = copysign(real(1), _salp0); // east-going? // omg12 = omg2 - omg1 real omg12 = outmask & LONG_UNROLL ? E * (sig12 - (atan2( ssig2, csig2) - atan2( _ssig1, _csig1)) + (atan2(E * somg2, comg2) - atan2(E * _somg1, _comg1))) : atan2(somg2 * _comg1 - comg2 * _somg1, comg2 * _comg1 + somg2 * _somg1); real lam12 = omg12 + _aA3c * ( sig12 + (Geodesic::SinCosSeries(true, ssig2, csig2, _cC3a, nC3_-1) - _bB31)); real lon12 = lam12 / Math::degree(); lon2 = outmask & LONG_UNROLL ? _lon1 + lon12 : Math::AngNormalize(Math::AngNormalize(_lon1) + Math::AngNormalize(lon12)); } if (outmask & LATITUDE) lat2 = Math::atan2d(sbet2, _f1 * cbet2); if (outmask & AZIMUTH) azi2 = Math::atan2d(salp2, calp2); if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) { real B22 = Geodesic::SinCosSeries(true, ssig2, csig2, _cC2a, nC2_), AB2 = (1 + _aA2m1) * (B22 - _bB21), J12 = (_aA1m1 - _aA2m1) * sig12 + (AB1 - AB2); if (outmask & REDUCEDLENGTH) // Add parens around (_csig1 * ssig2) and (_ssig1 * csig2) to ensure // accurate cancellation in the case of coincident points. m12 = _b * ((dn2 * (_csig1 * ssig2) - _dn1 * (_ssig1 * csig2)) - _csig1 * csig2 * J12); if (outmask & GEODESICSCALE) { real t = _k2 * (ssig2 - _ssig1) * (ssig2 + _ssig1) / (_dn1 + dn2); M12 = csig12 + (t * ssig2 - csig2 * J12) * _ssig1 / _dn1; M21 = csig12 - (t * _ssig1 - _csig1 * J12) * ssig2 / dn2; } } if (outmask & AREA) { real B42 = Geodesic::SinCosSeries(false, ssig2, csig2, _cC4a, nC4_); real salp12, calp12; if (_calp0 == 0 || _salp0 == 0) { // alp12 = alp2 - alp1, used in atan2 so no need to normalize salp12 = salp2 * _calp1 - calp2 * _salp1; calp12 = calp2 * _calp1 + salp2 * _salp1; // We used to include here some patch up code that purported to deal // with nearly meridional geodesics properly. However, this turned out // to be wrong once _salp1 = -0 was allowed (via // Geodesic::InverseLine). In fact, the calculation of {s,c}alp12 // was already correct (following the IEEE rules for handling signed // zeros). So the patch up code was unnecessary (as well as // dangerous). } else { // tan(alp) = tan(alp0) * sec(sig) // tan(alp2-alp1) = (tan(alp2) -tan(alp1)) / (tan(alp2)*tan(alp1)+1) // = calp0 * salp0 * (csig1-csig2) / (salp0^2 + calp0^2 * csig1*csig2) // If csig12 > 0, write // csig1 - csig2 = ssig12 * (csig1 * ssig12 / (1 + csig12) + ssig1) // else // csig1 - csig2 = csig1 * (1 - csig12) + ssig12 * ssig1 // No need to normalize salp12 = _calp0 * _salp0 * (csig12 <= 0 ? _csig1 * (1 - csig12) + ssig12 * _ssig1 : ssig12 * (_csig1 * ssig12 / (1 + csig12) + _ssig1)); calp12 = Math::sq(_salp0) + Math::sq(_calp0) * _csig1 * csig2; } S12 = _c2 * atan2(salp12, calp12) + _aA4 * (B42 - _bB41); } return arcmode ? s12_a12 : sig12 / Math::degree(); } void GeodesicLine::SetDistance(real s13) { _s13 = s13; real t; // This will set _a13 to NaN if the GeodesicLine doesn't have the // DISTANCE_IN capability. _a13 = GenPosition(false, _s13, 0u, t, t, t, t, t, t, t, t); } void GeodesicLine::SetArc(real a13) { _a13 = a13; // In case the GeodesicLine doesn't have the DISTANCE capability. _s13 = Math::NaN(); real t; GenPosition(true, _a13, DISTANCE, t, t, t, _s13, t, t, t, t); } void GeodesicLine::GenSetDistance(bool arcmode, real s13_a13) { arcmode ? SetArc(s13_a13) : SetDistance(s13_a13); } } // namespace GeographicLib geosphere/src/Accumulator.h0000644000176200001440000001754514323400747015472 0ustar liggesusers/** * \file Accumulator.hpp * \brief Header for GeographicLib::Accumulator class * * Copyright (c) Charles Karney (2010-2020) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ **********************************************************************/ #if !defined(GEOGRAPHICLIB_ACCUMULATOR_HPP) #define GEOGRAPHICLIB_ACCUMULATOR_HPP 1 #include "Constants.h" namespace GeographicLib { /** * \brief An accumulator for sums * * This allows many numbers of floating point type \e T to be added together * with twice the normal precision. Thus if \e T is double, the effective * precision of the sum is 106 bits or about 32 decimal places. * * The implementation follows J. R. Shewchuk, * Adaptive Precision * Floating-Point Arithmetic and Fast Robust Geometric Predicates, * Discrete & Computational Geometry 18(3) 305--363 (1997). * * Approximate timings (summing a vector) * - double: 2ns * - Accumulator: 23ns * * In the documentation of the member functions, \e sum stands for the value * currently held in the accumulator. * * Example of use: * \include example-Accumulator.cpp **********************************************************************/ template class GEOGRAPHICLIB_EXPORT Accumulator { private: // _s + _t accumulators for the sum. T _s, _t; // Same as Math::sum, but requires abs(u) >= abs(v). This isn't currently // used. static T fastsum(T u, T v, T& t) { GEOGRAPHICLIB_VOLATILE T s = u + v; GEOGRAPHICLIB_VOLATILE T vp = s - u; t = v - vp; return s; } void Add(T y) { // Here's Shewchuk's solution... T u; // hold exact sum as [s, t, u] // Accumulate starting at least significant end y = Math::sum(y, _t, u); _s = Math::sum(y, _s, _t); // Start is _s, _t decreasing and non-adjacent. Sum is now (s + t + u) // exactly with s, t, u non-adjacent and in decreasing order (except for // possible zeros). The following code tries to normalize the result. // Ideally, we want _s = round(s+t+u) and _u = round(s+t+u - _s). The // following does an approximate job (and maintains the decreasing // non-adjacent property). Here are two "failures" using 3-bit floats: // // Case 1: _s is not equal to round(s+t+u) -- off by 1 ulp // [12, -1] - 8 -> [4, 0, -1] -> [4, -1] = 3 should be [3, 0] = 3 // // Case 2: _s+_t is not as close to s+t+u as it shold be // [64, 5] + 4 -> [64, 8, 1] -> [64, 8] = 72 (off by 1) // should be [80, -7] = 73 (exact) // // "Fixing" these problems is probably not worth the expense. The // representation inevitably leads to small errors in the accumulated // values. The additional errors illustrated here amount to 1 ulp of the // less significant word during each addition to the Accumulator and an // additional possible error of 1 ulp in the reported sum. // // Incidentally, the "ideal" representation described above is not // canonical, because _s = round(_s + _t) may not be true. For example, // with 3-bit floats: // // [128, 16] + 1 -> [160, -16] -- 160 = round(145). // But [160, 0] - 16 -> [128, 16] -- 128 = round(144). // if (_s == 0) // This implies t == 0, _s = u; // so result is u else _t += u; // otherwise just accumulate u to t. } T Sum(T y) const { Accumulator a(*this); a.Add(y); return a._s; } public: /** * Construct from a \e T. This is not declared explicit, so that you can * write Accumulator a = 5;. * * @param[in] y set \e sum = \e y. **********************************************************************/ Accumulator(T y = T(0)) : _s(y), _t(0) { static_assert(!std::numeric_limits::is_integer, "Accumulator type is not floating point"); } /** * Set the accumulator to a number. * * @param[in] y set \e sum = \e y. **********************************************************************/ Accumulator& operator=(T y) { _s = y; _t = 0; return *this; } /** * Return the value held in the accumulator. * * @return \e sum. **********************************************************************/ T operator()() const { return _s; } /** * Return the result of adding a number to \e sum (but don't change \e * sum). * * @param[in] y the number to be added to the sum. * @return \e sum + \e y. **********************************************************************/ T operator()(T y) const { return Sum(y); } /** * Add a number to the accumulator. * * @param[in] y set \e sum += \e y. **********************************************************************/ Accumulator& operator+=(T y) { Add(y); return *this; } /** * Subtract a number from the accumulator. * * @param[in] y set \e sum -= \e y. **********************************************************************/ Accumulator& operator-=(T y) { Add(-y); return *this; } /** * Multiply accumulator by an integer. To avoid loss of accuracy, use only * integers such that \e n × \e T is exactly representable as a \e T * (i.e., ± powers of two). Use \e n = −1 to negate \e sum. * * @param[in] n set \e sum *= \e n. **********************************************************************/ Accumulator& operator*=(int n) { _s *= n; _t *= n; return *this; } /** * Multiply accumulator by a number. The fma (fused multiply and add) * instruction is used (if available) in order to maintain accuracy. * * @param[in] y set \e sum *= \e y. **********************************************************************/ Accumulator& operator*=(T y) { using std::fma; T d = _s; _s *= y; d = fma(y, d, -_s); // the error in the first multiplication _t = fma(y, _t, d); // add error to the second term return *this; } /** * Reduce accumulator to the range [-y/2, y/2]. * * @param[in] y the modulus. **********************************************************************/ Accumulator& remainder(T y) { using std::remainder; _s = remainder(_s, y); Add(0); // This renormalizes the result. return *this; } /** * Test equality of an Accumulator with a number. **********************************************************************/ bool operator==(T y) const { return _s == y; } /** * Test inequality of an Accumulator with a number. **********************************************************************/ bool operator!=(T y) const { return _s != y; } /** * Less operator on an Accumulator and a number. **********************************************************************/ bool operator<(T y) const { return _s < y; } /** * Less or equal operator on an Accumulator and a number. **********************************************************************/ bool operator<=(T y) const { return _s <= y; } /** * Greater operator on an Accumulator and a number. **********************************************************************/ bool operator>(T y) const { return _s > y; } /** * Greater or equal operator on an Accumulator and a number. **********************************************************************/ bool operator>=(T y) const { return _s >= y; } }; } // namespace GeographicLib #endif // GEOGRAPHICLIB_ACCUMULATOR_HPP geosphere/vignettes/0000755000176200001440000000000014334225370014246 5ustar liggesusersgeosphere/vignettes/geosphere.Rnw0000644000176200001440000004033113472155746016733 0ustar liggesusers\documentclass{article} \usepackage{natbib} \usepackage{graphics} \usepackage{amsmath} \usepackage{indentfirst} \usepackage[utf8]{inputenc} \usepackage{hyperref} \usepackage{hanging} %\VignetteIndexEntry{Introduction to the geosphere package} \SweaveOpts{keep.source=TRUE} \SweaveOpts{png=TRUE, pdf=FALSE} \SweaveOpts{resolution=100} \begin{document} <>= options(keep.source = TRUE, width = 60) foo <- packageDescription("geosphere") @ \title{Introduction to the "geosphere" package \\ (Version \Sexpr{foo$Version})} \author{Robert J. Hijmans} \maketitle \section{Introduction} This vignette describes the R package '\verb@geosphere@'. The package implements spherical trigonometry functions for geographic applications. Many of the functions have applications in navigation, but others are more general, or have no relation to navigation at all. There is a number of functions to compute distance and direction (= bearing, azimuth, course) along great circles (= shortest distance on a sphere, or "as the crow flies") and along rhumb lines (lines of constant bearing). There are also functions that compute distances on a spheroid. Other functions include the computation of the location of an object at a given direction and distance; and the area, perimeter, and centroid of a spherical polygon. Geographic locations must be specified in longitude and latitude (and in that order!) in degrees (i.e., NOT in radians). Degrees are (obviously) in decimal notation. Thus 12 degrees, 10 minutes, 30 seconds = 12 + 10/60 + 30/3600 = 12.175 degrees. The southern and western hemispheres have a negative sign. The default unit of distance is meter; but this can be adjusted by supplying a different radius 'r' to functions. Directions are expressed in degrees (N = 0 and 360, E = 90, S = 180, and W = 270 degrees). If arguments of functions that take several arguments (e.g. points, bearings, radius of the earth), do not have the same length (for vectors) or number of rows (for matrices) the shorter arguments are re-cycled. Many functions in this package are based on formulae provided by Ed Williams (\url{http://www.edwilliams.org/ftp/avsig/avform.txt}, and partly on javascript implementations of these formulae by Chris Veness (\url{http://www.movable-type.co.uk/scripts/latlong.html} ) Most geodesic computations (for a spheroid rather than a sphere) use the GeographicLib by C.F.F. Karney (\url{http://geographiclib.sourceforge.net/}. \section{Great circle distance} There are four different functions to compute distance between two points. These are, in order of increasing complexity of the algorithm, the 'Spherical law of cosines', 'Haversine' (Sinnott, 1984), 'Vincenty Sphere' and 'Vincenty Ellipsoid' (Vincenty, 1975) methods. The first three assume the earth to be a sphere, while the 'Vincenty Ellipsoid' assumes it is an ellipsoid (which is closer to the truth). The results from the first three functions are identical for practical purposes. The Haversine ('half-versed-sine') formula was published by R.W. Sinnott in 1984, although it has been known for much longer. At that time computational precision was lower than today (15 digits precision). With current precision, the spherical law of cosines formula appears to give equally good results down to very small distances. If you want greater accuracy, you could use the distVincentyEllipsoid method. Below the differences between the three spherical methods are illustrated. At very short distances, there are small differences between the 'law of the Cosine' and the other two methods. There are even smaller differences between the 'Haversine' and 'Vincenty Sphere' methods at larger distances. <>= library(geosphere) Lon <- c(1:9/1000, 1:9/100, 1:9/10, 1:90*2) Lat <- c(1:9/1000, 1:9/100, 1:9/10, 1:90) dcos <- distCosine(c(0,0), cbind(Lon, Lat)) dhav <- distHaversine(c(0,0), cbind(Lon, Lat)) dvsp <- distVincentySphere(c(0,0), cbind(Lon, Lat)) par(mfrow=(c(1,2))) plot(log(dcos), dcos-dhav, col='red', ylim=c(-1e-05, 1e-05), xlab="Log 'Law of Cosines' distance (m)", ylab="Law of Cosines minus Haversine distance") plot(log(dhav), dhav-dvsp, col='blue', xlab="Log 'Haversine' distance (m)", ylab="Vincenty Sphere minus Haversine distance") @ The difference with the 'Vincenty Ellipsoid' method is more pronounced. In the example below (using the default WGS83 ellipsoid), the difference is about 0.3% at very small distances, and 0.15% at larger distances. <>= dvse <- distVincentyEllipsoid(c(0,0), cbind(Lon, Lat)) plot(dvsp/1000, (dvsp-dvse)/1000, col='blue', xlab='Vincenty Sphere Distance (km)', ylab="Difference between 'Vincenty Sphere' and 'Vincenty Ellipsoid' methods (km)") @ For the most precise distance computation use the 'distGeo' function. \section{Points on great circles} Points on a great circle are returned by the function 'greatCircle', using two points on the great circle to define it, and an additional argument to indicate how many points should be returned. You can also use greatCircleBearing, and provide starting points and bearing as arguments. gcIntermediate only returns points on the great circle that are on the track of shortest distance between the two points defining the great circle; and midPoint computes the point half-way between the two points. You can use onGreatCircle to test whether a point is on a great circle between two other points. <>= LA <- c(-118.40, 33.95) NY <- c(-73.78, 40.63) data(wrld) plot(wrld, type='l') gc <- greatCircle(LA, NY) lines(gc, lwd=2, col='blue') gci <- gcIntermediate(LA, NY) lines(gci, lwd=4, col='green') points(rbind(LA, NY), col='red', pch=20, cex=2) mp <- midPoint(LA, NY) onGreatCircle(LA,NY, rbind(mp,c(0,0))) points(mp, pch='*', cex=3, col='orange') greatCircleBearing(LA, brng=270, n=10) @ \section{Point at distance and bearing} Function destPoint returns the location of point given a point of origin, and a distance and bearing. Its perhaps obvious use in georeferencing locations of distant sitings. It can also be used to make circular polygons (with a fixed radius, but in longitude/latitude coordinates) <>= destPoint(LA, b=65, d=100000) circle=destPoint(c(0,80), b=1:365, d=1000000) circle2=destPoint(c(0,80), b=1:365, d=500000) circle3=destPoint(c(0,80), b=1:365, d=100000) plot(circle, type='l') polygon(circle, col='blue', border='black', lwd=4) polygon(circle2, col='red', lwd=4, border='orange') polygon(circle3, col='white', lwd=4, border='black') @ \section{Maximum latitude on a great circle} You can use the functions illustrated below to find out what the maximum latitude is that a great circle will reach; at what latitude it crosses a specified longitude; or at what longitude it crosses a specified latitude. From the map below it appears that Clairaut's formula, used in gcMaxLat is not very accurate. Through optimization with function greatCircle, a more accurate value was found. The southern-most point is the antipode (a point at the opposite end of the world) of the northern-most point. <>= ml <- gcMaxLat(LA, NY) lat0 <- gcLat(LA, NY, lon=0) lon0 <- gcLon(LA, NY, lat=0) plot(wrld, type='l') lines(gc, lwd=2, col='blue') points(ml, col='red', pch=20, cex=2) points(cbind(0, lat0), pch=20, cex=2, col='yellow') points(t(rbind(lon0, 0)), pch=20, cex=2, col='green' ) f <- function(lon){gcLat(LA, NY, lon)} opt <- optimize(f, interval=c(-180, 180), maximum=TRUE) points(opt$maximum, opt$objective, pch=20, cex=2, col='dark green' ) anti <- antipode(c(opt$maximum, opt$objective)) points(anti, pch=20, cex=2, col='dark blue' ) @ \section{Great circle intersections} Points of intersection of two great circles can be computed in two ways. We use a second great circle that connects San Francisco with Amsterdam. We first compute where they cross by defining the great circles using two points on it (gcIntersect). After that, we compute the same points using a start point and initial bearing (gcIntersectBearing). The two points where the great circles cross are antipodes. Antipodes are connected with an infinite number of great circles. <>= SF <- c(-122.44, 37.74) AM <- c(4.75, 52.31) gc2 <- greatCircle(AM, SF) plot(wrld, type='l') lines(gc, lwd=2, col='blue') lines(gc2, lwd=2, col='green') int <- gcIntersect(LA, NY, SF, AM) int antipodal(int[,1:2], int[,3:4]) points(rbind(int[,1:2], int[,3:4]), col='red', pch=20, cex=2) bearing1 <- bearing(LA, NY) bearing2 <- bearing(SF, AM) bearing1 bearing2 gcIntersectBearing(LA, bearing1, SF, bearing2) @ \section{Triangulation} Below is triangulation example. We have three locations (NY, LA, MS) and three directions (281, 60, 195) towards a target. Because we are on a sphere, there are two (antipodal) results. We only show one here (by only using int[,1:2]). We compute the centroid from the polygon defined with the three points. To accurately draw a spherical polygon, we can use makePoly. This function inserts intermediate points along the paths between the vertices provided (default is one point every 10 km). <>= MS <- c(-93.26, 44.98) gc1 <- greatCircleBearing(NY, 281) gc2 <- greatCircleBearing(MS, 195) gc3 <- greatCircleBearing(LA, 55) plot(wrld, type='l', xlim=c(-125, -70), ylim=c(20, 60)) lines(gc1, col='green') lines(gc2, col='blue') lines(gc3, col='red') int <- gcIntersectBearing(rbind(NY, NY, MS), c(281, 281, 195), rbind(MS, LA, LA), c(195, 55, 55)) int distm(rbind(int[,1:2], int[,3:4])) int <- int[,1:2] points(int) poly <- rbind(int, int[1,]) centr <- centroid(poly) poly2 <- makePoly(int) polygon(poly2, col='yellow') points(centr, pch='*', col='dark red', cex=2) @ \section{Bearing} Below we first compute the distance and bearing from Los Angeles (LA) to New York (NY). These are then used to compute the point from LA at that distance in that (initial) bearing (direction). Bearing changes continuously when traveling along a Great Circle. The final bearing, when approaching NY, is also given. <>= d <- distCosine(LA, NY) d b <- bearing(LA, NY) b destPoint(LA, b, d) NY finalBearing(LA, NY) @ \section{Getting off-track} What if we went off-course and were flying over Minneapolis (MS)? The closest point on the planned route (p) can be computed with the alongTrackDistance and destPoint functions. The distance from 'p' to MS can be computed with the dist2gc (distance to great circle, or cross-track distance) function. The light green line represents the along-track distance, and the dark green line represents the cross-track distance. <>= atd <- alongTrackDistance(LA, NY, MS) p <- destPoint(LA, b, atd) plot(wrld, type='l', xlim=c(-130,-60), ylim=c(22,52)) lines(gci, col='blue', lwd=2) points(rbind(LA, NY), col='red', pch=20, cex=2) points(MS[1], MS[2], pch=20, col='blue', cex=2) lines(gcIntermediate(LA, p), col='green', lwd=3) lines(gcIntermediate(MS, p), col='dark green', lwd=3) points(p, pch=20, col='red', cex=2) dist2gc(LA, NY, MS) distCosine(p, MS) @ \section{Distance to a polyline} The two function describe above are used in the dist2Line function that computes the shortest distance between a set of points and a set of spherical poly-lines (or polygons). <>= line <- rbind(c(-180,-20), c(-150,-10), c(-140,55), c(10, 0), c(-140,-60)) pnts <- rbind(c(-170,0), c(-75,0), c(-70,-10), c(-80,20), c(-100,-50), c(-100,-60), c(-100,-40), c(-100,-20), c(-100,-10), c(-100,0)) d = dist2Line(pnts, line) plot( makeLine(line), type='l') points(line) points(pnts, col='blue', pch=20) points(d[,2], d[,3], col='red', pch='x', cex=2) for (i in 1:nrow(d)) lines(gcIntermediate(pnts[i,], d[i,2:3], 10), lwd=2, col='green') @ \section{Rhumb lines} Rhumb (from the Spanish word for course, 'rumbo') lines are straight lines on a Mercator projection map (and at most latitudes pretty straight on an equirectangular projection (=unprojected lon/lat) map). They were used in navigation because it is easier to follow a constant compass bearing than to continually adjust direction as is needed to follow a great circle, even though rhumb lines are normally longer than great-circle (orthodrome) routes. Most rhumb lines will gradually spiral towards one of the poles. <>= NP <- c(0, 85) bearing(SF, NP) b <- bearingRhumb(SF, NP) b dc <- distCosine(SF, NP) dr <- distRhumb(SF, NP) dc / dr pr <- destPointRhumb(SF, b, d=round(dr/100) * 1:100) pc <- rbind(SF, gcIntermediate(SF, NP), NP) par(mfrow=c(1,2)) data(wrld) plot(wrld, type='l', xlim=c(-140,10), ylim=c(15,90), main='Equirectangular') lines(pr, col='blue') lines(pc, col='red') data(merc) plot(merc, type='l', xlim=c(-15584729, 1113195), ylim=c(2500000, 22500000), main='Mercator') lines(mercator(pr), col='blue') lines(mercator(pc), col='red') @ \section{Characterizing polygons} The package has functions to compute the area, perimeter, centroid, and 'span' of a spherical polygon. One approach to compute these measures is to project the polygons first. Here we directly compute them based on spherical coordinates (longitude / latitude), except for centroid, which is computed by projecting the data to the Mercator projection (and inversely projecting the result). The function makePoly inserts additional vertices into a spherical polygon such that it can be plotted (perhaps after first projecting it) more correctly in a plane. Vertices are inserted, where necessary, at a specified distance. The function is only beneficial for polygons with large inter-vertex distances (in terms of longitude), particularly at high latitudes. <>= pol <- rbind(c(-120,-20), c(-80,5), c(0, -20), c(-40,-60), c(-120,-20)) areaPolygon(pol) perimeter(pol) centroid(pol) span(pol, fun=max) nicepoly = makePoly(pol) plot(pol, xlab='longitude', ylab='latitude', cex=2, lwd=3, xlim=c(-140, 0)) lines(wrld, col='grey') lines(pol, col='red', lwd=2) lines(nicepoly, col='blue', lwd=2) points(centroid(pol), pch='*', cex=3, col='dark green') text(centroid(pol)-c(0,2.5), 'centroid') legend(-140, -48, c('planar','spherical'), lty=1, lwd=2, col=c('red', 'blue'), title='polygon type') @ \section{Sampling} Random or regular sampling of longitude/latitude values on the globe needs to consider that the globe is spherical. That is, if you would take random points for latitude between -90 and 90 and for longitude between -180 and 180, the density of points would be higher near the poles than near the equator. In contrast, functions 'randomCoordinates' and 'randomCoordinates' return samples that are spatially balanced. <>= plot(wrld, type='l', col='grey') a = randomCoordinates(500) points(a, col='blue', pch=20, cex=0.5) b = regularCoordinates(3) points(b, col='red', pch='x') @ \section{Daylength} You can compute daylenght according to the formula by Forsythe et al. (1995). For any day of the year (an integer between 1 and 365; or a 'Date' object. <>= as.Date(80, origin='2009-12-31') as.Date(172, origin='2009-12-31') plot(0:90, daylength(lat=0:90, doy=1), ylim=c(0,24), type='l', xlab='Latitude', ylab='Daylength', main='Daylength by latitude and day of year', lwd=2) lines(0:90, daylength(lat=0:90, doy=80), col='green', lwd=2) lines(0:90, daylength(lat=0:90, doy=172), col='blue', lwd=2) legend(0,24, c('1','80','172'), lty=1, lwd=2, col=c('black', 'green', 'blue'), title='Day of year') @ \section{References} \begin{hangparas}{3em}{1} \noindent Forsythe, W.C., E.J. Rykiel Jr., R.S. Stahl, H. Wu and R.M. Schoolfield, 1995. A model comparison for daylength as a function of latitude and day of the year. Ecological Modeling 80:87-95. \noindent Sinnott, R.W, 1984. Virtues of the Haversine. Sky and Telescope 68(2): 159 \noindent Vincenty, T. 1975. Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations. Survey Review 23(176): 88-93. Available here: \url{http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf} \end{hangparas} \end{document} geosphere/R/0000755000176200001440000000000014334225313012434 5ustar liggesusersgeosphere/R/centroid.R0000644000176200001440000000333113472155746014404 0ustar liggesusers# Author: Robert J. Hijmans # April 2010 # version 0.1 # license GPL3 # See http://local.wasp.uwa.edu.au/~pbourke/geometry/polyarea/ .basiccentroid <- function(p) { p2 <- rbind(p[-1,], p[1,]) P <- p[,1] * p2[,2] - p2[,1] * p[,2] area6 <- 6 * sum(P) / 2 x <- sum((p[,1] + p2[,1]) * P) y <- sum((p[,2] + p2[,2]) * P) return(cbind(x, y) / area6 ) } if (!isGeneric("centroid")) { setGeneric("centroid", function(x, ...) standardGeneric("centroid")) } setMethod("centroid", signature(x='data.frame'), function(x) { centroid(as.matrix(x)) }) setMethod("centroid", signature(x='matrix'), function(x) { x <- .pointsToMatrix(x, poly=TRUE) dif1 <- max(x[,1]) - min(x[,1]) rotated <- FALSE if (dif1 > 180) { x2 <- x x2[,1] <- x2[,1]%%(360) - 180 dif1 <- max(x[,1]) - min(x[,1]) dif2 <- max(x2[,1]) - min(x2[,1]) if (dif2 < dif1) { rotated <- TRUE x <- x2 } } x <- mercator(x, r=1) cenM <- .basiccentroid(x) cenG <- mercator(cenM, r=1, inverse=TRUE) if (rotated) { cenG[,1] <- cenG[,1] + 180 cenG[,1] <- .normalizeLonDeg(cenG[,1]) } rownames(cenG) <- NULL return(cenG) } ) setMethod("centroid", signature(x='SpatialPolygons'), function(x) { if ( isTRUE(is.projected(x)) ) { return( coordinates(x)) } x <- x@polygons n <- length(x) res <- matrix(nrow=n, ncol=2) for (i in 1:n) { parts <- length(x[[i]]@Polygons ) parea <- sapply(x[[i]]@Polygons, function(y){ methods::slot(y, "area")} ) hole <- sapply(x[[i]]@Polygons, function(y){ methods::slot(y, "hole")} ) parea[hole] <- -1 j <- which.max(parea) crd <- x[[i]]@Polygons[[j]]@coords res[i,] <- centroid(crd) } return(res) } ) geosphere/R/gcIntersect.R0000644000176200001440000000457613472155746015063 0ustar liggesusers# author Robert Hijmans # October 2009 # version 0.1 # license GPL3 # based on an alogrithm described by Ed Williams # http://www.edwilliams.org/intersect.htm # Not used #gete <- function(lon, lat) { # ex <- cos(lat)*cos(lon) # ey <- -cos(lat)*sin(lon) # ez <- sin(lat) # return(cbind(ex, ey, ez)) #} gcIntersect <- function(p1, p2, p3, p4) { #intersection of two great circles defined by pt1 to pt2 and pt3 to pt4. einv <- function(e) { lat <- atan2(e[,3], sqrt(e[,1]^2 + e[,2]^2)) lon <- atan2(-e[,2], e[,1]) return(cbind(lon, lat)) } eXe5 <- function(lon1, lat1, lon2, lat2) { ex <- sin(lat1-lat2) *sin((lon1+lon2)/2) *cos((lon1-lon2)/2) - sin(lat1+lat2) *cos((lon1+lon2)/2) *sin((lon1-lon2)/2) ey <- sin(lat1-lat2) *cos((lon1+lon2)/2) *cos((lon1-lon2)/2) + sin(lat1+lat2) *sin((lon1+lon2)/2) *sin((lon1-lon2)/2) ez <- cos(lat1)*cos(lat2)*sin(lon1-lon2) return( cbind(ex, ey, ez) ) } eXe3 <- function(e1, e2) { x <- e1[,2] * e2[,3] -e2[,2] *e1[,3] y <- e1[,3] *e2[,1] -e2[,3] *e1[,1] z <- e1[,1] *e2[,2] -e1[,2] *e2[,1] return(cbind(x,y,z)) } eSQRT <- function(e) { return(sqrt(e[,1]^2 + e[,2]^2 + e[,3]^2)) } p1 <- .pointsToMatrix(p1) p2 <- .pointsToMatrix(p2) p3 <- .pointsToMatrix(p3) p4 <- .pointsToMatrix(p4) p1 <- cbind(p1[,1], p1[,2], p2[,1], p2[,2]) p3 <- cbind(p3[,1], p3[,2], p4[,1], p4[,2]) p <- cbind(p1[,1], p1[,2], p1[,3], p1[,4], p3[,1], p3[,2], p3[,3], p3[,4]) p1 <- p[,1:2,drop=FALSE] p2 <- p[,3:4,drop=FALSE] p3 <- p[,5:6,drop=FALSE] p4 <- p[,7:8,drop=FALSE] res <- matrix(NA, nrow=nrow(p1), ncol=4) colnames(res) <- c('lon1', 'lat1', 'lon2', 'lat2') keep <- ! antipodal(p1, p2) | antipodal(p3, p4) keep <- keep & ! apply(p1 == p2, 1, sum) == 2 if (sum(keep) == 0) { return(res) } toRad <- pi / 180 p1 <- p1[keep, , drop=FALSE] * toRad p2 <- p2[keep, , drop=FALSE] * toRad p3 <- p3[keep, , drop=FALSE] * toRad p4 <- p4[keep, , drop=FALSE] * toRad e1Xe2 <- eXe5(p1[,1], p1[,2], p2[,1], p2[,2]) e3Xe4 <- eXe5(p3[,1], p3[,2], p4[,1], p4[,2]) ea <- e1Xe2 / eSQRT(e1Xe2) eb <- e3Xe4 / eSQRT(e3Xe4) eaXeb <- eXe3(ea, eb) ll <- einv(eaXeb) ll2 <- cbind(ll[,1] + pi, -ll[,2]) pts <- cbind(ll, ll2) pts[,1] <- .normalizeLonRad(pts[,1]) pts[,3] <- .normalizeLonRad(pts[,3]) res[keep,] <- pts / toRad return(res) } geosphere/R/greatCircle.R0000644000176200001440000000220413472155746015017 0ustar liggesusers# author Robert Hijmans # October 2009 # version 0.1 # license GPL greatCircle <- function(p1, p2, n=360, sp=FALSE) { p1 <- .pointsToMatrix(p1) p2 <- .pointsToMatrix(p2) p <- cbind(p1[,1], p1[,2], p2[,1], p2[,2], n) p1 <- p[,1:2] p2 <- p[,3:4] n <- pmax(round(p[,5]), 1) if (nrow(p) == 1) { lon <- (1:n * 360 / n) - 180 lat <- gcLat(p1, p2, lon) res <- cbind(lon,lat) if (sp) { lat <- gcLat(p1, p2, 180) res <- list(rbind(cbind(-180, lat), res)) res <- SpatialLines( list( Lines( list( Line (res)), ID=as.character(1)) ), CRS("+proj=longlat +ellps=WGS84")) } } else { res <- list() for (i in 1:nrow(p1)) { lon <- (1:n[i] * 360 / n[i]) - 180 lat <- gcLat(p1[i,], p2[i,], lon) res[[i]] <- cbind(lon, lat) } if (sp) { for (i in 1:length(res)) { lat <- gcLat(p1[i,], p2[i,], 180) res[[i]] <- rbind(cbind(-180, lat), res[[i]]) res[[i]] <- Lines( list( Line (res[[i]])), ID=as.character(i)) } res <- SpatialLines(res, CRS("+proj=longlat +ellps=WGS84")) } } return(res) } #greatCircle(rbind(cbind(5,52), cbind(5,15)), c(-120,37), n=12) geosphere/R/sampleAlong.R0000644000176200001440000000464113472155746015044 0ustar liggesusers# Based on code by Barry Rowlingson #http://r-sig-geo.2731867.n2.nabble.com/how-to-generate-perpendicular-transects-along-a-line-feature-td7583710.html # Some adaptations by Robert Hijmans # January 2016 # version 0.1 # License GPL3 .evenspace <- function(xy, sep, start=0, size, direction=FALSE){ dx <- c(0,diff(xy[,1])) dy <- c(0,diff(xy[,2])) dseg <- sqrt(dx^2+dy^2) dtotal <- cumsum(dseg) linelength <- sum(dseg) pos <- seq(start,linelength, by=sep) whichseg <- unlist(lapply(pos, function(x){sum(dtotal<=x)})) x0 <- xy[whichseg,1] y0 <- xy[whichseg,2] x1 <- xy[whichseg+1,1] y1 <- xy[whichseg+1,2] dtotal <- dtotal[whichseg] further <- pos - dtotal dseg <- dseg[whichseg+1] f <- further/dseg x <- x0 + f * (x1-x0) y <- y0 + f * (y1-y0) r <- data.frame(x, y) # if (direction) { # r$direction <- atan2(y0-y1,x0-x1) # } r } .transect <- function(pts, len){ pts$thetaT = pts$theta+pi/2 dx <- len*cos(pts$thetaT) dy <- len*sin(pts$thetaT) data.frame(x0 = pts$x + dx, y0 = pts$y + dy, x1 = pts$x - dx, y1 = pts$y -dy) } .sampleAlong <- function(x, interval) { if (inherits(x, 'SpatialPolygons')) { line <- methods::as(line, 'SpatialLines') } if (inherits(x, 'SpatialLines')) { requireNamespace('raster') x <- raster::geom(x) allpts <- NULL for (p in unique(x[, 'cump'])) { y <- x[x[, 'cump']==p, c('x', 'y')] pts <- .evenspace(y, interval, direction=FALSE) allpts <- rbind(allpts, pts) } return(allpts) } else { x <- .pointsToMatrix(x) .evenspace(x, interval, direction=FALSE) } } .sampleAlongPerpendicular <- function(x, interval, pdist, np=1 ) { if (inherits(x, 'SpatialPolygons')) { line <- methods::as(line, 'SpatialLines') } if (inherits(x, 'SpatialLines')) { requireNamespace('raster') x <- raster::geom(x) allpts <- NULL for (p in unique(x[, 'cump'])) { y <- x[x[, 'cump']==p, c('x', 'y')] tspts <- .evenspace(y, interval) pts <- NULL for (i in 1:np) { pts1 <- .transect(tspts, i * pdist) pts <- cbind(pts, pts1) } allpts <- rbind(allpts, pts) } return(allpts) } else { x <- .pointsToMatrix(x) y <- .evenspace(x, interval) pts <- NULL for (i in 1:np) { pts1 <- .transect(y, i * pdist) pts <- cbind(pts, pts1) } return(pts) } } geosphere/R/distVincentySphere.R0000644000176200001440000000150213472155746016425 0ustar liggesusers# Author: Robert J. Hijmans # Date : June 2008 # Version 0.8 (taken from Raster package) # Licence GPL v3 # Vincenty formula for a sphere # http://en.wikipedia.org/wiki/Great_circle_distance distVincentySphere <- function(p1, p2, r=6378137) { toRad <- pi / 180 p1 <- .pointsToMatrix(p1) * toRad if (missing(p2)) { p2 <- p1[-1, ,drop=FALSE] p1 <- p1[-nrow(p1), ,drop=FALSE] } else { p2 <- .pointsToMatrix(p2) * toRad } p <- cbind(p1[,1], p1[,2], p2[,1], p2[,2], as.vector(r)) lon1 <- p[,1] lat1 <- p[,2] lon2 <- p[,3] lat2 <- p[,4] x <- sqrt((cos(lat2) * sin(lon1-lon2))^2 + (cos(lat1) * sin(lat2) - sin(lat1) * cos(lat2) * cos(lon1-lon2))^2) y <- sin(lat1) * sin(lat2) + cos(lat1) * cos(lat2) * cos(lon1-lon2) dist <- p[,5] * atan2(x, y) return ( as.vector(dist )) } geosphere/R/dist2gc.R0000644000176200001440000000130613472155746014134 0ustar liggesusers# based on code by Ed Williams # Licence: GPL # http://www.edwilliams.org/avform.htm#XTE # Port to R by Robert Hijmans # October 2009 # version 0.1 # license GPL3 dist2gc <- function(p1, p2, p3, r=6378137, sign=FALSE) { toRad <- pi / 180 p1 <- .pointsToMatrix(p1) p2 <- .pointsToMatrix(p2) p3 <- .pointsToMatrix(p3) r <- as.vector(r) p <- cbind(p1[,1], p1[,2], p2[,1], p2[,2], p3[,1], p3[,2], r) p1 <- p[,1:2] p2 <- p[,3:4] p3 <- p[,5:6] r <- p[,7] tc <- bearing(p1, p2, a=r, f=0) * toRad tcp <- bearing(p1, p3, a=r, f=0) * toRad dp <- distCosine(p1, p3, r=1) xtr <- (asin(sin(tcp-tc) * sin(dp)) * r) xtr <- as.vector(xtr) if (!sign) xtr <- abs(xtr) xtr } geosphere/R/distHaversine.R0000644000176200001440000000500014032771416015367 0ustar liggesusers# author of original JavaScript code: Chris Vennes # (c) 2002-2009 Chris Veness # http://www.movable-type.co.uk/scripts/latlong.html # Licence: LGPL, without any warranty express or implied # Port to R by Robert Hijmans # October 2009 # version 0.1 # license GPL3 distHaversine <- function(p1, p2, r=6378137) { #* Haversine formula to calculate distance between two points specified by #* from: Haversine formula - R.W. Sinnott, "Virtues of the Haversine", #* Sky and Telescope, vol 68, no 2, 1984 #* http:#//www.census.gov/cgi-bin/geo/gisfaq?Q5.1 # source http://www.movable-type.co.uk/scripts/latlong.html # (c) 2002-2009 Chris Veness toRad <- pi / 180 p1 <- .pointsToMatrix(p1) * toRad if (missing(p2)) { p2 <- p1[-1, ,drop=FALSE] p1 <- p1[-nrow(p1), ,drop=FALSE] } else { p2 <- .pointsToMatrix(p2) * toRad } p = cbind(p1[,1], p1[,2], p2[,1], p2[,2], as.vector(r)) dLat <- p[,4]-p[,2] dLon <- p[,3]-p[,1] a <- (sin(dLat/2))^2 + cos(p[,2]) * cos(p[,4]) * (sin(dLon/2))^2 # to avoid values of 'a' that are a sliver above 1 # which may occur at antipodes # https://stackoverflow.com/questions/45889616/why-does-disthaversine-return-nan-for-some-pairs-of-coordinates# a <- pmin(a, 1) dist <- 2 * atan2(sqrt(a), sqrt(1-a)) * p[,5] return( as.vector(dist)) } # lon1 <- p[,1] # lat1 <- p[,2] # lon2 <- p[,3] # lat2 <- p[,4] # r <- p[,5] # dLat <- (lat2-lat1) # dLon <- (lon2-lon1) # a <- sin(dLat/2) * sin(dLat/2) + cos(lat1) * cos(lat2) * sin(dLon/2) * sin(dLon/2) # dist <- 2 * atan2(sqrt(a), sqrt(1-a)) * r .distHaversine2 <- function(p1, p2, r=6378137) { ## following wikipedia toRad <- pi / 180 p1 <- .pointsToMatrix(p1) * toRad if (missing(p2)) { p2 <- p1[-1, ,drop=FALSE] p1 <- p1[-nrow(p1), ,drop=FALSE] } else { p2 <- .pointsToMatrix(p2) * toRad } p = cbind(p1[,1], p1[,2], p2[,1], p2[,2], as.vector(r)) dLat <- p[,4]-p[,2] dLon <- p[,3]-p[,1] a <- (sin(dLat/2))^2 + cos(p[,2]) * cos(p[,4]) * (sin(dLon/2))^2 a <- pmin(a, 1) dist <- 2 * r * asin(sqrt(a)) return( as.vector(dist)) } # from Thierry de Meeus .distHaversine3 <- function(p1, p2, r=6378137) { toRad <- pi / 180 p1 <- .pointsToMatrix(p1) * toRad if (missing(p2)) { p2 <- p1[-1, ,drop=FALSE] p1 <- p1[-nrow(p1), ,drop=FALSE] } else { p2 <- .pointsToMatrix(p2) * toRad } p <- cbind(p1[,1], p1[,2], p2[,1], p2[,2], as.vector(r)) r*(pi/2-asin(sin((p[,4]))*sin((p[,2]))+cos((p[,3])-(p[,1]))*cos((p[,4]))*cos((p[,2])))) } geosphere/R/span.R0000644000176200001440000000443713472155746013546 0ustar liggesusers# Author: Robert J. Hijmans, r.hijmans@gmail.com # Date : April 2010 # Version 1 # Licence GPL v3 if (!isGeneric("span")) { setGeneric("span", function(x, ...) standardGeneric("span")) } setMethod("span", signature(x='matrix'), function(x, nbands='fixed', n=100, res=0.1, fun, r=6378137, ...) { dif1 <- max(x[,1]) - min(x[,1]) rotated <- FALSE if (dif1 > 180) { x2 <- x x2[,1] <- x2[,1] %% 360 - 180 dif1 <- max(x[,1]) - min(x[,1]) dif2 <- max(x2[,1]) - min(x2[,1]) if (dif2 < dif1) { rotated <- TRUE x <- x2 } } x <- SpatialPolygons(list(Polygons(list(Polygon(x)), 1))) if (missing(fun)) { x <- span(x, nbands=nbands, n=n, res=res, ...) } else { x <- span(x, nbands=nbands, n=n, res=res, fun=fun, ...) } if (rotated & missing(fun)) { x$longitude = x$longitude + 180 } return(x) } ) setMethod("span", signature(x='SpatialPolygons'), function(x, nbands='fixed', n=100, res=0.1, fun, ...) { if (!requireNamespace('raster')) {stop('you need to install the "raster" package to use this function')} if (! nbands %in% c('fixed', 'variable')) { stop('bandwidth should be "fixed" or "variable"') } if (nbands == 'fixed') { n = max(n, 1) } else { if (res <= 0) { stop('res should be larger than zero') } } npol <- length(x@polygons) lonspan <- list() latspan <- list() lon <- list() lat <- list() for (i in 1:npol) { pp <- x[i,] rs <- raster::raster(pp) if (nbands == 'fixed') { dim(rs) <- c(n, n) } else { raster::res(rs) <- res } latitude <- raster::yFromRow(rs, 1:nrow(rs)) longitude <- raster::xFromCol(rs, 1:ncol(rs)) xd <- distGeo(cbind(0,latitude), cbind(raster::xres(rs),latitude), ...) yd <- distGeo(cbind(0,0), cbind(0,raster::yres(rs)), ...) rs <- raster::rasterize(pp, rs, silent=TRUE) rs <- raster::getValues(rs, format='matrix') latspan[[i]] <- as.vector(apply(rs, 1, sum, na.rm=TRUE) * yd) lonspan[[i]] <- as.vector(apply(rs, 2, sum, na.rm=TRUE) * xd) lat[[i]] <- latitude lon[[i]] <- longitude } if (! missing(fun)) { lon = sapply(lonspan, fun) lat = sapply(latspan, fun) return(cbind(lon, lat)) } else { return(c(lonspan=lonspan, latspan=latspan, longitude=lon, latitude=lat)) } } ) geosphere/R/destPoint.R0000644000176200001440000000127014161534117014533 0ustar liggesusers# Author: Robert J. Hijmans # Date : May 2015 # Licence GPL v3 destPoint <- function(p, b, d, a=6378137, f=1/298.257223563, ...) { # calculate destination point given start point, initial bearing (deg) and distance (m) r <- list(...)$r if (!is.null(r)) { # for backwards compatibility return( .old_destPoint(p, b, d, r=r) ) } b <- as.vector(b) d <- as.vector(d) p <- .pointsToMatrix(p) p <- cbind(p[,1], p[,2], b, d) r <- .geodesic(as.double(p[,1]), as.double(p[,2]), as.double(p[,3]), as.double(p[,4]), as.double(a), as.double(f)) r <- matrix(r, ncol=3, byrow=TRUE) colnames(r) <- c('lon', 'lat', 'finalbearing') return(r[, 1:2, drop=FALSE]) } geosphere/R/finalBearing.R0000644000176200001440000000111014161534151015132 0ustar liggesusers# Robert Hijmans # October 2009 # version 0.1 # Licence: GPL3 finalBearing <- function(p1, p2, a=6378137, f=1/298.257223563, sphere=FALSE) { if (sphere) { # for backwards compatibility return(.old_bearing(p2, p1) ) } p1 <- .pointsToMatrix(p1) p2 <- .pointsToMatrix(p2) p <- cbind(p1[,1], p1[,2], p2[,1], p2[,2]) r <- .inversegeodesic(as.double(p[,1]), as.double(p[,2]), as.double(p[,3]), as.double(p[,4]), as.double(a[1]), as.double(f[1])) r <- matrix(r, ncol=3, byrow=TRUE) # colnames(r) <- c('lon', 'lat', 'finalbearing') return(r[, 3]) } geosphere/R/alongTrack.R0000644000176200001440000000155313472155746014666 0ustar liggesusers# based on code by Ed Williams # licence GPL # http://www.edwilliams.org/avform.htm#XTE # Port to R by Robert Hijmans # October 2009 # version 0.1 # license GPL3 alongTrackDistance <- function(p1, p2, p3, r=6378137) { toRad <- pi / 180 p1 <- .pointsToMatrix(p1) p2 <- .pointsToMatrix(p2) p3 <- .pointsToMatrix(p3) p <- cbind(p1[,1], p1[,2], p2[,1], p2[,2], p3[,1], p3[,2], as.vector(r)) p1 <- p[,1:2,drop=FALSE] p2 <- p[,3:4,drop=FALSE] p3 <- p[,5:6,drop=FALSE] r = p[,7] tc <- bearing(p1, p2) * toRad tcp <- bearing(p1, p3) * toRad dp <- distCosine(p1, p3, r=1) xtr <- asin(sin(tcp-tc) * sin(dp)) # +1/-1 for ahead/behind [lat1,lon1] bearing <- sign(cos(tc - tcp)) dist <- bearing * acos(cos(dp) / cos(xtr)) * r if (is.vector(dist)) { dist <- matrix(dist) } colnames(dist) <- 'distance' return(abs(dist)) } geosphere/R/destPointRhumb.R0000644000176200001440000000235313472155746015547 0ustar liggesusers# based on JavaScript code by Chris Vennes # (c) 2002-2009 Chris Veness # http://www.movable-type.co.uk/scripts/latlong.html # Licence: LGPL, without any warranty express or implied # see http://www.edwilliams.org/avform.htm#Rhumb # for the original formulae # Robert Hijmans # October 2009 # version 0.1 # license GPL3 destPointRhumb <- function(p, b, d, r=6378137) { toRad <- pi / 180 b <- as.vector(b) d <- as.vector(d) r <- as.vector(r) p <- .pointsToMatrix(p) p <- cbind(p[,1], p[,2], b, d, r) r <- p[,5] d <- p[,4] / r #angular distance in radians b <- p[,3] * toRad lon1 <- p[,1] * toRad lat1 <- p[,2] lat1[lat1==90|lat1==-90] <- NA lat1 <- lat1 * toRad lat2 <- lat1 + d * cos(b) dLat <- lat2-lat1 dPhi <- log( tan(lat2/2 + pi/4) / tan(lat1/2 + pi/4) ) i <- abs(dLat) > 1e-10 q <- vector(length=length(i)) q[i] <- dLat[i]/dPhi[i] q[!i] <- cos(lat1[!i]) dLon <- d * sin(b) / q # check for points past the pole../ i <- (abs(lat2) > pi/2) & lat2 > 0 lat2[i] <- pi-lat2[i] i <- (abs(lat2) > pi/2) & lat2 <= 0 lat2[i] <- (pi-lat2[i]) lon2 <- (lon1+dLon+pi)%%(2*pi) - pi res <- cbind(lon2, lat2) / toRad colnames(res) <- c('lon', 'lat') return(res) } geosphere/R/antipodal.R0000644000176200001440000000126314161435516014542 0ustar liggesusers# Author: Robert J. Hijmans # October 2009 # version 1.0 # license GPL3 antipodal <- function(p1, p2, tol=1e-9) { p1 <- .pointsToMatrix(p1) p2 <- .pointsToMatrix(p2) p <- cbind(p1[,1], p1[,2], p2[,1], p2[,2]) p[,c(1,3)] <- .normalizeLonDeg(p[,c(1,3)]) diflon <- abs(p[,1] - p[,3]) diflat <- abs(p[,2] + p[,4]) ## FIX by Gareth Davies # (diflat < tol) & (diflon > (180 - tol)) ## FIX by Jonathan Rynd # (diflat < tol) & (abs(diflon%%360 - 180) < tol) (diflat < tol) & ((cos(p[,2] * pi/180) * abs(diflon%%360 - 180)) < tol) } antipode <- function(p) { p <- .pointsToMatrix(p) p[,1] <- .normalizeLonDeg(p[,1] + 180) p[,2] <- -p[,2] return( p ) } geosphere/R/gcLat.R0000644000176200001440000000201413472155746013624 0ustar liggesusers# author Robert Hijmans # October 2009 # version 0.1 # license GPL gcLat <- function(p1, p2, lon) { # Intermediate points on a great circle # source: http://www.edwilliams.org/avform.htm toRad <- pi / 180 d <- distCosine(p1, p2) p1 <- .pointsToMatrix(p1) p2 <- .pointsToMatrix(p2) p <- cbind(p1[,1], p1[,2], p2[,1], p2[,2], as.vector(lon)) p1 <- p[,1:2,drop=FALSE] p2 <- p[,3:4,drop=FALSE] lon <- p[,5] res <- rep(NA, nrow(p)) notanti <- ! antipodal(p1, p2) lon1 <- p1[,1] * toRad lat1 <- p1[,2] * toRad lon2 <- p2[,1] * toRad lat2 <- p2[,2] * toRad lon <- lon * toRad # cannot compute this for a meridian notmeridians <- ! sin(lon1-lon2)==0 keep <- notanti & notmeridians if (sum(keep) == 0) { return(res) } lon1 <- lon1[keep] lat1 <- lat1[keep] lon2 <- lon2[keep] lat2 <- lat2[keep] lon <- lon[keep] res[keep] <- atan((sin(lat1)*cos(lat2)*sin(lon-lon2) -sin(lat2)*cos(lat1)*sin(lon-lon1))/(cos(lat1)*cos(lat2)*sin(lon1-lon2))) return(res / toRad) } geosphere/R/makePoly.R0000644000176200001440000000740713472155746014366 0ustar liggesusers# author Robert Hijmans # April 2010 # version 0.1 # license GPL .makeSinglePoly <- function(p, interval=10000, ...) { res <- p[1,] for (i in 1:(nrow(p)-1)) { if (! isTRUE( all.equal(p[i,], p[i+1,]) )) { d <- distGeo(p[i,], p[i+1,], ...) n <- floor(d / interval) if (n > 0) { pts <- gcIntermediate(p[i,],p[i+1,], n) pts <- rbind(p[i,], pts, p[i+1,]) res <- rbind(res, pts, p[i+1,]) } else { res <- rbind(res, p[i+1,]) } } } if (nrow(res) < 3) stop('cannot make a valid polygon') return(res) } .makeSingleLine <- function(p, interval=10000, ...) { res <- p[1,] for (i in 1:(nrow(p)-1)) { if (! isTRUE( all.equal(p[i,], p[i+1,]) )) { d <- distGeo(p[i,], p[i+1,], ...) n <- floor(d / interval) if (n > 0) { pts <- gcIntermediate(p[i,],p[i+1,], n) pts <- rbind(p[i,], pts, p[i+1,]) res <- rbind(res, pts, p[i+1,]) } else { res <- rbind(res, p[i+1,]) } } } if (nrow(res) < 2) stop('cannot make a valid line') return(res) } makePoly <- function(p, interval=10000, sp=FALSE, ...) { if (inherits(p, 'SpatialPolygons')) { test <- !is.projected(p) if (! isTRUE (test) ) { if (is.na(test)) { warning('Coordinate reference system of SpatialPolygons object is not set. Assuming it is degrees (longitude/latitude)!') } else { stop('Points are projected. They should be in degrees (longitude/latitude)') } # or rather transform them ....? } x <- p@polygons n <- length(x) polys = list() for (i in 1:n) { parts <- length(x[[i]]@Polygons ) partlist <- list() for (j in 1:parts) { crd <- x[[i]]@Polygons[[j]]@coords crd <- .makeSinglePoly(crd, interval=interval, ...) partlist[[j]] <- Polygon(crd) } polys[[i]] <- Polygons(partlist, i) } polys <- SpatialPolygons(polys) if (inherits(p, 'SpatialPolygonsDataFrame')) { rownames(p@data) <- 1:nrow(p@data) polys <- SpatialPolygonsDataFrame(polys, p@data) } polys@proj4string <- p@proj4string return(polys) } else { p <- .pointsToMatrix(p) if (nrow(p) < 3) { stop('cannot make a polygon (insufficent number of vertices)') } if (! isTRUE(all.equal(p[1,], p[nrow(p),]))) { p <- rbind(p, p[1,]) } res <- .makeSinglePoly(p, interval=interval, ...) if (sp) { res <- SpatialPolygons(list(Polygons(list(Polygon(res)), 1))) res@proj4string <- CRS("+proj=longlat +datum=WGS84") } return(res) } } makeLine <- function(p, interval=10000, sp=FALSE, ...) { if (inherits(p, 'SpatialLines')) { test <- !is.projected(p) if (! isTRUE (test) ) { if (is.na(test)) { warning('Coordinate reference system of SpatialPolygons object is not set. Assuming it is degrees (longitude/latitude)!') } else { stop('Points are projected. They should be in degrees (longitude/latitude)') } # or rather transform them ....? } x = p@lines n = length(x) lines = list() for (i in 1:n) { parts = length(x[[i]]@Lines ) partlist = list() for (j in 1:parts) { crd = x[[i]]@Lines[[j]]@coords crd = .makeSingleLine(crd, interval=interval, ...) partlist[[j]] = Line(crd) } lines[[i]] = Lines(partlist, i) } lines <- SpatialLines(lines) if (inherits(p, 'SpatialLinesDataFrame')) { lines <- SpatialLinesDataFrame(lines, p@data) } lines@proj4string <- p@proj4string return(lines) } else { p <- .pointsToMatrix(p) if (nrow(p) < 3) { stop('cannot make a polygon (insufficent number of vertices)') } res <- .makeSingleLine(p, interval=interval, ...) if (sp) { res <- SpatialLines(list(Lines(list(Line(res)), 1))) res@proj4string <- CRS("+proj=longlat +datum=WGS84") } return(res) } } geosphere/R/onGreatCircle.R0000644000176200001440000000166313472155746015324 0ustar liggesusers# Author: Robert J. Hijmans # based on Dr. Rick's advice at: # http://mathforum.org/library/drmath/view/66114.html # August 2010 # version 1 # license GPL3 onGreatCircle <- function(p1, p2, p3, tol=0.0001) { # is p3 an intermediate points on a great circle defined by p1 and p2? toRad <- pi / 180 p1 <- .pointsToMatrix(p1) p2 <- .pointsToMatrix(p2) p3 <- .pointsToMatrix(p3) p <- cbind(p1[,1], p1[,2], p2[,1], p2[,2], p3[,1], p3[,2]) p1 <- p[,1:2, drop=FALSE] * toRad p2 <- p[,3:4, drop=FALSE] * toRad p3 <- p[,5:6, drop=FALSE] * toRad lon1 <- p1[,1] lat1 <- p1[,2] lon2 <- p2[,1] lat2 <- p2[,2] lon <- p3[,1] lat <- p3[,2] newlat <- atan((sin(lat1)*cos(lat2)*sin(lon-lon2) - sin(lat2)*cos(lat1)*sin(lon-lon1)) / (cos(lat1)*cos(lat2)*sin(lon1-lon2))) res <- abs(newlat - lat) < tol meridian <- p1[,1] == p2[,1] & p1[,1] == p3[,1] res[meridian] <- TRUE return(as.vector(res)) } geosphere/R/RcppExports.R0000644000176200001440000000123314166423400015047 0ustar liggesusers# Generated by using Rcpp::compileAttributes() -> do not edit by hand # Generator token: 10BE3573-1514-4C36-9D1C-5A225CD40393 .geodesic <- function(lon1, lat1, azi1, s12, a, f) { .Call(`_geosphere_geodesic`, lon1, lat1, azi1, s12, a, f) } .inversegeodesic <- function(lon1, lat1, lon2, lat2, a, f) { .Call(`_geosphere_inversegeodesic`, lon1, lat1, lon2, lat2, a, f) } .polygonarea <- function(lon, lat, a, f) { .Call(`_geosphere_polygonarea`, lon, lat, a, f) } .geod_intermediate <- function(lon1, lat1, lon2, lat2, n, distance, arc, a, f) { .Call(`_geosphere_geodesic_nodes`, lon1, lat1, lon2, lat2, n, distance, arc, a, f) } geosphere/R/geomedian.R0000644000176200001440000000310313472155746014522 0ustar liggesusers# Author: Robert J. Hijmans # March 2012 # version 1 # license GPL3 .geomedian <- function(xy, w=NULL) { xy <- .pointsToMatrix(xy) if (is.null(w)) { w <- 1 } else if (length(w) != nrow(xy)) { stop('length of weights not correct. It should be: ', nrow(xy)) } w <- w / sum(w) xyw <- cbind(xy, w) xy <- stats::na.omit(xyw) xy <- xyw[,1:2] w <- xyw[,3] est <- geomean(xy, w) fun <- function(p) { if (p[2] > 90 | p[2] < -90) { return(Inf) } else { p[1] = (p[1] + 180) %% 360 - 180 sum( distCosine(xy, p) * w) } } opt <- stats::optim(geomean(xy), fun) if (!is.null(opt$message)) { warning(opt$message) } return(opt$par) } ..geomedian_ndcor <- function(xy, w=NULL, threshold=100, maxiter=100) { requireNamespace('raster') if (inherits(xy, 'SpatialPolygons') | inherits(xy, 'SpatialPoints')) { stopifnot(raster::isLonLat(xy)) xy <- coordinates(xy) } if (is.null(w)) { w <- 1 } else if (length(w) != nrow(xy)) { stop('length of weights not correct. It should be: ', nrow(xy)) } w <- w / sum(w) xyw <- cbind(xy, w) xy <- stats::na.omit(xyw) xy <- xyw[,1:2] w <- xyw[,3] est <- geomean(xy, w) estold <- est iter = 1 while (TRUE) { d <- distCosine(xy, est) x <- sum(w*xy[,1] / d) / sum(w/d) y <- sum(w*xy[,2] / d) / sum(w/d) est <- cbind(x,y) dif <- distCosine(est, estold) if (dif < threshold) { return(est) } else if (iter > maxiter) { warning('maxiter reached') return(est) } estold <- est iter <- iter + 1 } } geosphere/R/distCosines.R0000644000176200001440000000242313472155746015065 0ustar liggesusers# Author: Robert J. Hijmans # Date : June 2008 # Licence GPL v3 # distance based on law of cosines # http://en.wikipedia.org/wiki/Great_circle_distance distCosine <- function(p1, p2, r=6378137) { p1 <- .pointsToMatrix(p1) if (missing(p2)) { p2 <- p1[-1, ,drop=FALSE] p1 <- p1[-nrow(p1), ,drop=FALSE] } else { p2 <- .pointsToMatrix(p2) } pp <- cbind(p1[,1], p1[,2], p2[,1], p2[,2], as.vector(r)) # remove identical points to avoid errors due to floating point math # problem reported by Bill Monahan i <- rowSums(abs(pp[, 1:2, drop=FALSE] - pp[, 3:4, drop=FALSE]) < .Machine$double.eps ^ 0.5) < 2 p <- pp[i, ,drop=FALSE] r <- rep(0, nrow(pp)) if (nrow(p) > 0) { p[,1:4] <- p[,1:4] * pi / 180 r[i] <- acos( sin(p[,2]) * sin(p[,4]) + cos(p[,2]) * cos(p[,4]) * cos(p[,1]-p[,3]) ) * p[,5] } r } # m = matrix(c(-58.65222,-19.65154,-52.985550,-1.484869, -69.652220, 7.348464, -69.652220,7.348464, -1,1 ,-1,1, -1,1.1,-1,1.1, -1,1.2,-1,1.2, -116.65220,72.01513,-121.48560,53.34847), ncol=4, byrow=T) # distCosine(m[,1:2], m[,3:4]) # n <- nrow(p) # d <- vector("double", n) # d <- .C('distance', as.integer(n), as.double(p[,1]), as.double(p[,2]), as.double(p[,3]), as.double(p[,4]), as.double(p[,5]), as.integer(1), d)[[8]] # return(d) geosphere/R/distVincentyEllipsoid.R0000644000176200001440000000573513472155746017137 0ustar liggesusers# author of original JavaScript code: Chris Vennes # (c) 2002-2009 Chris Veness # http://www.movable-type.co.uk/scripts/latlong.html # Licence: LGPL, without any warranty express or implied # Port to R by Robert Hijmans # October 2009 # version 0.1 # license GPL3 distVincentyEllipsoid <- function(p1, p2, a=6378137, b=6356752.3142, f=1/298.257223563) { #/* Vincenty Inverse Solution of Geodesics on the Ellipsoid (c) Chris Veness 2002-2009 #*/ #* Calculate geodesic distance (in m) between two points specified by latitude/longitude #* (in numeric degrees) using Vincenty inverse formula for ellipsoids # source http://www.movable-type.co.uk/scripts/latlong-vincenty.html # (c) 2002-2009 Chris Veness toRad <- pi / 180 p1 <- .pointsToMatrix(p1) * toRad if (missing(p2)) { p2 <- p1[-1, ,drop=FALSE] p1 <- p1[-nrow(p1), ,drop=FALSE] } else { p2 <- .pointsToMatrix(p2) * toRad } p = cbind(p1[,1], p1[,2], p2[,1], p2[,2], as.vector(a), as.vector(b), as.vector(f)) p1 = p[,1:2,drop=FALSE] p2 = p[,3:4,drop=FALSE] res <- vector(length=nrow(p1)) for (i in 1:dim(p1)[1]) { if ( any( is.na( c(p1[i,], p2[i,])))) { #improvement by George Wang and Sebastian P. Luque res[i] <- NA } else if (isTRUE(all.equal(p1[i,], p2[i,]))) { res[i] <- 0 } else { lon1 <- p1[i,1] lat1 <- p1[i,2] lon2 <- p2[i,1] lat2 <- p2[i,2] a = p[i,5] b = p[i,6] f = p[i,7] L <- (lon2-lon1) U1 <- atan((1-f) * tan(lat1)) U2 <- atan((1-f) * tan(lat2)) sinU1 <- sin(U1) cosU1 <- cos(U1) sinU2 <- sin(U2) cosU2 <- cos(U2) lambda <- L iterLimit <- 100 continue <- TRUE while (continue) { sinLambda <- sin(lambda) cosLambda <- cos(lambda) sinSigma <- sqrt((cosU2*sinLambda) * (cosU2*sinLambda) + (cosU1*sinU2-sinU1*cosU2*cosLambda) * (cosU1*sinU2-sinU1*cosU2*cosLambda)) cosSigma <- sinU1*sinU2 + cosU1*cosU2*cosLambda sigma <- atan2(sinSigma, cosSigma) sinAlpha <- cosU1 * cosU2 * sinLambda / sinSigma cosSqAlpha <- 1 - sinAlpha*sinAlpha cos2SigmaM <- cosSigma - 2*sinU1*sinU2/cosSqAlpha if (is.nan(cos2SigmaM)) cos2SigmaM <- 0 # equatorial line: cosSqAlpha=0 (par. 6) C <- f/16*cosSqAlpha*(4+f*(4-3*cosSqAlpha)) lambdaP <- lambda lambda <- L + (1-C) * f * sinAlpha * (sigma + C*sinSigma*(cos2SigmaM+C*cosSigma*(-1+2*cos2SigmaM*cos2SigmaM))) iterLimit <- iterLimit - 1 continue <- (abs(lambda-lambdaP) > 1e-12 && iterLimit > 0) } if (iterLimit==0) { res[i] <- NA # failed to converge } else { uSq <- cosSqAlpha * (a*a - b*b) / (b*b) A <- 1 + uSq/16384*(4096+uSq*(-768+uSq*(320-175*uSq))) B <- uSq/1024 * (256+uSq*(-128+uSq*(74-47*uSq))) deltaSigma <- B*sinSigma*(cos2SigmaM+B/4*(cosSigma*(-1+2*cos2SigmaM*cos2SigmaM)- B/6*cos2SigmaM*(-3+4*sinSigma*sinSigma)*(-3+4*cos2SigmaM*cos2SigmaM))) res[i] <- b*A*(sigma-deltaSigma) } } } return(as.vector(res)) } geosphere/R/perimeter.R0000644000176200001440000000344614172661303014565 0ustar liggesusers# Robert Hijmans # April 2010 # version 1 # License GPL3 if (!isGeneric("perimeter")) { setGeneric("perimeter", function(x, ...) standardGeneric("perimeter")) } setMethod("perimeter", signature(x='SpatialPolygons'), function(x, a=6378137, f=1/298.257223563, ...) { x <- x@polygons n <- length(x) res <- vector(length=n) for (i in 1:n) { parts <- length( x[[i]]@Polygons ) perim <- 0 for (j in 1:parts) { if (! x[[i]]@Polygons[[j]]@hole) { crd <- x[[i]]@Polygons[[j]]@coords perim <- perim + perimeter(crd, a=a, f=f, ...) } } res[i] <- perim } return(res) } ) setMethod("perimeter", signature(x='SpatialLines'), function(x, a=6378137, f=1/298.257223563, ...) { x <- x@lines n <- length(x) res <- vector(length=n) for (i in 1:n) { parts <- length( x[[i]]@Lines ) lng <- 0 for (j in 1:parts) { crd <- x[[i]]@Lines[[j]]@coords lng <- lng + perimeter(crd, a=a, f=f, ...) } res[i] <- lng } return(res) } ) setMethod("perimeter", signature(x='data.frame'), function(x, a=6378137, f=1/298.257223563, ...) { perimeter(as.matrix(x), a=a, f=f, ...) } ) setMethod("perimeter", signature(x='matrix'), function(x, a=6378137, f=1/298.257223563, ...) { r <- list(...)$r if (!is.null(r)) { # for backwards compatibility warning('removed argument "r" to use improved method') return( .old_perimeter(x, r=r) ) } n <- nrow(x) d <- .inversegeodesic(as.double(x[-n,1]), as.double(x[-n,2]), as.double(x[-1,1]), as.double(x[-1,2]), as.double(a), as.double(f)) sum(abs(x)) }) .old_perimeter <- function(x, r=6378137, ...) { x <- x[,1:2] if (isTRUE(all.equal(x[1,], x[nrow(x),]))) { x <- x[-nrow(x), ] } y <- rbind(x[-1,], x[1,]) d <- distHaversine(x, y, r=r) return(sum(d)) } geosphere/R/geodesic.R0000644000176200001440000000271714161534066014356 0ustar liggesusers# R implementation of # /* # * This is a C implementation of the geodesic algorithms described in # * # * C. F. F. Karney, # * Algorithms for geodesics, # * J. Geodesy 87, 43--55 (2013); # * https://dx.doi.org/10.1007/s00190-012-0578-z # * Addenda: http://geographiclib.sf.net/geod-addenda.html # * # * See the comments in geodesic.h for documentation. # * # * Copyright (c) Charles Karney (2012-2014) and licensed # * under the MIT/X11 License. For more information, see # * http://geographiclib.sourceforge.net/ # */ # # Robert Hijmans # May 2015 # version 1 # license GPL3 # Solve the direct geodesic problem. geodesic <- function(p, azi, d, a=6378137, f=1/298.257223563, ...) { p <- .pointsToMatrix(p) p <- cbind(p[,1], p[,2], azi, d) r <- .geodesic(as.double(p[,1]), as.double(p[,2]), as.double(p[,3]), as.double(p[,4]), as.double(a), as.double(f)) r <- matrix(r, ncol=3, byrow=TRUE) colnames(r) <- c('longitude', 'latitude', 'azimuth') r } # Solve the inverse geodesic problem. geodesic_inverse <- function(p1, p2, a=6378137, f=1/298.257223563, ...) { p1 <- .pointsToMatrix(p1) p2 <- .pointsToMatrix(p2) p <- cbind(p1[,1], p1[,2], p2[,1], p2[,2]) r <- .inversegeodesic(as.double(p[,1]), as.double(p[,2]), as.double(p[,3]), as.double(p[,4]), as.double(a), as.double(f)) r <- matrix(r, ncol=3, byrow=TRUE) colnames(r) <- c('distance', 'azimuth1', 'azimuth2') r } geosphere/R/midPoint.R0000644000176200001440000000262713472155746014367 0ustar liggesusers# Robert Hijmans # October 2009 # version 0.1 # License GPL3 midPoint <- function(p1, p2, a=6378137, f = 1/298.257223563) { # by Elias Pipping gi <- geodesic_inverse(p1, p2, a=a, f=f); destPoint(p1, gi[,'azimuth1'], gi[,'distance']/2, a = a, f = f) } .old_midPoint <- function(p1, p2) { # author of original JavaScript code: Chris Vennes # (c) 2002-2009 Chris Veness # http://www.movable-type.co.uk/scripts/latlong.html # Licence: LGPL, without any warranty express or implied # Much of the above based on formulae by Ed Williams # http://www.edwilliams.org/avform.htm # Port to R by Robert Hijmans # calculate midpoint of great circle line between p1 & p2. # see http:#//mathforum.org/library/drmath/view/51822.html for derivation # based on http://www.movable-type.co.uk/scripts/latlong.html # (c) 2002-2009 Chris Veness toRad <- pi / 180 p1 <- .pointsToMatrix(p1) * toRad p2 <- .pointsToMatrix(p2) * toRad p <- cbind(p1[,1], p1[,2], p2[,1], p2[,2]) lon1 <- p[,1] lat1 <- p[,2] lon2 <- p[,3] lat2 <- p[,4] dLon <- (lon2-lon1) Bx <- cos(lat2) * cos(dLon) By <- cos(lat2) * sin(dLon) lat <- atan2(sin(lat1) + sin(lat2), sqrt((cos(lat1) + Bx)*(cos(lat1) + Bx) + By*By ) ) lon <- lon1 + atan2(By, cos(lat1) + Bx) lon[is.nan(lon)] <- NA lat[is.nan(lat)] <- NA lon <- (lon+pi)%%(2*pi) - pi res <- cbind(lon, lat) / toRad return(res) } geosphere/R/greatCircleBearing.R0000644000176200001440000000054413472155746016314 0ustar liggesusers# author Robert Hijmans # April 2010 # version 0.1 # license GPL greatCircleBearing <- function(p, brng, n=360) { p <- .pointsToMatrix(p) p <- cbind(p[,1], p[,2], as.vector(brng), n) p2 <- destPoint(p[,1:2], p[,3], 10000000) return(greatCircle(p[,1:2], p2, n=p[,4])) } #greatCircleBearing(rbind(cbind(5,52), cbind(5,15)), 60, n=12) geosphere/R/randomCoordinates.R0000644000176200001440000000101713472155746016247 0ustar liggesusers# author Robert Hijmans # July 2010 # version 0.1 # license GPL # based on suggstions by Michael Orion # http://sci.tech-archive.net/Archive/sci.math/2005-09/msg04691.html randomCoordinates <- function(n) { z <- stats::runif(n) * 2 - 1 t <- stats::runif(n) * 2 * pi r <- sqrt(1-z^2) x <- r * cos(t) y <- r * sin(t) r <- sqrt(x^2 + y^2 + z^2) theta <- acos(z / r) phi <- atan2(y,x) lat <- theta * 180 / pi - 90 lon <- phi * 180 / pi return(cbind(lon,lat)) } geosphere/R/refEllipsoids.R0000644000176200001440000000263013472155746015402 0ustar liggesusers refEllipsoids <- function() { data.frame( ellipsoid = c('Airy (1930)', 'Australian National', 'Bessel 1841', 'Ethiopia, Indonesia, Japan, Korea', 'Namibia', 'Clarke 1866', 'Clarke 1880', 'Everest - Brunei & E. Malasia (Sabah & Sarawak)', 'Everest - India 1830', 'Everest - India 1956', 'Everest - Pakistan', 'Everest - W. Malasia and Singapore 1948', 'Everest - W. Malasia 1969', 'Geodetic Reference System 1980 (GRS 80)', 'Helmert 1906', 'Hough 1960', 'Indonesian 1974', 'International 1924', 'Krassovsky 1940', 'Modified Airy', 'Modified Fischer 1960 (South Asia)', 'South American 1969', 'World Geodetic System 1972 (WGS 72)', 'World Geodetic System 1984 (WGS 84)'), code = c('AA', 'AN', '??', 'BR', 'BN', 'CC', 'CD', 'EB', 'EA', 'EC', 'EF', 'EE', 'ED', 'RF', 'HE', 'HO', 'ID', 'IN', 'KA', 'AM', 'FA', 'SA', 'WD', 'WE'), invf = c(299.3249646, 298.25, 299.1528434, 299.1528128, 299.1528128, 294.9786982, 293.465, 300.8017, 300.8017, 300.8017, 300.8017, 300.8017, 300.8017, 298.2572221, 298.3, 297, 298.247, 297, 298.3, 299.3249646, 298.3, 298.25, 298.26, 298.2572236), a = c(6377563.396, 6378160, 6377397.155, 6377397.155, 6377483.865, 6378206.4, 6378249.145, 6377298.556, 6377276.345, 6377301.243, 6377309.613, 6377304.063, 6377295.664, 6378137, 6378200, 6378270, 6378160, 6378388, 6378245, 6377340.189, 6378155, 6378160, 6378135, 6378137), stringsAsFactors=FALSE ) } geosphere/R/distm.R0000644000176200001440000000106713472155746013721 0ustar liggesusers# Robert Hijmans # April 2010 # version 1 # License GPL3 .distm1 <- function(x, fun) { n = nrow(x) dm = matrix(0, ncol=n, nrow=n) if (n == 1) { return(dm) } for (i in 2:n) { j = 1:(i-1) dm[i,j] = fun(x[i,], x[j,]) } dm <- dm+t(dm) return(dm) } distm <- function(x, y, fun=distGeo) { x <- .pointsToMatrix(x) if (missing(y)) { return( .distm1(x, fun) ) } y <- .pointsToMatrix(y) n = nrow(x) m = nrow(y) dm = matrix(ncol=m, nrow=n) for (i in 1:n) { dm[i,] = fun(x[i,], y) } return(dm) } geosphere/R/distRhumb.R0000644000176200001440000000223113472155746014534 0ustar liggesusers# author of original JavaScript code: Chris Vennes # (c) 2002-2009 Chris Veness # http://www.movable-type.co.uk/scripts/latlong.html # Licence: LGPL, without any warranty express or implied # see http://www.edwilliams.org/avform.htm#Rhumb # for the original formulae # Port to R by Robert Hijmans # October 2009 # version 0.1 # license GPL3 distRhumb <- function(p1, p2, r=6378137) { # distance on a rhumb line toRad <- pi / 180 p1 <- .pointsToMatrix(p1) * toRad if (missing(p2)) { p2 <- p1[-1, ,drop=FALSE] p1 <- p1[-nrow(p1), ,drop=FALSE] } else { p2 <- .pointsToMatrix(p2) * toRad } p <- cbind(p1[,1], p1[,2], p2[,1], p2[,2], as.vector(r)) lon1 <- p[,1] lat1 <- p[,2] lon2 <- p[,3] lat2 <- p[,4] r <- p[,5] dLat <- (lat2-lat1) dLon <- abs(lon2-lon1) dPhi <- log(tan(lat2/2 + pi/4)/tan(lat1/2 + pi/4)) i <- abs(dLat) > 1e-10 q <- vector(length=length(i)) q[i] <- dLat[i]/dPhi[i] q[!i] <- cos(lat1[!i]) #// if dLon over 180 degrees take shorter rhumb across 180 degrees meridian: dLon[dLon > pi] <- 2*pi - dLon[dLon > pi] d <- sqrt(dLat*dLat + q*q*dLon*dLon) return(d * r) } geosphere/R/helper.R0000644000176200001440000000142113472155746014052 0ustar liggesusers# Author: Robert J. Hijmans # April 2010 # version 1 # license GPL3 .normalizeLonDeg <- function(x) { (x + 180) %% 360 - 180 } .normalizeLonRad <- function(x) { (x + pi) %% (2*pi) - pi } .isPolygon <- function(x, fix=FALSE) { x <- stats::na.omit(x) if (nrow(x) < 4) { stop('this is not a polygon (insufficent number of vertices)') } if (length(unique(x[,1]))==1) { stop('All longitudes are the same (not a polygon)') } if (length(unique(x[,2]))==1) { stop('All latitudes are the same (not a polygon)') } if (! all(!(is.na(x))) ) { stop('polygon has NA values)') } if (! isTRUE(all.equal(x[1,], x[nrow(x),]))) { stop('this is not a valid (closed) polygon. The first vertex is not equal to the last vertex') } return(x) } geosphere/R/gcLon.R0000644000176200001440000000262613472155746013645 0ustar liggesusers# author Robert Hijmans # October 2009 # version 0.1 # license GPL3 # based on #http://www.edwilliams.org/avform.htm#Par gcLon <- function(p1, p2, lat) { # longitudes at which a given great circle crosses a given parallel # source: http://www.edwilliams.org/avform.htm toRad <- pi / 180 p1 <- .pointsToMatrix(p1) p2 <- .pointsToMatrix(p2) p <- cbind(p1[,1], p1[,2], p2[,1], p2[,2], lat) p1 <- p[,1:2,drop=FALSE] p2 <- p[,3:4,drop=FALSE] lat <- p[,5] res <- matrix(NA, nrow=nrow(p1), ncol=2) colnames(res) <- c('lon1', 'lon2') anti <- ! antipodal(p1, p2) if (sum(anti) == 0) { return(res) } p1 <- p1[anti, ,drop=FALSE] * toRad p2 <- p2[anti, ,drop=FALSE] * toRad lon1 <- p1[,1] * -1 lat1 <- p1[,2] lon2 <- p2[,1] * -1 lat2 <- p2[,2] lat3 <- lat * toRad l12 <- lon1-lon2 A <- sin(lat1)*cos(lat2)*cos(lat3)*sin(l12) B <- sin(lat1)*cos(lat2)*cos(lat3)*cos(l12) - cos(lat1)*sin(lat2)*cos(lat3) C <- cos(lat1)*cos(lat2)*sin(lat3)*sin(l12) lon <- atan2(B,A) lon3 <- matrix(NA, nrow=length(lon1), ncol=2) i <- (abs(C) > sqrt(A^2 + B^2)) | (sqrt(A^2 + B^2) == 0) lon3[i,] <- NA i <- !i dlon <- rep(NA, length(A)) dlon[i] <- acos(C[i]/sqrt(A[i]^2+B[i]^2)) lon3[i,1] <- .normalizeLonRad(lon1[i]+dlon[i]+lon[i]) lon3[i,2] <- .normalizeLonRad(lon1[i]-dlon[i]+lon[i]) res[anti,] <- -1 * lon3 / toRad return(res) } geosphere/R/gcIntersectBearing.R0000644000176200001440000000442013472155746016337 0ustar liggesusers# author Chris Veness, Robert Hijmans # based on formulae by Ed Willians at # http://www.edwilliams.org/avform.htm#Intersection # October 2009 # version 0.1 # license GPL3 gcIntersectBearing <- function(p1, brng1, p2, brng2) { #crs13 true bearing from point 1 and the crs23 true bearing from point 2: toRad <- pi / 180 p1 <- .pointsToMatrix(p1) * toRad p2 <- .pointsToMatrix(p2) * toRad p <- cbind(p1[,1], p1[,2], p2[,1], p2[,2], as.vector(brng1), as.vector(brng2)) lon1 <- p[,1] lat1 <- p[,2] lon2 <- p[,3] lat2 <- p[,4] lat1[lat1==90|lat1==-90] <- NA lat2[lat2==90|lat2==-90] <- NA brng13 <- p[,5] * toRad brng23 <- p[,6] * toRad dLat <- lat2-lat1 dLon <- lon2-lon1 dist12 <- 2*asin( sqrt( sin(dLat/2)*sin(dLat/2) + cos(lat1)*cos(lat2)*sin(dLon/2)*sin(dLon/2) ) ) lat3 <- lon3 <- vector(length=length(nrow(lon1))) i <- rep(TRUE, length(dist12)) i[dist12 == 0] <- FALSE brngA <- acos( ( sin(lat2) - sin(lat1)*cos(dist12) ) / ( sin(dist12)*cos(lat1) ) ) brngA[is.na(brngA)] <- 0 # protect against rounding brngB <- acos( ( sin(lat1) - sin(lat2)*cos(dist12) ) / ( sin(dist12)*cos(lat2) ) ) g <- (sin(lon2-lon1) > 0) brng12 <- vector(length=length(g)) brng21 <- brng12 brng12[g] <- brngA[g] brng21[g] <- 2*pi - brngB[g] brng12[!g] <- 2*pi - brngA[!g] brng21[!g] <- brngB[!g] alpha1 <- (brng13 - brng12 + pi) %% (2*pi) - pi #// angle 2-1-3 alpha2 <- (brng21 - brng23 + pi) %% (2*pi) - pi #// angle 1-2-3 g <- sin(alpha1) == 0 & sin(alpha2) == 0 h <- (sin(alpha1) * sin(alpha2)) < 0 i <- !(g | h) & i lon3[!i] <- lat3[!i] <- NA alpha1 <- abs(alpha1) alpha2 <- abs(alpha2) alpha3 <- acos( -cos(alpha1)*cos(alpha2) + sin(alpha1)*sin(alpha2)*cos(dist12) ) dist13 <- atan2( sin(dist12)*sin(alpha1)*sin(alpha2), cos(alpha2)+cos(alpha1)*cos(alpha3) ) lat3[i] <- asin( sin(lat1[i])*cos(dist13[i]) + cos(lat1[i]) * sin(dist13[i]) * cos(brng13[i]) ) dLon13 <- atan2( sin(brng13)*sin(dist13)*cos(lat1), cos(dist13)-sin(lat1)*sin(lat3) ) lon3[i] <- lon1[i]+dLon13[i] lon3 <- (lon3+pi) %% (2*pi) - pi # // normalise to -180..180 degrees int <- cbind(lon3, lat3) / toRad colnames(int) <- c('lon', 'lat') int <- cbind(int, antipode(int)) rownames(int) <- NULL return(int) } geosphere/R/regularCoordinates.R0000644000176200001440000000173113472155746016433 0ustar liggesusers# author Robert Hijmans # July 2010 # version 0.1 # license GPL # Based on pascal code by Nils Haeck, simdesign.nl # http://mathforum.org/kb/message.jspa?messageID=3985660&tstart=0 regularCoordinates <- function(N) { N <- round(N) if (N < 1) {stop('N should be >= 1')} # subdivision angle beta <- 0.5 * pi / N # line segment length A <- 2 * sin(beta/2); # endcap points <- rbind(c(0, 0, 1), c(0, 0, -1)) # rings R <- sin(1:N * beta) Z <- cos(1:N * beta) M <- round(R * 2 * pi / A) for (i in 1:N) { j <- 0:(M[i]-1) Alpha <- j/M[i] * 2 * pi X <- cos(Alpha) * R[i] Y <- sin(Alpha) * R[i] points <- rbind(points, cbind(X, Y, Z[i])) if (i != N) { points <- rbind(points, cbind(X, Y, -Z[i])) } } r <- sqrt(points[,1]^2 + points[,2]^2 + points[,3]^2) theta <- acos(points[,3] / r) phi <- atan2(points[,2], points[,1]) lat <- theta * 180 / pi - 90 lon <- phi * 180 / pi return(cbind(lon,lat)) } geosphere/R/plotPoly.R0000644000176200001440000000356313472155746014426 0ustar liggesusers# Author: Robert Hijmans # April 2010 # version 0.1 # license GPL # inspired by an example in Software for Data Analysis by John Chambers (pp 250-1) # but adjusted to follow great circles, rather than straight (2D) lines. .doArrows <- function(p, line, fraction, length, interval, ...) { if (fraction >= 1) { graphics::lines(line, ...) } else { dist <- distGeo(p[-nrow(p),], p[-1,]) * (1 - fraction) bearing <- bearing(p[-nrow(p),], p[-1,]) p0 <- destPoint(p[-nrow(p),], bearing, dist) for (i in 1:nrow(p0)) { line = .makeSinglePoly(rbind(p0[i,], p[i+1,]), interval=interval) graphics::lines(line) } } bearing = finalBearing(p[-nrow(p),], p[-1,]) bearing = (bearing + 180) %% 360 pp = destPoint(p[-1,], bearing, interval) x0 <- pp[,1] y0 <- pp[,2] x1 <- p[,1][-1] y1 <- p[,2][-1] # delta = sqrt(mean((x1-x0)^2 + (y1-y0)^2, na.rm=TRUE)) # delta = delta * (par("pin")[1] / diff(range(x, na.rm=TRUE))) graphics::arrows(x0, y0, x1, y1, code=2, length=length, ...) } plotArrows <- function(p, fraction=0.9, length=0.15, first='', add=FALSE, ...) { asp=1 if (inherits(p, 'Spatial')) { bb = t(bbox(p)) interval = distm(bb)[2][1] / 1000 if (! add) { plot(bb, asp=asp, type='n') } p = p@polygons n = length(p) for (i in 1:n) { parts = length(p[[i]]@Polygons ) sumarea = 0 for (j in 1:parts) { pp = p[[i]]@Polygons[[j]]@coords line = .makeSinglePoly(pp, interval=interval) .doArrows(pp, line, fraction, length, interval=interval, ...) } graphics::points(pp[1,1], pp[1,2], pch=first, cex=2) } } else { interval = max(distm(p), na.rm=TRUE) / 1000 line = .makeSinglePoly(p, interval=interval) if (! add) { plot(line, asp=asp, type='n') } .doArrows(p, line=line, fraction, length, interval=interval, ...) graphics::points(p[1,1], p[1,2], pch=first, cex=2) } } geosphere/R/gcIntermediate.R0000644000176200001440000000632114172663435015520 0ustar liggesusers# author Robert Hijmans # October 2009 # version 0.1 # license GPL .interm <- function(p1, p2, n) { toRad <- pi / 180 if (antipodal(p1, p2)) { return(rep(Inf, nrow(p1))) } if (isTRUE(all.equal(p1, p2))) { return(cbind(rep(p1[,1], nrow(p1)), rep(p1[,2], nrow(p1)) )) } d <- distCosine(p1, p2, r=1) lon1 <- p1[,1] * toRad lat1 <- p1[,2] * toRad lon2 <- p2[,1] * toRad lat2 <- p2[,2] * toRad n <- max(round(n), 1) f <- 1:n / (n+1) A <- sin((1-f)*d) / sin(d) B <- sin(f*d) / sin(d) x <- A*cos(lat1)*cos(lon1) + B*cos(lat2)*cos(lon2) y <- A*cos(lat1)*sin(lon1) + B*cos(lat2)*sin(lon2) z <- A*sin(lat1) + B*sin(lat2) lat <- atan2(z,sqrt(x^2+y^2)) lon <- atan2(y,x) cbind(lon,lat)/toRad } .breakAtDateLine <- function(x) { r <- range(x[,1]) r <- r[2] - r[1] if (r > 200) { dif <- abs(x[-nrow(x),1] - x[-1,1]) tr <- which(dif==max(dif)) x1 <- x[1:tr, ,drop=FALSE] x2 <- x[(tr+1):nrow(x), ,drop=FALSE] if (x1[tr,1] < 0) { x1[tr,1] <- -180 x2[1,1] <- 180 } else { x1[tr,1] <- 180 x2[1,1] <- -180 } if (nrow(x1) <= 1) { res <- x2 } else if (nrow(x2) <= 1) { res <- x1 } else { res <- list(x1, x2) } return(res) } return(x) } gcIntermediate <- function( p1, p2, n=50, breakAtDateLine=FALSE, addStartEnd=FALSE, sp=FALSE, sepNA=FALSE) { # Intermediate points on a great circle # source: http://www.edwilliams.org/avform.htm p1 <- .pointsToMatrix(p1) p2 <- .pointsToMatrix(p2) p <- cbind(p1[,1], p1[,2], p2[,1], p2[,2], as.vector(n)) res <- list() for (i in 1:nrow(p)) { x <- .interm(p[i,1:2,drop=FALSE], p[i,3:4,drop=FALSE], p[i,5]) if (addStartEnd) { x <- rbind(p[i,1:2,drop=FALSE], x, p[i,3:4,drop=FALSE]) } if (breakAtDateLine) { res[[i]] <- .breakAtDateLine(x) } else { res[[i]] <- x } } if (sp) { for (i in 1:length(res)) { if (! is.list(res[[i]])) { res[[i]] <- Lines( list( Line (res[[i]])), ID=as.character(i)) } else { res[[i]] <- Lines( list( Line (res[[i]][[1]]), Line(res[[i]][[2]])), ID=as.character(i)) } } res <- SpatialLines(res, CRS("+proj=longlat +ellps=WGS84")) } else if (nrow(p) == 1 ) { res <- res[[1]] } else if (sepNA) { r <- res[[1]] for (i in 2:length(res)) { r <- rbind(r, c(NA,NA), res[[i]]) } return(r) } return(res) } .geodIntermediate <- function(p1, p2, n=50, breakAtDateLine=FALSE, addStartEnd=TRUE, sepNA=FALSE) { a=6378137 f=1/298.257223563 p1 <- .pointsToMatrix(p1) p2 <- .pointsToMatrix(p2) p <- cbind(p1[,1], p1[,2], p2[,1], p2[,2], as.vector(n)) res <- list() for (i in 1:nrow(p)) { x <- .geod_intermediate(p[i,1], p[i,2], p[i,3], p[i,4], p[i,5], -1, TRUE, a, f) x <- .interm(p[i,1:2,drop=FALSE], p[i,3:4,drop=FALSE], p[i,5]) if (!addStartEnd) { x <- x[-c(1, nrow(x)), ,drop=FALSE] } if (breakAtDateLine) { res[[i]] <- .breakAtDateLine(x) } else { res[[i]] <- x } } if (nrow(p) == 1 ) { res <- res[[1]] } else if (sepNA) { r <- res[[1]] for (i in 2:length(res)) { r <- rbind(r, c(NA,NA), res[[i]]) } return(r) } return(res) } geosphere/R/daylength.R0000644000176200001440000000326514334225267014554 0ustar liggesusers# Author: Robert J. Hijmans, r.hijmans@gmail.com # License GPL3 # Version 0.1 January 2009 daylength <- function(lat, doy) { if (inherits(doy, "Date") || inherits(doy, "character")) { doy <- as.character(doy) doy <- as.numeric(format(as.Date(doy), "%j")) } else { doy <- (doy-1) %% 365 + 1 } lat[lat > 90 | lat < -90] <- NA #Forsythe, William C., Edward J. Rykiel Jr., Randal S. Stahl, Hsin-i Wu and Robert M. Schoolfield, 1995. #A model comparison for daylength as a function of latitude and day of the year. Ecological Modeling 80:87-95. P <- asin(0.39795 * cos(0.2163108 + 2 * atan(0.9671396 * tan(0.00860*(doy-186))))) a <- (sin(0.8333 * pi/180) + sin(lat * pi/180) * sin(P)) / (cos(lat * pi/180) * cos(P)) a <- pmin(pmax(a, -1), 1) DL <- 24 - (24/pi) * acos(a) return(DL) } .daylength2 <- function(lat, doy) { if (inherits(doy, "Date") || inherits(doy, "character")) { doy <- as.character(doy) doy <- as.numeric(format(as.Date(doy), "%j")) } else { doy <- (doy-1) %% 365 + 1 } lat[lat > 90 | lat < -90] <- NA doy <- (doy-1) %% 365 + 1 # after Goudriaan and Van Laar RAD <- pi/180 # Sine and cosine of latitude (LAT) SINLAT <- sin(RAD * lat); COSLAT <- cos(RAD * lat); # Maximal sine of declination;} SINDCM <- sin(RAD * 23.45) #{Sine and cosine of declination (Eqns 3.4, 3.5);} SINDEC <- -SINDCM * cos(2*pi*(doy+10)/365) COSDEC <- sqrt(1-SINDEC*SINDEC); #The terms A and B according to Eqn 3.3;} A <- SINLAT*SINDEC; B <- COSLAT*COSDEC; C <- A/B; #Daylength according to Eqn 3.6; arcsin(c) = arctan(c/sqrt(c*c+1))} DAYL <- 12* (1+(2/pi)* atan(C/sqrt(C*C+1))) return(DAYL) } geosphere/R/dist2Line.R0000644000176200001440000000540513472155746014436 0ustar liggesusers# Author: George Wang & Robert J. Hijmans # August 2010 # version 1 # license GPL3 .spDistPoint2Line <- function(p, line, distfun) { test <- !is.projected(line) if (! isTRUE (test) ) { if (is.na(test)) { warning('Coordinate reference system of SpatialPolygons object is not set. Assuming it is degrees (longitude/latitude)!') } else { stop('Points are projected. They should be in degrees (longitude/latitude)') } # or rather transform them ....? } x <- line@lines n <- length(x) res <- matrix(nrow=nrow(p), ncol=4) colnames(res) <- c("distance","lon","lat","ID") res[] <- Inf for (i in 1:n) { parts <- length(x[[i]]@Lines ) for (j in 1:parts) { crd <- x[[i]]@Lines[[j]]@coords r <- cbind(dist2Line(p, crd, distfun), i) k <- r[,1] < res[,1] res[k, ] <- r[k, ] } } return(res) } dist2Line <- function(p, line, distfun=distGeo) { p <- .pointsToMatrix(p) if (inherits(line, 'SpatialPolygons')) { line <- methods::as(line, 'SpatialLines') } if (inherits(line, 'SpatialLines')) { return( .spDistPoint2Line(p, line, distfun) ) } line <- .pointsToMatrix(line) line1 <- line[-nrow(line), ,drop=FALSE] line2 <- line[-1, ,drop=FALSE] seglength <- distfun(line1, line2) res <- matrix(nrow=nrow(p), ncol=3) colnames(res) <- c("distance","lon","lat") for (i in 1:nrow(p)) { xy <- p[i,] # the shortest distance of a point to a great circle crossdist <- abs(dist2gc(line1, line2, xy)) # the alongTrackDistance is the length of the path along the great circle to the point of intersection # there are two, depending on which node you start # we want to use the min, but the max needs to be < segment length trackdist1 <- alongTrackDistance(line1, line2, xy) trackdist2 <- alongTrackDistance(line2, line1, xy) mintrackdist <- pmin(trackdist1, trackdist2) maxtrackdist <- pmax(trackdist1, trackdist2) crossdist[maxtrackdist >= seglength] <- NA # if the crossdist is NA, we use the distance to the nodes nodedist <- distfun(xy, line) warnopt = getOption('warn') options('warn'=-1) distmin1 <- min(nodedist, na.rm=TRUE) distmin2 <- min(crossdist, na.rm=TRUE) options('warn'= warnopt) if (distmin1 <= distmin2) { j <- which.min(nodedist) res[i,] <- c(distmin1, line[j,]) } else { j <- which.min(crossdist) # if else to determine from which node to start if (trackdist1[j] < trackdist2[j]) { bear <- bearing(line1[j,], line2[j,]) pt <- destPoint(line1[j,], bear, mintrackdist[j]) res[i,] <- c(crossdist[j], pt) } else { bear <- bearing(line2[j,], line1[j,]) pt <- destPoint(line2[j,], bear, mintrackdist[j]) res[i,] <- c(crossdist[j], pt) } } } return(res) } geosphere/R/areaPolygon.R0000644000176200001440000000557014161540713015050 0ustar liggesusers# Robert Hijmans # April 2010 # version 1 # license GPL3 if (!isGeneric("areaPolygon")) { setGeneric("areaPolygon", function(x, ...) standardGeneric("areaPolygon")) } setMethod('areaPolygon', signature(x='data.frame'), function(x, a=6378137, f=1/298.257223563, ...) { areaPolygon(as.matrix(x), a=a, f=f, ...) } ) setMethod('areaPolygon', signature(x='SpatialPolygons'), function(x, a=6378137, f=1/298.257223563, ...) { test <- is.projected(x) if ( isTRUE (test) ) { if (is.na(test)) { warning('Coordinate reference system of SpatialPolygons object is not set. Assuming it is degrees (longitude/latitude)!') } else { stop('The coordinate reference system is not longitude/latitude.') } # or rather transform them ....? } x <- x@polygons n <- length(x) res <- vector(length=n) for (i in 1:n) { parts <- length(x[[i]]@Polygons ) sumarea <- 0 for (j in 1:parts) { crd <- x[[i]]@Polygons[[j]]@coords ar <- areaPolygon(crd, a=a, f=f, ...) if (x[[i]]@Polygons[[j]]@hole) { sumarea <- sumarea - ar } else { sumarea <- sumarea + ar } } res[i] <- sumarea } return(res) } ) setMethod('areaPolygon', signature(x='matrix'), function(x, a=6378137, f=1/298.257223563, ...) { r <- list(...)$r if (!is.null(r)) { # for backwards compatibility warning('remove argument "r" to use an better algorithm') return( .old_areaPolygon(x, r=r) ) } # calling geographiclib x <- .polygonarea(as.double(x[,1]), as.double(x[,2]), as.double(a), as.double(f)) abs(x[3]) }) .old_areaPolygon <- function(x, r=6378137, ...) { # Based on code by Jason_Steven (http://forum.worldwindcentral.com/showthread.php?p=69704) # Reference: Bevis, M. and G. Cambareri, 1987. Computing the area of a spherical polygon of arbitrary shape. Mathematical Geology 19: 335-346 haversine <- function(y) { (1-cos(y))/2 } x <- .pointsToMatrix(x, poly=TRUE) x <- makePoly(x) # for some corner cases # rotate? dif1 <- max(x[,1]) - min(x[,1]) if (dif1 > 180) { x2 <- x x2[,1] <- x2[,1] %% 360 - 180 #dif1 <- max(x[,1]) - min(x[,1]) dif2 <- max(x2[,1]) - min(x2[,1]) if (dif2 < dif1) { x <- x2 } } x <- x * pi / 180 r <- r[1] j <- 1:nrow(x) k <- c(2:nrow(x), 1) i <- x[j,1] != x[k,1] j <- j[i] k <- k[i] lam1 <- x[j,1] lam2 <- x[k,1] beta1 <- x[j,2] beta2 <- x[k,2] cosB1 <- cos( beta1 ) cosB2 <- cos( beta2 ) hav <- haversine( beta2 - beta1 ) + cosB1 * cosB2 * haversine( lam2 - lam1 ) a <- 2 * asin( sqrt( hav ) ) b <- pi / 2 - beta2 c <- pi / 2 - beta1 s <- 0.5 * ( a + b + c ) t <- tan( s / 2 ) * tan( ( s - a ) / 2 ) * tan( ( s - b ) / 2 ) * tan( ( s - c ) / 2 ) excess <- abs( 4 * atan( sqrt( abs( t ) ) ) ) excess[lam2 < lam1] <- -excess[lam2 < lam1] arsum <- abs( sum( excess ) ) * r * r return(arsum ) } geosphere/R/bearing.R0000644000176200001440000000254414161534017014175 0ustar liggesusers# Author: Robert J. Hijmans # Date : March 2010 / May 2015 # Version 2.0 # Licence GPL v3 bearing <- function(p1, p2, a=6378137, f=1/298.257223563) { p1 <- .pointsToMatrix(p1) if (missing(p2)) { if (nrow(p1) < 2) { return(NA) } p2 <- p1[-1, ,drop=FALSE] p1 <- p1[-nrow(p1), ,drop=FALSE] addNA <- TRUE } else { p2 <- .pointsToMatrix(p2) addNA <- FALSE } p <- cbind(p1[,1], p1[,2], p2[,1], p2[,2]) r <- .inversegeodesic(as.double(p[,1]), as.double(p[,2]), as.double(p[,3]), as.double(p[,4]), as.double(a[1]), as.double(f[1])) r <- matrix(r, ncol=3, byrow=TRUE) if (addNA) { c(r[, 2], NA) } else { r[, 2] } } .old_bearing <- function(p1, p2) { toRad <- pi / 180 p1 <- .pointsToMatrix(p1) * toRad p2 <- .pointsToMatrix(p2) * toRad p <- cbind(p1[,1], p1[,2], p2[,1], p2[,2]) p1 <- p[, 1:2, drop=FALSE] p2 <- p[, 3:4, drop=FALSE] keep <- ! apply(p1 == p2, 1, sum) == 2 res <- rep(NA, length=nrow(p1)) if (sum(keep) == 0) { return(res) } p1 <- p1[keep, , drop=FALSE] p2 <- p2[keep, , drop=FALSE] dLon <- p2[,1] - p1[,1] y <- sin(dLon) * cos(p2[,2]) x <- cos(p1[,2]) * sin(p2[,2]) - sin(p1[,2]) * cos(p2[,2]) * cos(dLon) azm <- atan2(y, x) / toRad azm <- (azm+360) %% 360 i <- azm > 180 azm[i] <- -1 * (360 - azm[i]) res[keep] <- azm return(res) } geosphere/R/gcMaxLat.R0000644000176200001440000000255413472155746014303 0ustar liggesusers# Based on formulae by Ed Williams # http://www.edwilliams.org/avform.htm # Port to R by Robert Hijmans # October 2009 # version 0.1 # License GPL3 gcMaxLat <- function(p1, p2) { toRad <- pi / 180 p1 <- .pointsToMatrix(p1) p2 <- .pointsToMatrix(p2) p <- cbind(p1[,1], p1[,2], p2[,1], p2[,2]) p1 <- p[,1:2,drop=FALSE] p2 <- p[,3:4,drop=FALSE] anti <- antipodal(p1, p2) same <- apply(p1 == p2, 1, sum) == 2 use <- !(anti | same) res <- matrix(rep(NA, nrow(p1)*2), ncol=2) colnames(res) <- c('lon', 'lat') if (length(use)==0) { return(res) } pp1 <- p1[use, , drop=FALSE] pp2 <- p2[use, , drop=FALSE] b <- .old_bearing(pp1, pp2) * toRad lat <- pp1[,2] * toRad # Clairaut's formula : the maximum latitude of a great circle path, given a bearing and latitude on the great circle maxlat <- acos(abs(sin(b) * cos(lat))) / toRad ml <- maxlat - 0.000000000001 maxlon <- mean(gcLon(pp1, pp2, ml)) res[use,] <- cbind(maxlon, maxlat) # lon <- pp1[,1] * toRad # maxlon <- rep(NA, length(maxlat)) # i <- maxlat==0 # j <- b < pi & !i # k <- !j & !i # maxlon[j] <- lon[j] - atan2(cos(b[j]), sin(b[j]) * sin(lat[j])) # maxlon[k] <- lon[k] + pi - atan2(cos(b[k]), sin(b[k]) * sin(lat[k])) # maxlon <- -1 * ((maxlon+pi)%%(2*pi) - pi) # res[use,] <- cbind(maxlon, maxlat)/ toRad return(res) } geosphere/R/distGeo.R0000644000176200001440000000123014161534041014150 0ustar liggesusers# Author: Robert J. Hijmans # Date : May 2015 # Licence GPL v3 distGeo <- function(p1, p2, a=6378137, f=1/298.257223563) { p1 <- .pointsToMatrix(p1) if (missing(p2)) { if (nrow(p1) == 1) return(0) if (nrow(p1) == 0) return(NULL) p2 <- p1[-1, ,drop=FALSE] p1 <- p1[-nrow(p1), ,drop=FALSE] addNA <- TRUE } else { p2 <- .pointsToMatrix(p2) addNA <- FALSE } p <- cbind(p1[,1], p1[,2], p2[,1], p2[,2]) r <- .inversegeodesic(as.double(p[,1]), as.double(p[,2]), as.double(p[,3]), as.double(p[,4]), as.double(a), as.double(f)) r <- matrix(r, ncol=3, byrow=TRUE) if (addNA){ c(r[,1], NA) } else { r[,1] } } geosphere/R/horizon.R0000644000176200001440000000022513472155746014264 0ustar liggesusers horizon <- function(h, r=6378137) { x = cbind(as.vector(h), as.vector(r)) h = x[,1] r = x[,2] b = 0.8279 sqrt( 2 * r * h / b ) } geosphere/R/bearingRhumb.R0000644000176200001440000000221113472155746015176 0ustar liggesusers# author of original JavaScript code: Chris Vennes # (c) 2002-2009 Chris Veness # http://www.movable-type.co.uk/scripts/latlong.html # Licence: LGPL, without any warranty express or implied # see http://www.edwilliams.org/avform.htm#Rhumb # for the original formulae # Port to R by Robert Hijmans # October 2009 # version 0.1 # license GPL3 bearingRhumb <- function(p1, p2) { toRad <- pi / 180 p1 <- .pointsToMatrix(p1) * toRad p2 <- .pointsToMatrix(p2) * toRad p <- cbind(p1[,1], p1[,2], p2[,1], p2[,2]) p1 <- p[, 1:2, drop=FALSE] p2 <- p[, 3:4, drop=FALSE] keep <- ! apply(p1 == p2, 1, sum) == 2 res <- rep(NA, length=nrow(p1)) if (sum(keep) == 0) { return(res) } lon1 <- p1[keep, 1, drop=FALSE] lat1 <- p1[keep, 2, drop=FALSE] lon2 <- p2[keep, 1, drop=FALSE] lat2 <- p2[keep, 2, drop=FALSE] dLon <- (lon2-lon1) dPhi <- log(tan(lat2/2 + pi/4)/tan(lat1/2+pi/4)) i <- (abs(dLon) > pi) j <- i & dLon > 0 dLon[j] <- -(2*pi-dLon[j]) j <- i & dLon <= 0 dLon[j] <- dLon[j] <- (2*pi+dLon[j]) b <- atan2(dLon, dPhi) b <- b / toRad b <- (b+360) %% 360 res[keep] = b return(res) } geosphere/R/mercator.R0000644000176200001440000000077113472155746014416 0ustar liggesusers# Author: Robert J. Hijmans # April 2010 # version 0.1 # license GPL3 mercator <- function(p, inverse=FALSE, r=6378137) { toRad <- pi / 180 if (inverse) { p <- .pointsToMatrix(p, checkLonLat=FALSE) p[ ,2] <- pi/2 - 2 * atan(exp(-p[,2] / r)) p[ ,1] <- p[,1] / r colnames(p) <- c('lon', 'lat') return( p / toRad ) } else { p <- .pointsToMatrix(p) * toRad p[,2] <- log( tan(p[,2]) + (1 / cos(p[,2]))) p <- p * r colnames(p) <- c('x', 'y') return( p ) } } geosphere/R/distMeeus.R0000644000176200001440000000214213472155746014536 0ustar liggesusers# R code by Robert Hijmans # based on Java script code by # Stephen R. Schmitt (copyright, 2004) # http://web.archive.org/web/20070108024032/http://home.att.net/~srschmitt/script_greatcircle.html # algorithm taken from "Astronomical Algorithms" by Jean Meeus distMeeus <- function(p1, p2, a=6378137, f=1/298.257223563) { toRad <- pi / 180 p1 <- .pointsToMatrix(p1) * toRad if (missing(p2)) { p2 <- p1[-1, ,drop=FALSE] p1 <- p1[-nrow(p1), ,drop=FALSE] } else { p2 <- .pointsToMatrix(p2) * toRad } F <- ( p1[,2] + p2[,2] ) / 2 G <- ( p1[,2] - p2[,2] ) / 2 L <- ( p1[,1] - p2[,1] ) / 2 sinG2 <- ( sin( G ) )^2 cosG2 <- ( cos( G ) )^2 sinF2 <- ( sin( F ) )^2 cosF2 <- ( cos( F ) )^2 sinL2 <- ( sin( L ) )^2 cosL2 <- ( cos( L ) )^2 S <- sinG2 * cosL2 + cosF2 * sinL2 C <- cosG2 * cosL2 + sinF2 * sinL2 w <- atan( sqrt( S/C ) ) R <- sqrt( S*C )/w D <- 2 * w * a H1 <- (3*R - 1)/(2*C) H2 <- (3*R + 1)/(2*S) dst <- D*( 1 + f*H1*sinF2*cosG2 - f*H2*cosF2*sinG2 ) # remove NaN for when p1[i,]==p2[i,] dst[which(w==0)] <- 0 return ( as.vector(dst) ) } geosphere/R/pointsToMatrix.R0000644000176200001440000000431713472155746015606 0ustar liggesusers# Author: Robert J. Hijmans & Jacob van Etten # October 2009 # version 1 # license GPL3 .pointsToMatrix <- function(p, checkLonLat=TRUE, poly=FALSE) { if (inherits(p, 'SpatialPoints')) { test <- !is.projected(p) if (! isTRUE (test) ) { if (is.na(test)) { warning('Coordinate reference system of SpatialPoints object is not set. Assuming it is degrees (longitude/latitude)!') } else { stop('Points are projected. They should be in degrees (longitude/latitude)') } # or rather transform them ....? } p <- coordinates(p) } else if (is.data.frame(p)) { p <- as.matrix(p) } else if (is.vector(p)){ if (length(p) != 2) { stop('Wrong length for a vector, should be 2') } else { p <- matrix(p, ncol=2) } } else if (is.matrix(p)) { if (ncol(p) != 2) { stop( 'A points matrix should have 2 columns') } cn <- colnames(p) if (length(cn) == 2) { if (toupper(cn[1]) == 'Y' | toupper(cn[2]) == 'X') { warning('Suspect column names (x and y reversed?)') } if (toupper(substr(cn[1],1,3) == 'LAT' | toupper(substr(cn[2],1,3)) == 'LON')) { warning('Suspect column names (longitude and latitude reversed?)') } } } else { stop('points should be vectors of length 2, matrices with 2 columns, or inheriting from a SpatialPoints* object') } if (! is.numeric(p) ) { p[] <- as.numeric(p) } if (checkLonLat & nrow(p) > 0) { if (length(stats::na.omit(p[,1])) > 0) { if (min(p[,1], na.rm=TRUE) < -360) { stop('longitude < -360') } if (max(p[,1], na.rm=TRUE) > 360) { stop('longitude > 360') } if (min(p[,1], na.rm=TRUE) < -180) { warning('longitude < -180') } if (max(p[,1], na.rm=TRUE) > 180) { warning('longitude > 180') } } if (length(stats::na.omit(p[,2])) > 0) { if (min(p[,2], na.rm=TRUE) < -90) { stop('latitude < -90') } if (max(p[,2], na.rm=TRUE) > 90) { stop('latitude > 90') } } } if (poly) { if (! isTRUE(all.equal(p[1,], p[nrow(p),]))) { p <- rbind(p, p[1,]) } i <- p[-nrow(p),1] == p[-1,1] & p[-nrow(p),2] == p[-1,2] i <- which(isTRUE(i)) if (length(i) > 0) { p <- p[-i, ,drop=FALSE] } .isPolygon(p) } return(p) } geosphere/R/old_destPoint.R0000644000176200001440000000231313472155746015403 0ustar liggesusers# author of original JavaScript code: Chris Vennes # (c) 2002-2009 Chris Veness # http://www.movable-type.co.uk/scripts/latlong.html # Licence: LGPL, without any warranty express or implied # Based on formulae by Ed Williams # http://www.edwilliams.org/avform.htm # Port to R by Robert Hijmans # October 2009 # version 0.1 # License GPL3 .old_destPoint <- function(p, b, d, r=6378137) { # calculate destination point given start point, initial bearing (deg) and distance (km) # see http:#//www.edwilliams.org/avform.htm#LL # source http://www.movable-type.co.uk/scripts/latlong.html # (c) 2002-2009 Chris Veness toRad <- pi / 180 b = as.vector(b) d = as.vector(d) r = as.vector(r) p <- .pointsToMatrix(p) p = cbind(p[,1], p[,2], b, d, r) lon1 <- p[,1] * toRad lat1 <- p[,2] * toRad b <- p[,3] * toRad d = p[,4] r = p[,5] lat2 <- asin( sin(lat1)*cos(d/r) + cos(lat1)*sin(d/r)*cos(b) ) lon2 <- lon1 + atan2(sin(b)*sin(d/r)*cos(lat1), cos(d/r)-sin(lat1)*sin(lat2)) lon2 <- (lon2+pi)%%(2*pi) - pi #// normalise to -180...+180 lon2[is.nan(lon2)] <- NA lat2[is.nan(lat2)] <- NA res <- cbind(lon2, lat2) / toRad colnames(res) <- c('lon', 'lat') return(res) } geosphere/R/lengthLine.R0000644000176200001440000000150113472155746014663 0ustar liggesusers# Author: Robert J. Hijmans # August 2016 # version 1 # license GPL3 lengthLine <- function(line) { if (inherits(line, 'SpatialPolygons')) { requireNamespace('raster') line <- raster::geom(methods::as(line, 'SpatialLines')) } else if (inherits(line, 'SpatialLines')) { requireNamespace('raster') line <- raster::geom(line) } else { line <- cbind(object=1, part=1, cump=1, line[, 1:2]) colnames(line)[4:5] <- c('x', 'y') } ids <- unique(line[,1]) len <- rep(0, length(ids)) for (i in 1:length(ids)) { d <- line[line[,1] == ids[i], ] parts <- unique(d[,2]) for (p in parts) { dd <- d[d[,2] == p, ,drop=FALSE] for (j in 1:(nrow(dd)-1)) { len[i] <- len[i] + distGeo(dd[j, c('x', 'y'), drop=FALSE], dd[j+1, c('x', 'y'), drop=FALSE]) } } } return(len) } geosphere/R/geomean.R0000644000176200001440000000123513472155746014211 0ustar liggesusers# Author: Robert J. 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6c7010c04e7686a0204a6077c0fd024d *src/Rhumb.h 1eaf3edd3970b709c5d9bad53230e5bc *src/TransverseMercator.cpp 405c21a9998da0d1411f02f47bc038d0 *src/TransverseMercator.h c92e9a174ac3b5529402c9795b612356 *src/a_dist.c bd73d1d0cd6efd059299b606e3f86c99 *src/a_geodesic.cpp e7ed3cfd2ceb132ccd9e7c342ba7f0a0 *src/a_util.c 02b3ab8677d52f791681845435a5252a *src/a_util.h d554906e5117a31c97b4ed41eecafb09 *src/kissfft.h 5090e95e97f748525ed9905c51309b22 *vignettes/geosphere.Rnw geosphere/inst/0000755000176200001440000000000014334225370013213 5ustar liggesusersgeosphere/inst/doc/0000755000176200001440000000000014334225370013760 5ustar liggesusersgeosphere/inst/doc/geosphere.R0000644000176200001440000002031314334225370016063 0ustar liggesusers### R code from vignette source 'geosphere.Rnw' ################################################### ### code chunk number 1: foo ################################################### options(keep.source = TRUE, width = 60) foo <- packageDescription("geosphere") ################################################### ### code chunk number 2: geosphere.Rnw:58-71 ################################################### library(geosphere) Lon <- c(1:9/1000, 1:9/100, 1:9/10, 1:90*2) Lat <- c(1:9/1000, 1:9/100, 1:9/10, 1:90) dcos <- distCosine(c(0,0), cbind(Lon, Lat)) dhav <- distHaversine(c(0,0), cbind(Lon, Lat)) dvsp <- distVincentySphere(c(0,0), cbind(Lon, Lat)) par(mfrow=(c(1,2))) plot(log(dcos), dcos-dhav, col='red', ylim=c(-1e-05, 1e-05), xlab="Log 'Law of Cosines' distance (m)", ylab="Law of Cosines minus Haversine distance") plot(log(dhav), dhav-dvsp, col='blue', xlab="Log 'Haversine' distance (m)", ylab="Vincenty Sphere minus Haversine distance") ################################################### ### code chunk number 3: geosphere.Rnw:76-79 ################################################### dvse <- distVincentyEllipsoid(c(0,0), cbind(Lon, Lat)) plot(dvsp/1000, (dvsp-dvse)/1000, col='blue', xlab='Vincenty Sphere Distance (km)', ylab="Difference between 'Vincenty Sphere' and 'Vincenty Ellipsoid' methods (km)") ################################################### ### code chunk number 4: geosphere.Rnw:89-102 ################################################### LA <- c(-118.40, 33.95) NY <- c(-73.78, 40.63) data(wrld) plot(wrld, type='l') gc <- greatCircle(LA, NY) lines(gc, lwd=2, col='blue') gci <- gcIntermediate(LA, NY) lines(gci, lwd=4, col='green') points(rbind(LA, NY), col='red', pch=20, cex=2) mp <- midPoint(LA, NY) onGreatCircle(LA,NY, rbind(mp,c(0,0))) points(mp, pch='*', cex=3, col='orange') greatCircleBearing(LA, brng=270, n=10) ################################################### ### code chunk number 5: geosphere.Rnw:109-117 ################################################### destPoint(LA, b=65, d=100000) circle=destPoint(c(0,80), b=1:365, d=1000000) circle2=destPoint(c(0,80), b=1:365, d=500000) circle3=destPoint(c(0,80), b=1:365, d=100000) plot(circle, type='l') polygon(circle, col='blue', border='black', lwd=4) polygon(circle2, col='red', lwd=4, border='orange') polygon(circle3, col='white', lwd=4, border='black') ################################################### ### code chunk number 6: geosphere.Rnw:125-139 ################################################### ml <- gcMaxLat(LA, NY) lat0 <- gcLat(LA, NY, lon=0) lon0 <- gcLon(LA, NY, lat=0) plot(wrld, type='l') lines(gc, lwd=2, col='blue') points(ml, col='red', pch=20, cex=2) points(cbind(0, lat0), pch=20, cex=2, col='yellow') points(t(rbind(lon0, 0)), pch=20, cex=2, col='green' ) f <- function(lon){gcLat(LA, NY, lon)} opt <- optimize(f, interval=c(-180, 180), maximum=TRUE) points(opt$maximum, opt$objective, pch=20, cex=2, col='dark green' ) anti <- antipode(c(opt$maximum, opt$objective)) points(anti, pch=20, cex=2, col='dark blue' ) ################################################### ### code chunk number 7: geosphere.Rnw:146-161 ################################################### SF <- c(-122.44, 37.74) AM <- c(4.75, 52.31) gc2 <- greatCircle(AM, SF) plot(wrld, type='l') lines(gc, lwd=2, col='blue') lines(gc2, lwd=2, col='green') int <- gcIntersect(LA, NY, SF, AM) int antipodal(int[,1:2], int[,3:4]) points(rbind(int[,1:2], int[,3:4]), col='red', pch=20, cex=2) bearing1 <- bearing(LA, NY) bearing2 <- bearing(SF, AM) bearing1 bearing2 gcIntersectBearing(LA, bearing1, SF, bearing2) ################################################### ### code chunk number 8: geosphere.Rnw:169-189 ################################################### MS <- c(-93.26, 44.98) gc1 <- greatCircleBearing(NY, 281) gc2 <- greatCircleBearing(MS, 195) gc3 <- greatCircleBearing(LA, 55) plot(wrld, type='l', xlim=c(-125, -70), ylim=c(20, 60)) lines(gc1, col='green') lines(gc2, col='blue') lines(gc3, col='red') int <- gcIntersectBearing(rbind(NY, NY, MS), c(281, 281, 195), rbind(MS, LA, LA), c(195, 55, 55)) int distm(rbind(int[,1:2], int[,3:4])) int <- int[,1:2] points(int) poly <- rbind(int, int[1,]) centr <- centroid(poly) poly2 <- makePoly(int) polygon(poly2, col='yellow') points(centr, pch='*', col='dark red', cex=2) ################################################### ### code chunk number 9: geo5 ################################################### d <- distCosine(LA, NY) d b <- bearing(LA, NY) b destPoint(LA, b, d) NY finalBearing(LA, NY) ################################################### ### code chunk number 10: geosphere.Rnw:213-224 ################################################### atd <- alongTrackDistance(LA, NY, MS) p <- destPoint(LA, b, atd) plot(wrld, type='l', xlim=c(-130,-60), ylim=c(22,52)) lines(gci, col='blue', lwd=2) points(rbind(LA, NY), col='red', pch=20, cex=2) points(MS[1], MS[2], pch=20, col='blue', cex=2) lines(gcIntermediate(LA, p), col='green', lwd=3) lines(gcIntermediate(MS, p), col='dark green', lwd=3) points(p, pch=20, col='red', cex=2) dist2gc(LA, NY, MS) distCosine(p, MS) ################################################### ### code chunk number 11: geosphere.Rnw:232-241 ################################################### line <- rbind(c(-180,-20), c(-150,-10), c(-140,55), c(10, 0), c(-140,-60)) pnts <- rbind(c(-170,0), c(-75,0), c(-70,-10), c(-80,20), c(-100,-50), c(-100,-60), c(-100,-40), c(-100,-20), c(-100,-10), c(-100,0)) d = dist2Line(pnts, line) plot( makeLine(line), type='l') points(line) points(pnts, col='blue', pch=20) points(d[,2], d[,3], col='red', pch='x', cex=2) for (i in 1:nrow(d)) lines(gcIntermediate(pnts[i,], d[i,2:3], 10), lwd=2, col='green') ################################################### ### code chunk number 12: geosphere.Rnw:249-269 ################################################### NP <- c(0, 85) bearing(SF, NP) b <- bearingRhumb(SF, NP) b dc <- distCosine(SF, NP) dr <- distRhumb(SF, NP) dc / dr pr <- destPointRhumb(SF, b, d=round(dr/100) * 1:100) pc <- rbind(SF, gcIntermediate(SF, NP), NP) par(mfrow=c(1,2)) data(wrld) plot(wrld, type='l', xlim=c(-140,10), ylim=c(15,90), main='Equirectangular') lines(pr, col='blue') lines(pc, col='red') data(merc) plot(merc, type='l', xlim=c(-15584729, 1113195), ylim=c(2500000, 22500000), main='Mercator') lines(mercator(pr), col='blue') lines(mercator(pc), col='red') ################################################### ### code chunk number 13: geosphere.Rnw:277-291 ################################################### pol <- rbind(c(-120,-20), c(-80,5), c(0, -20), c(-40,-60), c(-120,-20)) areaPolygon(pol) perimeter(pol) centroid(pol) span(pol, fun=max) nicepoly = makePoly(pol) plot(pol, xlab='longitude', ylab='latitude', cex=2, lwd=3, xlim=c(-140, 0)) lines(wrld, col='grey') lines(pol, col='red', lwd=2) lines(nicepoly, col='blue', lwd=2) points(centroid(pol), pch='*', cex=3, col='dark green') text(centroid(pol)-c(0,2.5), 'centroid') legend(-140, -48, c('planar','spherical'), lty=1, lwd=2, col=c('red', 'blue'), title='polygon type') ################################################### ### code chunk number 14: geosphere.Rnw:299-304 ################################################### plot(wrld, type='l', col='grey') a = randomCoordinates(500) points(a, col='blue', pch=20, cex=0.5) b = regularCoordinates(3) points(b, col='red', pch='x') ################################################### ### code chunk number 15: geosphere.Rnw:313-321 ################################################### as.Date(80, origin='2009-12-31') as.Date(172, origin='2009-12-31') plot(0:90, daylength(lat=0:90, doy=1), ylim=c(0,24), type='l', xlab='Latitude', ylab='Daylength', main='Daylength by latitude and day of year', lwd=2) lines(0:90, daylength(lat=0:90, doy=80), col='green', lwd=2) lines(0:90, daylength(lat=0:90, doy=172), col='blue', lwd=2) legend(0,24, c('1','80','172'), lty=1, lwd=2, col=c('black', 'green', 'blue'), title='Day of year') geosphere/inst/doc/geosphere.Rnw0000644000176200001440000004033113472155746016445 0ustar liggesusers\documentclass{article} \usepackage{natbib} \usepackage{graphics} \usepackage{amsmath} \usepackage{indentfirst} \usepackage[utf8]{inputenc} \usepackage{hyperref} \usepackage{hanging} %\VignetteIndexEntry{Introduction to the geosphere package} \SweaveOpts{keep.source=TRUE} \SweaveOpts{png=TRUE, pdf=FALSE} \SweaveOpts{resolution=100} \begin{document} <>= options(keep.source = TRUE, width = 60) foo <- packageDescription("geosphere") @ \title{Introduction to the "geosphere" package \\ (Version \Sexpr{foo$Version})} \author{Robert J. Hijmans} \maketitle \section{Introduction} This vignette describes the R package '\verb@geosphere@'. The package implements spherical trigonometry functions for geographic applications. Many of the functions have applications in navigation, but others are more general, or have no relation to navigation at all. There is a number of functions to compute distance and direction (= bearing, azimuth, course) along great circles (= shortest distance on a sphere, or "as the crow flies") and along rhumb lines (lines of constant bearing). There are also functions that compute distances on a spheroid. Other functions include the computation of the location of an object at a given direction and distance; and the area, perimeter, and centroid of a spherical polygon. Geographic locations must be specified in longitude and latitude (and in that order!) in degrees (i.e., NOT in radians). Degrees are (obviously) in decimal notation. Thus 12 degrees, 10 minutes, 30 seconds = 12 + 10/60 + 30/3600 = 12.175 degrees. The southern and western hemispheres have a negative sign. The default unit of distance is meter; but this can be adjusted by supplying a different radius 'r' to functions. Directions are expressed in degrees (N = 0 and 360, E = 90, S = 180, and W = 270 degrees). If arguments of functions that take several arguments (e.g. points, bearings, radius of the earth), do not have the same length (for vectors) or number of rows (for matrices) the shorter arguments are re-cycled. Many functions in this package are based on formulae provided by Ed Williams (\url{http://www.edwilliams.org/ftp/avsig/avform.txt}, and partly on javascript implementations of these formulae by Chris Veness (\url{http://www.movable-type.co.uk/scripts/latlong.html} ) Most geodesic computations (for a spheroid rather than a sphere) use the GeographicLib by C.F.F. Karney (\url{http://geographiclib.sourceforge.net/}. \section{Great circle distance} There are four different functions to compute distance between two points. These are, in order of increasing complexity of the algorithm, the 'Spherical law of cosines', 'Haversine' (Sinnott, 1984), 'Vincenty Sphere' and 'Vincenty Ellipsoid' (Vincenty, 1975) methods. The first three assume the earth to be a sphere, while the 'Vincenty Ellipsoid' assumes it is an ellipsoid (which is closer to the truth). The results from the first three functions are identical for practical purposes. The Haversine ('half-versed-sine') formula was published by R.W. Sinnott in 1984, although it has been known for much longer. At that time computational precision was lower than today (15 digits precision). With current precision, the spherical law of cosines formula appears to give equally good results down to very small distances. If you want greater accuracy, you could use the distVincentyEllipsoid method. Below the differences between the three spherical methods are illustrated. At very short distances, there are small differences between the 'law of the Cosine' and the other two methods. There are even smaller differences between the 'Haversine' and 'Vincenty Sphere' methods at larger distances. <>= library(geosphere) Lon <- c(1:9/1000, 1:9/100, 1:9/10, 1:90*2) Lat <- c(1:9/1000, 1:9/100, 1:9/10, 1:90) dcos <- distCosine(c(0,0), cbind(Lon, Lat)) dhav <- distHaversine(c(0,0), cbind(Lon, Lat)) dvsp <- distVincentySphere(c(0,0), cbind(Lon, Lat)) par(mfrow=(c(1,2))) plot(log(dcos), dcos-dhav, col='red', ylim=c(-1e-05, 1e-05), xlab="Log 'Law of Cosines' distance (m)", ylab="Law of Cosines minus Haversine distance") plot(log(dhav), dhav-dvsp, col='blue', xlab="Log 'Haversine' distance (m)", ylab="Vincenty Sphere minus Haversine distance") @ The difference with the 'Vincenty Ellipsoid' method is more pronounced. In the example below (using the default WGS83 ellipsoid), the difference is about 0.3% at very small distances, and 0.15% at larger distances. <>= dvse <- distVincentyEllipsoid(c(0,0), cbind(Lon, Lat)) plot(dvsp/1000, (dvsp-dvse)/1000, col='blue', xlab='Vincenty Sphere Distance (km)', ylab="Difference between 'Vincenty Sphere' and 'Vincenty Ellipsoid' methods (km)") @ For the most precise distance computation use the 'distGeo' function. \section{Points on great circles} Points on a great circle are returned by the function 'greatCircle', using two points on the great circle to define it, and an additional argument to indicate how many points should be returned. You can also use greatCircleBearing, and provide starting points and bearing as arguments. gcIntermediate only returns points on the great circle that are on the track of shortest distance between the two points defining the great circle; and midPoint computes the point half-way between the two points. You can use onGreatCircle to test whether a point is on a great circle between two other points. <>= LA <- c(-118.40, 33.95) NY <- c(-73.78, 40.63) data(wrld) plot(wrld, type='l') gc <- greatCircle(LA, NY) lines(gc, lwd=2, col='blue') gci <- gcIntermediate(LA, NY) lines(gci, lwd=4, col='green') points(rbind(LA, NY), col='red', pch=20, cex=2) mp <- midPoint(LA, NY) onGreatCircle(LA,NY, rbind(mp,c(0,0))) points(mp, pch='*', cex=3, col='orange') greatCircleBearing(LA, brng=270, n=10) @ \section{Point at distance and bearing} Function destPoint returns the location of point given a point of origin, and a distance and bearing. Its perhaps obvious use in georeferencing locations of distant sitings. It can also be used to make circular polygons (with a fixed radius, but in longitude/latitude coordinates) <>= destPoint(LA, b=65, d=100000) circle=destPoint(c(0,80), b=1:365, d=1000000) circle2=destPoint(c(0,80), b=1:365, d=500000) circle3=destPoint(c(0,80), b=1:365, d=100000) plot(circle, type='l') polygon(circle, col='blue', border='black', lwd=4) polygon(circle2, col='red', lwd=4, border='orange') polygon(circle3, col='white', lwd=4, border='black') @ \section{Maximum latitude on a great circle} You can use the functions illustrated below to find out what the maximum latitude is that a great circle will reach; at what latitude it crosses a specified longitude; or at what longitude it crosses a specified latitude. From the map below it appears that Clairaut's formula, used in gcMaxLat is not very accurate. Through optimization with function greatCircle, a more accurate value was found. The southern-most point is the antipode (a point at the opposite end of the world) of the northern-most point. <>= ml <- gcMaxLat(LA, NY) lat0 <- gcLat(LA, NY, lon=0) lon0 <- gcLon(LA, NY, lat=0) plot(wrld, type='l') lines(gc, lwd=2, col='blue') points(ml, col='red', pch=20, cex=2) points(cbind(0, lat0), pch=20, cex=2, col='yellow') points(t(rbind(lon0, 0)), pch=20, cex=2, col='green' ) f <- function(lon){gcLat(LA, NY, lon)} opt <- optimize(f, interval=c(-180, 180), maximum=TRUE) points(opt$maximum, opt$objective, pch=20, cex=2, col='dark green' ) anti <- antipode(c(opt$maximum, opt$objective)) points(anti, pch=20, cex=2, col='dark blue' ) @ \section{Great circle intersections} Points of intersection of two great circles can be computed in two ways. We use a second great circle that connects San Francisco with Amsterdam. We first compute where they cross by defining the great circles using two points on it (gcIntersect). After that, we compute the same points using a start point and initial bearing (gcIntersectBearing). The two points where the great circles cross are antipodes. Antipodes are connected with an infinite number of great circles. <>= SF <- c(-122.44, 37.74) AM <- c(4.75, 52.31) gc2 <- greatCircle(AM, SF) plot(wrld, type='l') lines(gc, lwd=2, col='blue') lines(gc2, lwd=2, col='green') int <- gcIntersect(LA, NY, SF, AM) int antipodal(int[,1:2], int[,3:4]) points(rbind(int[,1:2], int[,3:4]), col='red', pch=20, cex=2) bearing1 <- bearing(LA, NY) bearing2 <- bearing(SF, AM) bearing1 bearing2 gcIntersectBearing(LA, bearing1, SF, bearing2) @ \section{Triangulation} Below is triangulation example. We have three locations (NY, LA, MS) and three directions (281, 60, 195) towards a target. Because we are on a sphere, there are two (antipodal) results. We only show one here (by only using int[,1:2]). We compute the centroid from the polygon defined with the three points. To accurately draw a spherical polygon, we can use makePoly. This function inserts intermediate points along the paths between the vertices provided (default is one point every 10 km). <>= MS <- c(-93.26, 44.98) gc1 <- greatCircleBearing(NY, 281) gc2 <- greatCircleBearing(MS, 195) gc3 <- greatCircleBearing(LA, 55) plot(wrld, type='l', xlim=c(-125, -70), ylim=c(20, 60)) lines(gc1, col='green') lines(gc2, col='blue') lines(gc3, col='red') int <- gcIntersectBearing(rbind(NY, NY, MS), c(281, 281, 195), rbind(MS, LA, LA), c(195, 55, 55)) int distm(rbind(int[,1:2], int[,3:4])) int <- int[,1:2] points(int) poly <- rbind(int, int[1,]) centr <- centroid(poly) poly2 <- makePoly(int) polygon(poly2, col='yellow') points(centr, pch='*', col='dark red', cex=2) @ \section{Bearing} Below we first compute the distance and bearing from Los Angeles (LA) to New York (NY). These are then used to compute the point from LA at that distance in that (initial) bearing (direction). Bearing changes continuously when traveling along a Great Circle. The final bearing, when approaching NY, is also given. <>= d <- distCosine(LA, NY) d b <- bearing(LA, NY) b destPoint(LA, b, d) NY finalBearing(LA, NY) @ \section{Getting off-track} What if we went off-course and were flying over Minneapolis (MS)? The closest point on the planned route (p) can be computed with the alongTrackDistance and destPoint functions. The distance from 'p' to MS can be computed with the dist2gc (distance to great circle, or cross-track distance) function. The light green line represents the along-track distance, and the dark green line represents the cross-track distance. <>= atd <- alongTrackDistance(LA, NY, MS) p <- destPoint(LA, b, atd) plot(wrld, type='l', xlim=c(-130,-60), ylim=c(22,52)) lines(gci, col='blue', lwd=2) points(rbind(LA, NY), col='red', pch=20, cex=2) points(MS[1], MS[2], pch=20, col='blue', cex=2) lines(gcIntermediate(LA, p), col='green', lwd=3) lines(gcIntermediate(MS, p), col='dark green', lwd=3) points(p, pch=20, col='red', cex=2) dist2gc(LA, NY, MS) distCosine(p, MS) @ \section{Distance to a polyline} The two function describe above are used in the dist2Line function that computes the shortest distance between a set of points and a set of spherical poly-lines (or polygons). <>= line <- rbind(c(-180,-20), c(-150,-10), c(-140,55), c(10, 0), c(-140,-60)) pnts <- rbind(c(-170,0), c(-75,0), c(-70,-10), c(-80,20), c(-100,-50), c(-100,-60), c(-100,-40), c(-100,-20), c(-100,-10), c(-100,0)) d = dist2Line(pnts, line) plot( makeLine(line), type='l') points(line) points(pnts, col='blue', pch=20) points(d[,2], d[,3], col='red', pch='x', cex=2) for (i in 1:nrow(d)) lines(gcIntermediate(pnts[i,], d[i,2:3], 10), lwd=2, col='green') @ \section{Rhumb lines} Rhumb (from the Spanish word for course, 'rumbo') lines are straight lines on a Mercator projection map (and at most latitudes pretty straight on an equirectangular projection (=unprojected lon/lat) map). They were used in navigation because it is easier to follow a constant compass bearing than to continually adjust direction as is needed to follow a great circle, even though rhumb lines are normally longer than great-circle (orthodrome) routes. Most rhumb lines will gradually spiral towards one of the poles. <>= NP <- c(0, 85) bearing(SF, NP) b <- bearingRhumb(SF, NP) b dc <- distCosine(SF, NP) dr <- distRhumb(SF, NP) dc / dr pr <- destPointRhumb(SF, b, d=round(dr/100) * 1:100) pc <- rbind(SF, gcIntermediate(SF, NP), NP) par(mfrow=c(1,2)) data(wrld) plot(wrld, type='l', xlim=c(-140,10), ylim=c(15,90), main='Equirectangular') lines(pr, col='blue') lines(pc, col='red') data(merc) plot(merc, type='l', xlim=c(-15584729, 1113195), ylim=c(2500000, 22500000), main='Mercator') lines(mercator(pr), col='blue') lines(mercator(pc), col='red') @ \section{Characterizing polygons} The package has functions to compute the area, perimeter, centroid, and 'span' of a spherical polygon. One approach to compute these measures is to project the polygons first. Here we directly compute them based on spherical coordinates (longitude / latitude), except for centroid, which is computed by projecting the data to the Mercator projection (and inversely projecting the result). The function makePoly inserts additional vertices into a spherical polygon such that it can be plotted (perhaps after first projecting it) more correctly in a plane. Vertices are inserted, where necessary, at a specified distance. The function is only beneficial for polygons with large inter-vertex distances (in terms of longitude), particularly at high latitudes. <>= pol <- rbind(c(-120,-20), c(-80,5), c(0, -20), c(-40,-60), c(-120,-20)) areaPolygon(pol) perimeter(pol) centroid(pol) span(pol, fun=max) nicepoly = makePoly(pol) plot(pol, xlab='longitude', ylab='latitude', cex=2, lwd=3, xlim=c(-140, 0)) lines(wrld, col='grey') lines(pol, col='red', lwd=2) lines(nicepoly, col='blue', lwd=2) points(centroid(pol), pch='*', cex=3, col='dark green') text(centroid(pol)-c(0,2.5), 'centroid') legend(-140, -48, c('planar','spherical'), lty=1, lwd=2, col=c('red', 'blue'), title='polygon type') @ \section{Sampling} Random or regular sampling of longitude/latitude values on the globe needs to consider that the globe is spherical. That is, if you would take random points for latitude between -90 and 90 and for longitude between -180 and 180, the density of points would be higher near the poles than near the equator. In contrast, functions 'randomCoordinates' and 'randomCoordinates' return samples that are spatially balanced. <>= plot(wrld, type='l', col='grey') a = randomCoordinates(500) points(a, col='blue', pch=20, cex=0.5) b = regularCoordinates(3) points(b, col='red', pch='x') @ \section{Daylength} You can compute daylenght according to the formula by Forsythe et al. (1995). For any day of the year (an integer between 1 and 365; or a 'Date' object. <>= as.Date(80, origin='2009-12-31') as.Date(172, origin='2009-12-31') plot(0:90, daylength(lat=0:90, doy=1), ylim=c(0,24), type='l', xlab='Latitude', ylab='Daylength', main='Daylength by latitude and day of year', lwd=2) lines(0:90, daylength(lat=0:90, doy=80), col='green', lwd=2) lines(0:90, daylength(lat=0:90, doy=172), col='blue', lwd=2) legend(0,24, c('1','80','172'), lty=1, lwd=2, col=c('black', 'green', 'blue'), title='Day of year') @ \section{References} \begin{hangparas}{3em}{1} \noindent Forsythe, W.C., E.J. Rykiel Jr., R.S. Stahl, H. Wu and R.M. Schoolfield, 1995. A model comparison for daylength as a function of latitude and day of the year. Ecological Modeling 80:87-95. \noindent Sinnott, R.W, 1984. Virtues of the Haversine. Sky and Telescope 68(2): 159 \noindent Vincenty, T. 1975. Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations. Survey Review 23(176): 88-93. Available here: \url{http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf} \end{hangparas} \end{document} geosphere/inst/doc/geosphere.pdf0000644000176200001440000067650414335012510016445 0ustar liggesusers%PDF-1.5 % 1 0 obj << /Type /ObjStm /Length 4416 /Filter /FlateDecode /N 84 /First 673 >> stream x\ksDZ_1"~b罛Jʒr亾q P -sfvz @eRT 1=>=2UVi*+ozTQEN/իJJkM"Cptg{2mC hq2ޣqLXYٖڢXemh,<(.*ہO)ۣ2.]bhZ9tbYV9ЉS ^zS.zF:~{ʷl* z8C;DAء$BR aZ0DU-ZN LT:*"A!m❊}b-~;j9{h.vTGޡTaR8%-&t6;L5 b u-jPv=dV{j^cAB>u`nFUǥ9V(V?jbܬOX+?>?ɳb3[ZңɷWkV}5]&Wil^jh?W3ֺ%sXVQu/ѓ՟&?6úY.&6W/wՇftru>[ $Coj̝m3̬i]b7mo`QM|o#wL^NI9[6Vzr9\.͇ˇ bkMb]`q%-/Vӫ&2b,fÙ^k Rh1x14k2`p = vԳ&ۯV˳325ys4uN&ݏM^oחwԵn}U]S.Ζw|ftZoX64BNNZq8V|t-ڴ-t]^Nzc<,Xq 4^ gĖIs1+a[P-)WжT+MA;u{ۂmAfMRܻfAf\_ >z햠GSI5q*3j5f nAC4ܳGO1htG ."-VHe6aly !|3?_:yL^&NGb Ln.t4͇a03\hK5uױoa`Ú=Vo !P%䊘ܳ2φ{_|.2^l^ᎷD"(hUoh},~Fqݫ[EUx(|^,<6d;U_0"U=2Lwݞtw01N|X;Ǯg!Ľ*Z#wG[Nh!d.\ԡRn `ֽn\1w_?9hw U;p}-U>TU8Zw09{'pzՇRO>4<{/V 0-.b96_&~x/_<VYsu6Jx4&O:oxcAo۾s;NCm~ݚ!;Zp&#B5W?gk O)B2)[Ipf/ j巕߲nַ9v4wE &;ZxZ `N, ˂-:6BXpyl³g -<[`A-Y.6FMǷl $e`"Ev-md\Yx%o;1TrA;r߱.0v#}UO^rA9zKRҤҦҥҧ22KejS[V:թNmujS[ښ֤&5LM)9!8z8bK K+S,.a$0_IáR^Vxpi+h;$`^0 /m O[QNx;ᅧbRXAܣF>  DT_ C{Z‚cL6/HP/xW@IgLt9MСpYvu>)z#$rc- fq# 7bLZ`Ɋ`B 1r8QFL,#v;͈}{,0ʜbȬ6mĘ, İLb0QB SG 2b0Y,C# 6{gt͈53peĘ2NRe;]>h0i &'3k H.ᔻޘ؄FO),p4Ccv s0:4X5i|ÄT̅u*ͤnz+7W 6ML&gk.ˑZ`k FA6]>@29U\SS0twSڦJV^ZŖ#;?Fՙ ( &U3if|F(;ޥEAW[,a"Ns_ ƍPHm6B!VmBhDq( GX>a#Pܗ#?B!x ^%5 ^E<R%x#jĩb8UTlTq]8Uhq, &[q8UqqTũ"SE8.Nq\*8Uqqũ"SE8.Nq\*x/VվX@5̗cFSp/Z i HZAL- FtX1ّXhTmڶrmm[V3LGNڢS0Vl3̀Օkk6à[:זckKjb.mo4DȍN|Cm=|<( mjqm\UW(t}楏 )^76{5=?c)=tm6O 9i)ཱིUb2? #`T o \B|SQFh~V=ñWD3֠*/7w1=ԏKtCpj`.g̥m8XVU079©s BV[s{EU0W\Ws= V\~Ys`̵0Y~@uU պ*j]H ZWR"ToO@U ՊFІomX  1@bPA.G0 UӊA,Ρz! @]ޯ9mkp7Xpo3N7` plyK3fg%};j{55ɰ+]}SM=kTثOjk|^[o%݀N-nh0j~]GI:ec}tdc}tdc}tdmIl&f-Y􄘵bROtI'f-81kiuUǎg@] *VBljț&}_8]vKkrXj [DnŔ5}9Pgx9Y ^'?nYϫ*"`޲}Z_لGճw[%ű+XOƖQG2;~qiэ^dL_-ۻ caqn1, =i0aqn1,=Pmvpam8qF**  fx0ވŘ8\ 9JTKu &` ח:t9QOPvLqvb}۞({*A O 0964(K7L iv[,%5IIlͯn|$-bs\)Qs!vya=HL{6L/5޼_ԓYͦfdZcmǶc6/of?{W;f4}^˷bo W؍kBӪ<$_MrӡIJL{|s9W\MO/)9endstream endobj 86 0 obj << /Filter /FlateDecode /Length 1932 >> stream x}XK۸W( 3[$k'z6C6 aS fvT FMf16a#DUe&TUl?6ɣ ͥowJh=a<ƙL8OfIfÏ2R]"E%KV{ſn22nKd [?x#L%hl)8T*T_{0!H? 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ODϵ/:eT!"0wGO{LTayJȄ=WR>/Ǽ#q28ZR7e.e+cް"o^5 NTK0񜄿 k}FTdtH]^Z^S}-"E^$g{z;h#7%sҺjY>ywQ>ykni-PTY \#%*Vb!O ^词0>g cJ&5څ7ȉ@";܆CVvFjŧ QYUf>Jղ}&Yn"`#Snd0:`oH2}i"m;2iFOT}$/;!G U  ,]B2bW#߆yNb21 ?`:rK& 듏VØv S3b5" D|O:5X>ae8 ۑZ=G3=av2q`J[|攃^pe}E-Z@)yqnhޱF":JLV?#Cq 緔" XAaDںG>0 YZgeosphere/man/0000755000176200001440000000000014334225312013005 5ustar liggesusersgeosphere/man/area.Rd0000644000176200001440000000405214131145600014201 0ustar liggesusers\name{areaPolygon} \Rdversion{1.1} \alias{areaPolygon} \alias{areaPolygon,matrix-method} \alias{areaPolygon,data.frame-method} \alias{areaPolygon,SpatialPolygons-method} \title{ Area of a longitude/latitude polygon } \description{ Compute the area of a polygon in angular coordinates (longitude/latitude) on an ellipsoid. } \usage{ \S4method{areaPolygon}{matrix}(x, a=6378137, f=1/298.257223563, ...) \S4method{areaPolygon}{SpatialPolygons}(x, a=6378137, f=1/298.257223563, ...) } \arguments{ \item{x}{longitude/latitude of the points forming a polygon; Must be a matrix or data.frame of 2 columns (first one is longitude, second is latitude) or a SpatialPolygons* object} \item{a}{major (equatorial) radius of the ellipsoid} \item{f}{ellipsoid flattening. The default value is for WGS84 } \item{...}{Additional arguments. None implemented} } \value{ area in square meters } \note{ Use raster::area for polygons that have a planar (projected) coordinate reference system. } \author{ This function calls GeographicLib code by C.F.F. Karney } \references{ C.F.F. Karney, 2013. Algorithms for geodesics, J. Geodesy 87: 43-55. \doi{10.1007/s00190-012-0578-z}. Addenda: \url{https://geographiclib.sourceforge.io/geod-addenda.html}. Also see \url{https://geographiclib.sourceforge.io/} } \seealso{ \code{ \link[geosphere]{centroid}, \link[geosphere]{perimeter} } } \examples{ p <- rbind(c(-180,-20), c(-140,55), c(10, 0), c(-140,-60), c(-180,-20)) areaPolygon(p) # Be careful with very large polygons, as they may not be what they seem! # For example, if you wanted a polygon to compute the area equal to about 1/4 of the ellipsoid # this won't work: b <- matrix(c(-180, 0, 90, 90, 0, 0, -180, 0), ncol=2, byrow=TRUE) areaPolygon(b) # Becausee the shortest path between (-180,0) and (0,0) is # over one of the poles, not along the equator! # Inserting a point along the equator fixes that b <- matrix(c(-180, 0, 0, 0, -90,0, -180, 0), ncol=2, byrow=TRUE) areaPolygon(b) } \keyword{ spatial } geosphere/man/bearingRhumb.Rd0000644000176200001440000000217014131210620015670 0ustar liggesusers\name{bearingRhumb} \Rdversion{1.1} \alias{bearingRhumb} \title{ Rhumbline direction } \description{ Bearing (direction of travel; true course) along a rhumb line (loxodrome) between two points. } \usage{ bearingRhumb(p1, p2) } \arguments{ \item{p1}{longitude/latitude of point(s). Can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{p2}{as above} } \value{ A direction (bearing) in degrees } \references{ \url{https://www.edwilliams.org/avform147.htm#Rhumb} \url{https://en.wikipedia.org/wiki/Rhumb_line} } \author{ Chris Veness and Robert Hijmans, based on formulae by Ed Williams } \note{ Unlike most great circles, a rhumb line is a line of constant bearing (direction), i.e. tracks of constant true course. The meridians and the equator are both rhumb lines and great circles. Rhumb lines approaching a pole become a tightly wound spiral. } \seealso{ \code{ \link[geosphere]{bearing}, \link[geosphere]{distRhumb} } } \examples{ bearingRhumb(c(10,10),c(20,20)) } \keyword{ spatial } geosphere/man/finalDirection.Rd0000644000176200001440000000261714131145601016231 0ustar liggesusers\name{finalBearing} \Rdversion{1.1} \alias{finalBearing} \title{ Final direction } \description{ Get the final direction (bearing) when arriving at \code{p2} after starting from \code{p1} and following the shortest path on an ellipsoid (following a geodetic) or on a sphere (following a great circle). } \usage{ finalBearing(p1, p2, a=6378137, f=1/298.257223563, sphere=FALSE) } \arguments{ \item{p1}{longitude/latitude of point(s). Can be a vector of two numbers, a matrix of 2 columns (first column is longitude, second column is latitude) or a SpatialPoints* object} \item{p2}{as above} \item{a}{major (equatorial) radius of the ellipsoid. The default value is for WGS84 } \item{f}{ellipsoid flattening. The default value is for WGS84 } \item{sphere}{logical. If \code{TRUE}, the bearing is computed for a sphere, instead of for an ellipsoid} } \value{ A vector of directions (bearings) in degrees } \examples{ bearing(c(10,10),c(20,20)) finalBearing(c(10,10),c(20,20)) } \author{ This function calls GeographicLib code by C.F.F. Karney } \references{ C.F.F. Karney, 2013. Algorithms for geodesics, J. Geodesy 87: 43-55. \doi{10.1007/s00190-012-0578-z}. Addenda: \url{https://geographiclib.sourceforge.io/geod-addenda.html}. Also see \url{https://geographiclib.sourceforge.io/} } \seealso{ \code{ \link[geosphere]{bearing} } } \keyword{ spatial } geosphere/man/distVincentySphere.Rd0000644000176200001440000000241514131143521017124 0ustar liggesusers\name{distVincentySphere} \Rdversion{1.1} \alias{distVincentySphere} \title{ 'Vincenty' (sphere) great circle distance } \description{ The shortest distance between two points (i.e., the 'great-circle-distance' or 'as the crow flies'), according to the 'Vincenty (sphere)' method. This method assumes a spherical earth, ignoring ellipsoidal effects and it is less accurate then the \code{distVicentyEllipsoid} method. } \usage{ distVincentySphere(p1, p2, r=6378137) } \arguments{ \item{p1}{longitude/latitude of point(s). Can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{p2}{as above; or missing, in which case the sequential distance between the points in p1 is computed} \item{r}{radius of the earth; default = 6378137 m} } \value{ Distance value in the same unit as \code{r} (default is meters) } \references{ \url{https://en.wikipedia.org/wiki/Great_circle_distance} } \author{ Robert Hijmans } \seealso{ \code{\link[geosphere]{distGeo}, \link[geosphere]{distVincentyEllipsoid}, \link[geosphere]{distHaversine}, \link[geosphere]{distCosine}, \link{distMeeus}} } \examples{ distVincentySphere(c(0,0),c(90,90)) } \keyword{ spatial } geosphere/man/greatCircleBearing.Rd0000644000176200001440000000152114131210620016776 0ustar liggesusers\name{greatCircleBearing} \Rdversion{1.1} \alias{greatCircleBearing} \title{ Great circle } \description{ Get points on a great circle as defined by a point and an initial bearing } \usage{ greatCircleBearing(p, brng, n=360) } \arguments{ \item{p}{longitude/latitude of a single point. Can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{brng}{bearing} \item{n}{The requested number of points on the great circle} } \value{ A matrix of points, or a list of matrices (e.g., if multiple bearings are supplied) } \references{ \url{https://www.edwilliams.org/avform147.htm#Int} } \author{ Robert Hijmans based on formulae by Ed Williams } \examples{ greatCircleBearing(c(5,52), 45, n=12) } \keyword{ spatial } geosphere/man/perimeter.Rd0000644000176200001440000000321114131145601015262 0ustar liggesusers\name{perimeter} \Rdversion{1.1} \alias{perimeter} \alias{perimeter,matrix-method} \alias{perimeter,data.frame-method} \alias{perimeter,SpatialPolygons-method} \alias{perimeter,SpatialLines-method} \title{ Compute the perimeter of a longitude/latitude polygon } \description{ Compute the perimeter of a polygon (or the length of a line) with longitude/latitude coordinates, on an ellipsoid (WGS84 by default) } \usage{ \S4method{perimeter}{matrix}(x, a=6378137, f=1/298.257223563, ...) \S4method{perimeter}{SpatialPolygons}(x, a=6378137, f=1/298.257223563, ...) \S4method{perimeter}{SpatialLines}(x, a=6378137, f=1/298.257223563, ...) } \arguments{ \item{x}{Longitude/latitude of the points forming a polygon or line; Must be a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPolygons* or SpatialLines* object} \item{a}{major (equatorial) radius of the ellipsoid. The default value is for WGS84 } \item{f}{ellipsoid flattening. The default value is for WGS84 } \item{...}{Additional arguments. None implemented} } \value{ Numeric. The perimeter or length in m. } \seealso{ \code{ \link{areaPolygon}, \link[geosphere]{centroid} } } \author{ This function calls GeographicLib code by C.F.F. Karney } \references{ C.F.F. Karney, 2013. Algorithms for geodesics, J. Geodesy 87: 43-55. \doi{10.1007/s00190-012-0578-z}. Addenda: \url{https://geographiclib.sourceforge.io/geod-addenda.html}. Also see \url{https://geographiclib.sourceforge.io/} } \examples{ xy <- rbind(c(-180,-20), c(-140,55), c(10, 0), c(-140,-60), c(-180,-20)) perimeter(xy) } \keyword{ spatial } geosphere/man/distm.Rd0000644000176200001440000000200014131143521014400 0ustar liggesusers\name{distm} \Rdversion{1.1} \alias{distm} \title{ Distance matrix } \description{ Distance matrix of a set of points, or between two sets of points } \usage{ distm(x, y, fun=distGeo) } \arguments{ \item{x}{longitude/latitude of point(s). Can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{y}{Same as \code{x}. If missing, y is the same as x} \item{fun}{A function to compute distances (e.g., distCosine or distGeo)} } \value{ Matrix of distances } \references{ \url{https://en.wikipedia.org/wiki/Great_circle_distance} } \author{ Robert Hijmans } \seealso{ \code{\link[geosphere]{distGeo}, \link[geosphere]{distCosine}, \link[geosphere]{distHaversine}, \link[geosphere]{distVincentySphere}, \link[geosphere]{distVincentyEllipsoid}} } \examples{ xy <- rbind(c(0,0),c(90,90),c(10,10),c(-120,-45)) distm(xy) xy2 <- rbind(c(0,0),c(10,-10)) distm(xy, xy2) } \keyword{ spatial } geosphere/man/horizon.Rd0000644000176200001440000000137514131210620014761 0ustar liggesusers\name{horizon} \alias{horizon} \title{Distance to the horizon} \description{ Empirical function to compute the distance to the horizon from a given altitude. The earth is assumed to be smooth, i.e. mountains and other obstacles are ignored. } \usage{ horizon(h, r=6378137) } \arguments{ \item{h}{altitude, numeric >= 0. Should have the same unit as r} \item{r}{radius of the earth; default value is 6378137 m} } \value{ Distance in units of \code{h} (default is meters) } \references{ \url{https://www.edwilliams.org/avform147.htm#Horizon} Bowditch, 1995. American Practical Navigator. Table 12. } \author{ Robert J. Hijmans } \examples{ horizon(1.80) # me horizon(324) # Eiffel tower } \keyword{ spatial } geosphere/man/makepoly.Rd0000644000176200001440000000221713472155746015136 0ustar liggesusers\name{makePoly} \alias{makePoly} \alias{makeLine} \title{Add vertices to a polygon or line} \description{ Make a polygon or line by adding intermedate points (vertices) on the great circles inbetween the points supplied. This can be relevant when vertices are relatively far apart. It can make the shape of the object to be accurate, when plotted on a plane. \code{makePoly} will also close the polygon if needed. } \usage{ makePoly(p, interval=10000, sp=FALSE, ...) makeLine(p, interval=10000, sp=FALSE, ...) } \arguments{ \item{p}{a 2-column matrix (longitude/latitude) or a SpatialPolygons or SpatialLines object} \item{interval}{maximum interval of points, in units of r} \item{sp}{Logical. If \code{TRUE}, a SpatialPolygons object is retunred (depends on the 'sp' package)} \item{...}{additional arguments passed to distGeo} } \value{ A matrix } \author{Robert J. Hijmans } \examples{ pol <- rbind(c(-180,-20), c(-160,5), c(-60, 0), c(-160,-60), c(-180,-20)) plot(pol) lines(pol, col='red', lwd=3) pol2 = makePoly(pol, interval=100000) lines(pol2, col='blue', lwd=2) } \keyword{methods} \keyword{spatial} geosphere/man/gcIntersectBearing.Rd0000644000176200001440000000231414131210620017025 0ustar liggesusers\name{gcIntersectBearing} \Rdversion{1.1} \alias{gcIntersectBearing} \title{ Intersections of two great circles } \description{ Get the two points where two great cricles cross each other. In this function, great circles are defined by a points and an initial bearing. In function \code{ \link[geosphere]{gcIntersect}} they are defined by two sets of points. } \usage{ gcIntersectBearing(p1, brng1, p2, brng2) } \arguments{ \item{p1}{longitude/latitude of point(s). Can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{brng1}{Bearing from p1} \item{p2}{As above. Should have same length as p1, or a single point (or vice versa when p1 is a single point} \item{brng2}{Bearing from p2} } \value{ a matrix with four columns (two points) } \seealso{ \code{ \link[geosphere]{gcIntersect} } } \references{ \url{https://www.edwilliams.org/avform147.htm#Intersection} \url{https://www.movable-type.co.uk/scripts/latlong.html} } \author{ Chris Veness and Robert Hijmans based on code by Ed Williams } \examples{ gcIntersectBearing(c(10,0), 10, c(-10,0), 10) } \keyword{ spatial } geosphere/man/distCosine.Rd0000644000176200001440000000211514131143521015373 0ustar liggesusers\name{distCosine} \Rdversion{1.1} \alias{distCosine} \title{ 'Law of cosines' great circle distance } \description{ The shortest distance between two points (i.e., the 'great-circle-distance' or 'as the crow flies'), according to the 'law of the cosines'. This method assumes a spherical earth, ignoring ellipsoidal effects. } \usage{ distCosine(p1, p2, r=6378137) } \arguments{ \item{p1}{longitude/latitude of point(s). Can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{p2}{as above} \item{r}{radius of the earth; default = 6378137 m} } \value{ Vector of distances in the same unit as \code{r} (default is meters) } \references{ \url{https://en.wikipedia.org/wiki/Great_circle_distance} } \author{ Robert Hijmans } \seealso{ \code{\link[geosphere]{distGeo}, \link[geosphere]{distHaversine}, \link[geosphere]{distVincentySphere}, \link[geosphere]{distVincentyEllipsoid}, \link{distMeeus}} } \examples{ distCosine(c(0,0),c(90,90)) } \keyword{ spatial } geosphere/man/plotArrows.Rd0000644000176200001440000000211513472155746015466 0ustar liggesusers\name{plotArrows} \alias{plotArrows} \title{Plot} \description{ Plot polygons with arrow heads on each line segment, pointing towards the next vertex. This shows the direction of each line segment. } \usage{ plotArrows(p, fraction=0.9, length=0.15, first='', add=FALSE, ...) } \arguments{ \item{p}{Polygons (either a 2 column matrix or data.frame; or a SpatialPolygons* object} \item{fraction}{numeric between 0 and 1. When smaller then 1, interrupted lines are drawn} \item{length}{length of the edges of the arrow head (in inches)} \item{first}{Character to plot on first (and last) vertex } \item{add}{Logical. If \code{TRUE}, the plot is added to an existing plot} \item{...}{Additional arguments, see Details} } \author{Robert J. Hijmans} \note{ Based on an example in Software for Data Analysis by John Chambers (pp 250-251) but adjusted such that the line segments follow great circles between vertices. } \examples{ pol <- rbind(c(-180,-20), c(-160,5), c(-60, 0), c(-160,-60), c(-180,-20)) plotArrows(pol) } \keyword{methods} \keyword{spatial} geosphere/man/distRhumb.Rd0000644000176200001440000000272114131143521015233 0ustar liggesusers\name{distRhumb} \Rdversion{1.1} \alias{distRhumb} \title{ Distance along a rhumb line } \description{ A rhumb line (loxodrome) is a path of constant bearing (direction), which crosses all meridians at the same angle. } \usage{ distRhumb(p1, p2, r=6378137) } \arguments{ \item{p1}{longitude/latitude of point(s). Can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{p2}{as above; or missing, in which case the sequential distance between the points in p1 is computed} \item{r}{radius of the earth; default = 6378137 m} } \details{ Rhumb (from the Spanish word for course, 'rumbo') lines are straight lines on a Mercator projection map. They were used in navigation because it is easier to follow a constant compass bearing than to continually adjust the bearing as is needed to follow a great circle, even though rhumb lines are normally longer than great-circle (orthodrome) routes. Most rhumb lines will gradually spiral towards one of the poles. } \value{ distance in units of r (default=meters) } \references{ \url{https://www.movable-type.co.uk/scripts/latlong.html} } \author{ Robert Hijmans and Chris Veness } \seealso{ \code{\link[geosphere]{distCosine}, \link[geosphere]{distHaversine}, \link[geosphere]{distVincentySphere}, \link[geosphere]{distVincentyEllipsoid}} } \examples{ distRhumb(c(10,10),c(20,20)) } \keyword{ spatial } geosphere/man/dist2line.Rd0000644000176200001440000000340614131141731015171 0ustar liggesusers\name{dist2Line} \Rdversion{1.1} \alias{dist2Line} \title{ Distance between points and lines or the border of polygons. } \description{ The shortest distance between points and polylines or polygons. } \usage{dist2Line(p, line, distfun=distGeo) } \arguments{ \item{p}{longitude/latitude of point(s). Can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{line}{longitude/latitude of line as a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialLines* or SpatialPolygons* object} \item{distfun}{A distance function, such as \link[geosphere]{distGeo}} } \value{ matrix with distance and lon/lat of the nearest point on the line. Distance is in the same unit as \code{r} in the \code{distfun}(default is meters). If \code{line} is a \code{Spatial*} object, the ID (index) of (one of) the nearest objects is also returned. Thus if the objects are polygons and the point is inside a polygon the function may return the ID of a neighboring polygon that shares the nearest border. You can use the \code{intersect} function in packages \code{terra}. } \author{ George Wang and Robert Hijmans } \seealso{ \code{\link{dist2gc}, \link{alongTrackDistance} } } \examples{ line <- rbind(c(-180,-20), c(-150,-10), c(-140,55), c(10, 0), c(-140,-60)) pnts <- rbind(c(-170,0), c(-75,0), c(-70,-10), c(-80,20), c(-100,-50), c(-100,-60), c(-100,-40), c(-100,-20), c(-100,-10), c(-100,0)) d = dist2Line(pnts, line) plot( makeLine(line), type='l') points(line) points(pnts, col='blue', pch=20) points(d[,2], d[,3], col='red', pch='x') for (i in 1:nrow(d)) lines(gcIntermediate(pnts[i,], d[i,2:3], 10), lwd=2) } \keyword{ spatial } geosphere/man/lengthLine.Rd0000644000176200001440000000112414131141671015363 0ustar liggesusers\name{lengthLine} \alias{lengthLine} \title{ Length of lines } \description{ Compute the length of lines } \usage{ lengthLine(line) } \arguments{ \item{line}{longitude/latitude of line as a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialLines* or SpatialPolygons* object} } \value{ length (in meters) for each line } \seealso{ For planar coordinates, see the terra or sf packages } \examples{ line <- rbind(c(-180,-20), c(-150,-10), c(-140,55), c(10, 0), c(-140,-60)) d <- lengthLine(line) } \keyword{ spatial } geosphere/man/geosphere-package.Rd0000644000176200001440000000443014131210620016636 0ustar liggesusers\name{geosphere-package} \alias{geosphere-package} \alias{geosphere} \docType{package} \title{Geosphere} \description{ This package implements functions that compute various aspects of distance, direction, area, etc. for geographic (geodetic) coordinates. Some of the functions are based on an ellipsoid (spheroid) model of the world, other functions use a (simpler, but less accuarate) spherical model. Functions using an ellipsoid can be recognized by having arguments to specify the ellipsoid's radius and flattening (\code{a} and \code{f}). By setting the value for \code{f} to zero, the ellipsoid becomes a sphere. There are also functions to compute intersections of of rhumb lines. There are also functions to compute the distance between points and polylines, and to characterize spherical polygons; for random sampling on a sphere, and to compute daylength. See the vignette \code{vignette('geosphere')} for examples. Geographic locations must be specified in latitude and longitude in degrees (NOT radians). Degrees are (obviously) in decimal notation. Thus 12 degrees, 30 minutes, 10 seconds = 12 + 30/60 + 10/3600 = 12.50278 degrees. The Southern and Western hemispheres have a negative sign. The default unit of distance is meter; but this can be adjusted by supplying a different radius \code{r} to functions. Directions are expressed in degrees (North = 0 and 360, East = 90, Sout = 180, and West = 270 degrees). } \author{ Robert Hijmans, using code by C.F.F. Karney and Chris Veness; formulas by Ed Williams; and with contributions from George Wang, Elias Pipping and others. Maintainer: Robert J. Hijmans } \references{ C.F.F. Karney, 2013. Algorithms for geodesics, J. Geodesy 87: 43-55. \doi{10.1007/s00190-012-0578-z}. Addenda: \url{https://geographiclib.sourceforge.io/geod-addenda.html}. Also see \url{https://geographiclib.sourceforge.io/} \url{https://www.edwilliams.org/avform147.htm} \url{https://www.movable-type.co.uk/scripts/latlong.html} \url{https://en.wikipedia.org/wiki/Great_circle_distance} \url{https://mathworld.wolfram.com/SphericalTrigonometry.html} } \section{Acknowledgements}{ David Purdy, Bill Monahan and others for suggestions to improve the package. } \keyword{ package } \keyword{ spatial } geosphere/man/gcMaxLat.Rd0000644000176200001440000000231714131143521014773 0ustar liggesusers\name{gcMaxLat} \Rdversion{1.1} \alias{gcMaxLat} \title{ Highest latitude on a great circle } \description{ What is northern most point that will be reached when following a great circle? Computed with Clairaut's formula. The southern most point is the \code{\link[geosphere]{antipode}} of the northern-most point. This does not seem to be very precise; and you could use optimization instead to find this point (see examples) } \usage{ gcMaxLat(p1, p2) } \arguments{ \item{p1}{longitude/latitude of point(s). Can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{p2}{as above} } \value{ A matrix with coordinates (longitude/latitude) } \references{ \url{https://www.edwilliams.org/ftp/avsig/avform.txt} \url{https://www.movable-type.co.uk/scripts/latlong.html} } \author{ Ed Williams, Chris Veness, Robert Hijmans } \seealso{ \code{\link[geosphere]{gcLat}, \link[geosphere]{gcLon}} } \examples{ gcMaxLat(c(5,52), c(-120,37)) # Another way to get there: f <- function(lon){gcLat(c(5,52), c(-120,37), lon)} optimize(f, interval=c(-180, 180), maximum=TRUE) } \keyword{ spatial } geosphere/man/gcIntersect.Rd0000644000176200001440000000171014131143521015541 0ustar liggesusers\name{gcIntersect} \Rdversion{1.1} \alias{gcIntersect} \title{ Intersections of two great circles } \description{ Get the two points where two great cricles cross each other. Great circles are defined by two points on it. } \usage{ gcIntersect(p1, p2, p3, p4) } \arguments{ \item{p1}{Longitude/latitude of a single point, in degrees; can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{p2}{As above} \item{p3}{As above} \item{p4}{As above} } \value{ two points for each pair of great circles } \seealso{ \code{ \link[geosphere]{gcIntersectBearing} } } \references{ \url{https://www.edwilliams.org/intersect.htm} } \author{ Robert Hijmans, based on equations by Ed Williams (see reference) } \examples{ p1 <- c(5,52); p2 <- c(-120,37); p3 <- c(-60,0); p4 <- c(0,70) gcIntersect(p1,p2,p3,p4) } \keyword{ spatial } geosphere/man/bearing.Rd0000644000176200001440000000300614131145600014676 0ustar liggesusers\name{bearing} \Rdversion{1.1} \alias{bearing} \title{ Direction of travel } \description{ Get the initial bearing (direction; azimuth) to go from point \code{p1} to point \code{p2} (in longitude/latitude) following the shortest path on an ellipsoid (geodetic). Note that the bearing of travel changes continuously while going along the path. A route with constant bearing is a rhumb line (see \code{\link[geosphere]{bearingRhumb}}). } \usage{ bearing(p1, p2, a=6378137, f=1/298.257223563) } \arguments{ \item{p1}{longitude/latitude of point(s). Can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{p2}{as above. Can also be missing, in which case the bearing is computed going from the first point to the next and continuing along the following points} \item{a}{major (equatorial) radius of the ellipsoid. The default value is for WGS84 } \item{f}{ellipsoid flattening. The default value is for WGS84 } } \note{use \code{f=0} to get a bearing on a sphere (great circle)} \value{ Bearing in degrees } \author{ Robert Hijmans } \references{ C.F.F. Karney, 2013. Algorithms for geodesics, J. Geodesy 87: 43-55. \doi{10.1007/s00190-012-0578-z}. Addenda: \url{https://geographiclib.sourceforge.io/geod-addenda.html}. Also see \url{https://geographiclib.sourceforge.io/} } \seealso{ \code{ \link[geosphere]{bearingRhumb} } } \examples{ bearing(c(10,10),c(20,20)) } \keyword{ spatial } geosphere/man/distGeo.Rd0000644000176200001440000000551614131145601014676 0ustar liggesusers\name{distGeo} \Rdversion{1.1} \alias{distGeo} \title{ Distance on an ellipsoid (the geodesic) } \description{ Highly accurate estimate of the shortest distance between two points on an ellipsoid (default is WGS84 ellipsoid). The shortest path between two points on an ellipsoid is called the geodesic. } \usage{ distGeo(p1, p2, a=6378137, f=1/298.257223563) } \arguments{ \item{p1}{longitude/latitude of point(s). Can be a vector of two numbers, a matrix of 2 columns (first column is longitude, second column is latitude) or a SpatialPoints* object} \item{p2}{as above; or missing, in which case the sequential distance between the points in p1 is computed} \item{a}{numeric. Major (equatorial) radius of the ellipsoid. The default value is for WGS84 } \item{f}{numeric. Ellipsoid flattening. The default value is for WGS84 } } \value{ Vector of distances in meters } \details{ Parameters from the WGS84 ellipsoid are used by default. It is the best available global ellipsoid, but for some areas other ellipsoids could be preferable, or even necessary if you work with a printed map that refers to that ellipsoid. Here are parameters for some commonly used ellipsoids. Also see the \code{\link{refEllipsoids}} function. \tabular{rlll}{ \tab \code{ ellipsoid } \tab \code{ a } \tab \code{ f } \cr \tab \code{ WGS84 } \tab \code{ 6378137 } \tab \code{ 1/298.257223563 } \cr \tab \code{ GRS80 } \tab \code{ 6378137 } \tab \code{ 1/298.257222101 } \cr \tab \code{ GRS67 } \tab \code{ 6378160 } \tab \code{ 1/298.25 } \cr \tab \code{ Airy 1830 } \tab \code{ 6377563.396 } \tab \code{ 1/299.3249646 } \cr \tab \code{ Bessel 1841 } \tab \code{ 6377397.155 } \tab \code{ 1/299.1528434 } \cr \tab \code{ Clarke 1880 } \tab \code{ 6378249.145 } \tab \code{ 1/293.465 } \cr \tab \code{ Clarke 1866 } \tab \code{ 6378206.4 } \tab \code{ 1/294.9786982 } \cr \tab \code{ International 1924 } \tab \code{ 6378388 } \tab \code{ 1/297 } \cr \tab \code{ Krasovsky 1940 } \tab \code{ 6378245 } \tab \code{ 1/298.2997381 } \cr } more info: \url{https://en.wikipedia.org/wiki/Reference_ellipsoid} } \author{ This function calls GeographicLib code by C.F.F. Karney } \references{ C.F.F. Karney, 2013. Algorithms for geodesics, J. Geodesy 87: 43-55. \doi{10.1007/s00190-012-0578-z}. Addenda: \url{https://geographiclib.sourceforge.io/geod-addenda.html}. Also see \url{https://geographiclib.sourceforge.io/} } \seealso{ \code{\link[geosphere]{distCosine}, \link[geosphere]{distHaversine}, \link[geosphere]{distVincentySphere}, \link[geosphere]{distVincentyEllipsoid}, \link{distMeeus}} } \examples{ distGeo(c(0,0),c(90,90)) } \keyword{ spatial } geosphere/man/gcLon.Rd0000644000176200001440000000143414131210620014327 0ustar liggesusers\name{gcLon} \Rdversion{1.1} \alias{gcLon} \title{ Longitude on a Great Circle } \description{ Longitudes at which a great circle crosses a latitude (parallel) } \usage{ gcLon(p1, p2, lat) } \arguments{ \item{p1}{longitude/latitude of point(s). Can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{p2}{as above} \item{lat}{a latitude} } \value{ vector of two numbers (longitudes) } \references{ \url{https://www.edwilliams.org/avform147.htm#Intersection} } \author{ Robert Hijmans based on code by Ed Williams } \seealso{ \code{\link[geosphere]{gcLat}, \link[geosphere]{gcMaxLat}} } \examples{ gcLon(c(5,52), c(-120,37), 40) } \keyword{ spatial } geosphere/man/onGreatCircle.Rd0000644000176200001440000000160613472155746016037 0ustar liggesusers\name{onGreatCircle} \Rdversion{1.1} \alias{onGreatCircle} \title{ Is a point on a given great circle? } \description{ Test if a point is on a great circle defined by two other points. } \usage{ onGreatCircle(p1, p2, p3, tol=0.0001) } \arguments{ \item{p1}{Longitude/latitude of the first point defining a great circle, in degrees; can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{p2}{as above for the second point} \item{p3}{the point(s) to be tested if they are on the great circle or not} \item{tol}{numeric. maximum distance from the great circle (in degrees) that is tolerated to be considered on the circle} } \value{ logical } \author{ Robert Hijmans } \examples{ onGreatCircle(c(0,0), c(30,30), rbind(c(-10 -11.33812), c(10,20))) } \keyword{ spatial } geosphere/man/gcLat.Rd0000644000176200001440000000142114131210620014313 0ustar liggesusers\name{gcLat} \Rdversion{1.1} \alias{gcLat} \title{ Latitude on a Great Circle } \description{ Latitude at which a great circle crosses a longitude } \usage{ gcLat(p1, p2, lon) } \arguments{ \item{p1}{Longitude/latitude of a single point, in degrees; can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{p2}{As above} \item{lon}{Longitude} } \value{ A numeric (latitude) } \references{ \url{https://www.edwilliams.org/avform147.htm#Int} } \author{ Robert Hijmans based on a formula by Ed Williams } \seealso{ \code{\link[geosphere]{gcLon}, \link[geosphere]{gcMaxLat}} } \examples{ gcLat(c(5,52), c(-120,37), lon=-120) } \keyword{ spatial } geosphere/man/destPointRhumb.Rd0000644000176200001440000000214614131210620016235 0ustar liggesusers\name{destPointRhumb} \Rdversion{1.1} \alias{destPointRhumb} \title{ Destination along a rhumb line } \description{ Calculate the destination point when travelling along a 'rhumb line' (loxodrome), given a start point, direction, and distance. } \usage{ destPointRhumb(p, b, d, r = 6378137) } \arguments{ \item{p}{longitude/latitude of point(s). Can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{b}{bearing (direction) in degrees} \item{d}{distance; in the same unit as \code{r} (default is meters)} \item{r}{radius of the earth; default = 6378137 m} } \value{ Coordinates (longitude/latitude) of a point } \references{ \url{https://www.edwilliams.org/avform147.htm#Rhumb} \url{https://www.movable-type.co.uk/scripts/latlong.html} \url{https://en.wikipedia.org/wiki/Rhumb_line} } \author{ Chris Veness; ported to R by Robert Hijmans } \seealso{ \code{ \link[geosphere]{destPoint} } } \examples{ destPointRhumb(c(0,0), 30, 100000, r = 6378137) } \keyword{ spatial } geosphere/man/dist2gc.Rd0000644000176200001440000000267014131143521014634 0ustar liggesusers\name{dist2gc} \Rdversion{1.1} \alias{dist2gc} \title{ Cross Track Distance } \description{ Compute the distance of a point to a great-circle path (also referred to as the cross track distance or cross track error). The great circle is defined by \code{p1} and \code{p2}, while \code{p3} is the point away from the path. } \usage{ dist2gc(p1, p2, p3, r=6378137, sign=FALSE) } \arguments{ \item{p1}{Start of great circle path. longitude/latitude of point(s). Can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{p2}{End of great circle path. As above} \item{p3}{Point away from the great cricle path. As for p2} \item{r}{radius of the earth; default = 6378137} \item{sign}{logical. If \code{TRUE}, a negative sign is used to indicated that the points are to the left of the great circle} } \value{ A distance in units of \code{r} (default is meters) If \code{sign=TRUE}, the sign indicates which side of the path p3 is on. Positive means right of the course from p1 to p2, negative means left. } \author{ Ed Williams and Robert Hijmans } \seealso{ \code{\link{dist2Line}, \link{alongTrackDistance} } } \references{ \url{https://www.movable-type.co.uk/scripts/latlong.html} \url{https://www.edwilliams.org/ftp/avsig/avform.txt} } \examples{ dist2gc(c(0,0),c(90,90),c(80,80)) } \keyword{ spatial } geosphere/man/distHaversine.Rd0000644000176200001440000000342114131143521016100 0ustar liggesusers\name{distHaversine} \Rdversion{1.1} \alias{distHaversine} \title{ 'Haversine' great circle distance } \description{ The shortest distance between two points (i.e., the 'great-circle-distance' or 'as the crow flies'), according to the 'haversine method'. This method assumes a spherical earth, ignoring ellipsoidal effects. } \usage{distHaversine(p1, p2, r=6378137) } \arguments{ \item{p1}{longitude/latitude of point(s). Can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{p2}{as above; or missing, in which case the sequential distance between the points in p1 is computed} \item{r}{radius of the earth; default = 6378137 m} } \details{ The Haversine ('half-versed-sine') formula was published by R.W. Sinnott in 1984, although it has been known for much longer. At that time computational precision was lower than today (15 digits precision). With current precision, the spherical law of cosines formula appears to give equally good results down to very small distances. If you want greater accuracy, you could use the \code{\link[geosphere]{distVincentyEllipsoid}} method. } \value{ Vector of distances in the same unit as \code{r} (default is meters) } \references{ Sinnott, R.W, 1984. Virtues of the Haversine. Sky and Telescope 68(2): 159 \url{https://www.movable-type.co.uk/scripts/latlong.html} \url{https://en.wikipedia.org/wiki/Great_circle_distance} } \author{ Chris Veness and Robert Hijmans } \seealso{ \code{\link[geosphere]{distGeo}, \link[geosphere]{distCosine}, \link[geosphere]{distVincentySphere}, \link[geosphere]{distVincentyEllipsoid}, \link{distMeeus}} } \examples{ distHaversine(c(0,0),c(90,90)) } \keyword{ spatial } geosphere/man/destPoint.Rd0000644000176200001440000000354214131145600015245 0ustar liggesusers\name{destPoint} \Rdversion{1.1} \alias{destPoint} \title{ Destination given bearing (direction) and distance } \description{ Given a start point, initial bearing (direction), and distance, this function computes the destination point travelling along a the shortest path on an ellipsoid (the geodesic). } \usage{ destPoint(p, b, d, a=6378137, f=1/298.257223563, ...) } \arguments{ \item{p}{Longitude and Latitude of point(s), in degrees. Can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{b}{numeric. Bearing (direction) in degrees} \item{d}{numeric. Distance in meters} \item{a}{major (equatorial) radius of the ellipsoid. The default value is for WGS84 } \item{f}{ellipsoid flattening. The default value is for WGS84 } \item{...}{additional arguments. If an argument 'r' is supplied, this is taken as the radius of the earth (e.g. 6378137 m) and computations are for a sphere (great circle) instead of an ellipsoid (geodetic). This is for backwards compatibility only} } \note{ Direction changes continuously when travelling along a geodesic. Therefore, the final direction is not the same as the initial direction. You can compute the final direction with \code{\link{finalBearing}} (see examples, below) } \value{ A pair of coordinates (longitude/latitude) } \author{ This function calls GeographicLib code by C.F.F. Karney } \references{ C.F.F. Karney, 2013. Algorithms for geodesics, J. Geodesy 87: 43-55. \doi{10.1007/s00190-012-0578-z}. Addenda: \url{https://geographiclib.sourceforge.io/geod-addenda.html}. Also see \url{https://geographiclib.sourceforge.io/} } \examples{ p <- cbind(5,52) d <- destPoint(p,30,10000) d #final direction, when arriving at endpoint: finalBearing(d, p) } \keyword{ spatial } geosphere/man/refEllipsoids.Rd0000644000176200001440000000117513472155746016123 0ustar liggesusers\name{refEllipsoids} \alias{refEllipsoids} \title{Reference ellipsoids} \description{ This function returns a data.frame with parameters \code{a} (semi-major axis) and \code{1/f} (inverse flattening) for a set of reference ellipsoids. } \usage{ refEllipsoids() } \note{ To compute parameter \code{b} you can do } \value{ data.frame } \seealso{ \code{ \link[geosphere]{area}, \link[geosphere]{perimeter} } } \author{Robert J. Hijmans } \examples{ e <- refEllipsoids() e[e$code=='WE', ] #to compute semi-minor axis b: e$b <- e$a - e$a / e$invf } \keyword{methods} \keyword{spatial} geosphere/man/distVincentyEllipsoid.Rd0000644000176200001440000000701614131143521017624 0ustar liggesusers\name{distVincentyEllipsoid} \Rdversion{1.1} \alias{distVincentyEllipsoid} \title{ 'Vincenty' (ellipsoid) great circle distance } \description{ The shortest distance between two points (i.e., the 'great-circle-distance' or 'as the crow flies'), according to the 'Vincenty (ellipsoid)' method. This method uses an ellipsoid and the results are very accurate. The method is computationally more intensive than the other great-circled methods in this package. } \usage{ distVincentyEllipsoid(p1, p2, a=6378137, b=6356752.3142, f=1/298.257223563) } \arguments{ \item{p1}{longitude/latitude of point(s), in degrees 1; can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{p2}{as above; or missing, in which case the sequential distance between the points in p1 is computed} \item{a}{Equatorial axis of ellipsoid} \item{b}{Polar axis of ellipsoid} \item{f}{Inverse flattening of ellipsoid} } \details{ The WGS84 ellipsoid is used by default. It is the best available global ellipsoid, but for some areas other ellipsoids could be preferable, or even necessary if you work with a printed map that refers to that ellipsoid. Here are parameters for some commonly used ellipsoids: \tabular{rllll}{ \tab \code{ ellipsoid } \tab \code{ a } \tab \code{ b } \tab \code{ f } \cr \tab \code{ WGS84 } \tab \code{ 6378137 } \tab \code{ 6356752.3142 } \tab \code{ 1/298.257223563 } \cr \tab \code{ GRS80 } \tab \code{ 6378137 } \tab \code{ 6356752.3141 } \tab \code{ 1/298.257222101 } \cr \tab \code{ GRS67 } \tab \code{ 6378160 } \tab \code{ 6356774.719 } \tab \code{ 1/298.25 } \cr \tab \code{ Airy 1830 } \tab \code{ 6377563.396 } \tab \code{ 6356256.909 } \tab \code{ 1/299.3249646 } \cr \tab \code{ Bessel 1841 } \tab \code{ 6377397.155 } \tab \code{ 6356078.965 } \tab \code{ 1/299.1528434 } \cr \tab \code{ Clarke 1880 } \tab \code{ 6378249.145 } \tab \code{ 6356514.86955 } \tab \code{ 1/293.465 } \cr \tab \code{ Clarke 1866 } \tab \code{ 6378206.4 } \tab \code{ 6356583.8 } \tab \code{ 1/294.9786982 } \cr \tab \code{ International 1924 } \tab \code{ 6378388 } \tab \code{ 6356911.946 } \tab \code{ 1/297 } \cr \tab \code{ Krasovsky 1940 } \tab \code{ 6378245 } \tab \code{ 6356863 } \tab \code{ 1/298.2997381 } \cr } \code{a} is the 'semi-major axis', and \code{b} is the 'semi-minor axis' of the ellipsoid. \code{f} is the flattening. Note that \code{f = (a-b)/a} more info: \url{https://en.wikipedia.org/wiki/Reference_ellipsoid} } \value{ Distance value in the same units as the ellipsoid (default is meters) } \references{ Vincenty, T. 1975. Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations. Survey Review Vol. 23, No. 176, pp88-93. Available here: \url{https://www.movable-type.co.uk/scripts/latlong-vincenty.html} \url{https://en.wikipedia.org/wiki/Great_circle_distance} } \author{ Chris Veness and Robert Hijmans } \seealso{ \code{\link[geosphere]{distGeo}, \link{distVincentySphere}, \link{distHaversine}, \link{distCosine}, \link{distMeeus}} } \examples{ distVincentyEllipsoid(c(0,0),c(90,90)) # on a 'Clarke 1880' ellipsoid distVincentyEllipsoid(c(0,0),c(90,90), a=6378249.145, b=6356514.86955, f=1/293.465) } \keyword{ spatial } geosphere/man/distMeeus.Rd0000644000176200001440000000546614131143521015245 0ustar liggesusers\name{distMeeus} \Rdversion{1.1} \alias{distMeeus} \title{ 'Meeus' great circle distance } \description{ The shortest distance between two points on an ellipsoid (the 'geodetic'), according to the 'Meeus' method. \code{\link{distGeo}} should be more accurate. } \usage{ distMeeus(p1, p2, a=6378137, f=1/298.257223563) } \arguments{ \item{p1}{longitude/latitude of point(s), in degrees 1; can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{p2}{as above; or missing, in which case the sequential distance between the points in p1 is computed} \item{a}{numeric. Major (equatorial) radius of the ellipsoid. The default value is for WGS84 } \item{f}{numeric. Ellipsoid flattening. The default value is for WGS84 } } \details{ Parameters from the WGS84 ellipsoid are used by default. It is the best available global ellipsoid, but for some areas other ellipsoids could be preferable, or even necessary if you work with a printed map that refers to that ellipsoid. Here are parameters for some commonly used ellipsoids: \tabular{rlll}{ \tab \code{ ellipsoid } \tab \code{ a } \tab \code{ f } \cr \tab \code{ WGS84 } \tab \code{ 6378137 } \tab \code{ 1/298.257223563 } \cr \tab \code{ GRS80 } \tab \code{ 6378137 } \tab \code{ 1/298.257222101 } \cr \tab \code{ GRS67 } \tab \code{ 6378160 } \tab \code{ 1/298.25 } \cr \tab \code{ Airy 1830 } \tab \code{ 6377563.396 } \tab \code{ 1/299.3249646 } \cr \tab \code{ Bessel 1841 } \tab \code{ 6377397.155 } \tab \code{ 1/299.1528434 } \cr \tab \code{ Clarke 1880 } \tab \code{ 6378249.145 } \tab \code{ 1/293.465 } \cr \tab \code{ Clarke 1866 } \tab \code{ 6378206.4 } \tab \code{ 1/294.9786982 } \cr \tab \code{ International 1924 } \tab \code{ 6378388 } \tab \code{ 1/297 } \cr \tab \code{ Krasovsky 1940 } \tab \code{ 6378245 } \tab \code{ 1/298.2997381 } \cr } more info: \url{https://en.wikipedia.org/wiki/Reference_ellipsoid} } \value{ Distance value in the same units as parameter \code{a} of the ellipsoid (default is meters) } \note{ This algorithm is also used in the \code{spDists} function in the sp package } \references{ Meeus, J., 1999 (2nd edition). Astronomical algoritms. Willman-Bell, 477p. } \author{ Robert Hijmans, based on a script by Stephen R. Schmitt } \seealso{ \code{\link[geosphere]{distGeo}, \link{distVincentyEllipsoid}, \link{distVincentySphere}, \link{distHaversine}, \link{distCosine}} } \examples{ distMeeus(c(0,0),c(90,90)) # on a 'Clarke 1880' ellipsoid distMeeus(c(0,0),c(90,90), a=6378249.145, f=1/293.465) } \keyword{ spatial } geosphere/man/antipode.Rd0000644000176200001440000000222714131143521015076 0ustar liggesusers\name{antipode} \Rdversion{1.1} \alias{antipode} \alias{antipodal} \title{ Antipodes } \description{ Compute an antipode, or check whether two points are antipodes. Antipodes are places on Earth that are diametrically opposite to one another; and could be connected by a straight line through the centre of the Earth. Antipodal points are connected by an infinite number of great circles (e.g. the meridians connecting the poles), and can therefore not be used in some great circle based computations. } \usage{ antipode(p) antipodal(p1, p2, tol=1e-9) } \arguments{ \item{p}{Longitude/latitude of a single point, in degrees; can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{p1}{as above} \item{p2}{as above} \item{tol}{tolerance for equality} } \value{ antipodal points or a logical value (\code{TRUE} if antipodal) } \references{ \url{https://en.wikipedia.org/wiki/Antipodes} } \author{ Robert Hijmans } \examples{ antipode(rbind(c(5,52), c(-120,37), c(-60,0), c(0,70))) antipodal(c(0,0), c(180,0)) } \keyword{ spatial } geosphere/man/greatCircle.Rd0000644000176200001440000000163114131210620015510 0ustar liggesusers\name{greatCircle} \Rdversion{1.1} \alias{greatCircle} \title{ Great circle } \description{ Get points on a great circle as defined by the shortest distance between two specified points } \usage{ greatCircle(p1, p2, n=360, sp=FALSE) } \arguments{ \item{p1}{longitude/latitude of point(s). Can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{p2}{as above} \item{n}{The requested number of points on the Great Circle} \item{sp}{Logical. Return a SpatialLines object?} } \value{ A matrix of points, or a list of such matrices (e.g., if multiple bearings are supplied) } \references{ \url{https://www.edwilliams.org/avform147.htm#Int} } \author{ Robert Hijmans, based on a formula provided by Ed Williams } \examples{ greatCircle(c(5,52), c(-120,37), n=36) } \keyword{ spatial } geosphere/man/data.Rd0000644000176200001440000000053613472155746014230 0ustar liggesusers\name{wrld} \alias{wrld} \alias{merc} \docType{data} \title{World countries} \description{ world coastline and country outlines in longitude/latitude (wrld) and in Mercator projection (merc). } \usage{ data(wrld) data(merc) } \source{ Derived from the wrld_simpl data set in package maptools } \keyword{datasets} geosphere/man/geomean.Rd0000644000176200001440000000141313472155746014725 0ustar liggesusers\name{geomean} \docType{methods} \alias{geomean} \title{ Mean location of sperhical coordinates } \description{ mean location for spherical (longitude/latitude) coordinates that deals with the angularity. I.e., the mean of longitudes -179 and 178 is 179.5 } \usage{ geomean(xy, w) } \arguments{ \item{xy}{matrix with two columns (longitude/latitude), or a SpatialPoints or SpatialPolygons object with a longitude/latitude CRS} \item{w}{weights (vector of numeric values, with a length that is equal to the number of spatial features in \code{x}} } \value{ Ccoordinate pair (numeric) } \examples{ xy <- cbind(x=c(-179,179, 177), y=c(12,14,16)) xy geomean(xy) } \author{Robert J. Hijmans} \keyword{methods} \keyword{spatial} geosphere/man/daylength.Rd0000644000176200001440000000172114253023226015254 0ustar liggesusers\name{daylength} \alias{daylength} \title{ Daylength } \description{ Compute daylength (photoperiod) for a latitude and date. } \usage{ daylength(lat, doy) } \arguments{ \item{lat}{latitude, in degrees. I.e. between -90.0 and 90.0 } \item{doy}{integer, day of the year (1..365) for common (non-leap) years; or an object of class Date; or a character that can be coerced into a date, using 'yyyy-mm-dd' format, e.g. '1982-11-23' } } \value{ Daylength in hours } \references{ Forsythe, William C., Edward J. Rykiel Jr., Randal S. Stahl, Hsin-i Wu and Robert M. Schoolfield, 1995. A model comparison for daylength as a function of latitude and day of the year. Ecological Modeling 80:87-95. } \author{ Robert J. Hijmans } \examples{ daylength(-25, '2010-10-10') daylength(45, 1:365) # average monthly daylength dl <- daylength(45, 1:365) tapply(dl, rep(1:12, c(31,28,31,30,31,30,31,31,30,31,30,31)), mean) } \keyword{ spatial } geosphere/man/geodesic.Rd0000644000176200001440000000663114131145601015061 0ustar liggesusers\name{geodesic} \Rdversion{1.1} \alias{geodesic} \alias{geodesic_inverse} \title{ geodesic and inverse geodesic problem } \description{ Highly accurate estimate of the 'geodesic problem' (find location and azimuth at arrival when departing from a location, given an direction (azimuth) at departure and distance) and the 'inverse geodesic problem' (find the distance between two points and the azimuth of departure and arrival for the shortest path. Computations are for an ellipsoid (default is WGS84 ellipsoid). This is a direct implementation of the the GeographicLib code by C.F.F. Karney that is also used in several other functions in this package (for example, in \code{\link{distGeo}} and \code{\link{areaPolygon}}). } \usage{ geodesic(p, azi, d, a=6378137, f=1/298.257223563, ...) geodesic_inverse(p1, p2, a=6378137, f=1/298.257223563, ...) } \arguments{ \item{p}{longitude/latitude of point(s). Can be a vector of two numbers, a matrix of 2 columns (first column is longitude, second column is latitude) or a SpatialPoints* object} \item{p1}{as above} \item{p2}{as above} \item{azi}{numeric. Azimuth of departure in degrees} \item{d}{numeric. Distance in meters} \item{a}{numeric. Major (equatorial) radius of the ellipsoid. The default value is for WGS84 } \item{f}{numeric. Ellipsoid flattening. The default value is for WGS84 } \item{...}{additional arguments (none implemented)} } \value{ Three column matrix with columns 'longitude', 'latitude', 'azimuth' (geodesic); or 'distance' (in meters), 'azimuth1' (of departure), 'azimuth2' (of arrival) (geodesic_inverse) } \details{ Parameters from the WGS84 ellipsoid are used by default. It is the best available global ellipsoid, but for some areas other ellipsoids could be preferable, or even necessary if you work with a printed map that refers to that ellipsoid. Here are parameters for some commonly used ellipsoids. \tabular{rlll}{ \tab \code{ ellipsoid } \tab \code{ a } \tab \code{ f } \cr \tab \code{ WGS84 } \tab \code{ 6378137 } \tab \code{ 1/298.257223563 } \cr \tab \code{ GRS80 } \tab \code{ 6378137 } \tab \code{ 1/298.257222101 } \cr \tab \code{ GRS67 } \tab \code{ 6378160 } \tab \code{ 1/298.25 } \cr \tab \code{ Airy 1830 } \tab \code{ 6377563.396 } \tab \code{ 1/299.3249646 } \cr \tab \code{ Bessel 1841 } \tab \code{ 6377397.155 } \tab \code{ 1/299.1528434 } \cr \tab \code{ Clarke 1880 } \tab \code{ 6378249.145 } \tab \code{ 1/293.465 } \cr \tab \code{ Clarke 1866 } \tab \code{ 6378206.4 } \tab \code{ 1/294.9786982 } \cr \tab \code{ International 1924 } \tab \code{ 6378388 } \tab \code{ 1/297 } \cr \tab \code{ Krasovsky 1940 } \tab \code{ 6378245 } \tab \code{ 1/298.2997381 } \cr } more info: \url{https://en.wikipedia.org/wiki/Reference_ellipsoid} } \author{ This function calls GeographicLib code by C.F.F. Karney } \references{ C.F.F. Karney, 2013. Algorithms for geodesics, J. Geodesy 87: 43-55. \doi{10.1007/s00190-012-0578-z}. Addenda: \url{https://geographiclib.sourceforge.io/geod-addenda.html}. Also see \url{https://geographiclib.sourceforge.io/} } \seealso{ \code{\link{distGeo}} } \examples{ geodesic(cbind(0,0), 30, 1000000) geodesic_inverse(cbind(0,0), cbind(90,90)) } \keyword{ spatial } geosphere/man/mercator.Rd0000644000176200001440000000151213472155746015126 0ustar liggesusers\name{mercator} \Rdversion{1.1} \alias{mercator} \title{ Mercator projection } \description{ Transform longitude/latiude points to the Mercator projection. The main purpose of this function is to compute centroids, and to illustrate rhumb lines in the vignette. } \usage{ mercator(p, inverse=FALSE, r=6378137) } \arguments{ \item{p}{longitude/latitude of point(s). Can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{inverse}{Logical. If \code{TRUE}, do the inverse projection (from Mercator to longitude/latitude} \item{r}{Numeric. Radius of the earth; default = 6378137 m} } \value{ matrix } \author{ Robert Hijmans } \examples{ a = mercator(c(5,52)) a mercator(a, inverse=TRUE) } \keyword{ spatial } geosphere/man/intermediate.Rd0000644000176200001440000000263214131210620015740 0ustar liggesusers\name{intermediate} \Rdversion{1.1} \alias{gcIntermediate} \title{ Intermediate points on a great circle (sphere) } \description{ Get intermediate points (way points) between the two locations with longitude/latitude coordinates. gcIntermediate is based on a spherical model of the earth and internally uses \code{\link{distCosine}}. } \usage{ gcIntermediate(p1, p2, n=50, breakAtDateLine=FALSE, addStartEnd=FALSE, sp=FALSE, sepNA) } \arguments{ \item{p1}{longitude/latitude of a single point, in degrees. This can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{p2}{as for \code{p1}} \item{n}{integer. The desired number of intermediate points} \item{breakAtDateLine}{logical. Return two matrices if the dateline is crossed?} \item{addStartEnd}{logical. Add p1 and p2 to the result?} \item{sp}{logical. Return a SpatialLines object?} \item{sepNA}{logical. Rather than as a list, return the values as a two column matrix with lines seperated by a row of NA values? (for use in 'plot')} } \value{ matrix or list with intermediate points } \references{ \url{https://www.edwilliams.org/avform147.htm#Intermediate} } \author{ Robert Hijmans based on code by Ed Williams (great circle) } \examples{ gcIntermediate(c(5,52), c(-120,37), n=6, addStartEnd=TRUE) } \keyword{ spatial } geosphere/man/randomCoordinates.Rd0000644000176200001440000000211314131210415016735 0ustar liggesusers\name{randomCoordinates} \Rdversion{1.1} \alias{randomCoordinates} \alias{regularCoordinates} \title{ Random or regularly distributed coordinates on the globe } \description{ randomCoordinates returns a 'uniform random sample' in the sense that the probability that a point is drawn from any region is equal to the area of that region divided by the area of the entire sphere. This would not happen if you took a random uniform sample of longitude and latitude, as the sample would be biased towards the poles. regularCoordiaates returns a set of coordinates that are regularly distributed on the globe. } \usage{ randomCoordinates(n) regularCoordinates(N) } \arguments{ \item{n}{Sample size (number of points (coordinate pairs)) } \item{N}{Number of 'parts' in which the earth is subdived ) } } \value{ Matrix of lon/lat coordiantes } \author{ Robert Hijmans, based on code by Nils Haeck (regularCoordinates) and on suggstions by Michael Orion (randomCoordinates) } \examples{ randomCoordinates(3) regularCoordinates(1) } \keyword{ spatial } geosphere/man/span.Rd0000644000176200001440000000322513472155746014256 0ustar liggesusers\name{span} \alias{span} \alias{span,SpatialPolygons-method} \alias{span,matrix-method} \title{Span of polygons} \description{ Compute the approximate surface span of polygons in longitude and latitude direction. Span is computed by rasterizing the polygons; and precision increases with the number of 'scan lines'. You can either use a fixed number of scan lines for each polygon, or a fixed band-width. } \usage{ span(x, ...) } \arguments{ \item{x}{a SpatialPolygons* object or a 2-column matrix (longitude/latitude)} \item{...}{Additional arguments, see Details} } \details{ The following additional arguments can be passed, to replace default values for this function \tabular{rll}{ \tab \code{nbands} \tab Character. Method to determine the number of bands to 'scan' the polygon. Either 'fixed' or 'variable' \cr \tab \code{n} \tab Integer >= 1. If \code{nbands='fixed'}, how many bands should be used \cr \tab \code{res} \tab Numeric. If \code{nbands='variable'}, what should the bandwidth be (in degrees)? \cr \tab \code{fun} \tab Logical. A function such as mean or min. Mean computes the average span \cr \tab \code{...} \tab further additional arguments passed to distGeo\cr } } \value{ A list, or a matrix if a function \code{fun} is specified. Values are in the units of \code{r} (default is meter) } \author{Robert J. Hijmans } \examples{ pol <- rbind(c(-180,-20), c(-160,5), c(-60, 0), c(-160,-60), c(-180,-20)) plot(pol) lines(pol) # lon and lat span in m span(pol, fun=max) x <- span(pol) max(x$latspan) mean(x$latspan) plot(x$longitude, x$lonspan) } \keyword{methods} \keyword{spatial} geosphere/man/centroid.Rd0000644000176200001440000000255213472325401015111 0ustar liggesusers\name{centroid} \alias{centroid} \alias{centroid,matrix-method} \alias{centroid,data.frame-method} \alias{centroid,SpatialPolygons-method} \title{Centroid of spherical polygons} \description{ Compute the centroid of longitude/latitude polygons. Unlike other functions in this package, there is no spherical trigonometry involved in the implementation of this function. Instead, the function projects the polygon to the (conformal) Mercator coordinate reference system, computes the centroid, and then inversely projects it to longitude and latitude. This approach fails for polygons that include one of the poles (and is rather biased for anything close to the poles). The function should work for polygons that cross the -180/180 meridian (date line). } \usage{ centroid(x, ...) } \arguments{ \item{x}{SpatialPolygons* object, or a 2-column matrix or data.frame reprenting a single polgyon (longitude/latitude)} \item{...}{Additional arguments. None implemented} } \note{ For multi-part polygons, the centroid of the largest part is returned. } \value{ A matrix (longitude/latitude) } \seealso{ \code{ \link[geosphere]{area}, \link[geosphere]{perimeter} } } \author{Robert J. Hijmans } \examples{ pol <- rbind(c(-180,-20), c(-160,5), c(-60, 0), c(-160,-60), c(-180,-20)) centroid(pol) } \keyword{methods} \keyword{spatial} geosphere/man/midPoint.Rd0000644000176200001440000000137013472155746015077 0ustar liggesusers\name{midPoint} \Rdversion{1.1} \alias{midPoint} \title{Mid-point} \description{ Find the point half-way between two points along an ellipsoid } \usage{ midPoint(p1, p2, a=6378137, f = 1/298.257223563) } \arguments{ \item{p1}{longitude/latitude of point(s). Can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{p2}{As above} \item{a}{major (equatorial) radius of the ellipsoid} \item{f}{ellipsoid flattening. The default value is for WGS84 } } \value{ matrix with coordinate pairs } \author{ Elias Pipping and Robert Hijmans } \examples{ midPoint(c(0,0),c(90,90)) midPoint(c(0,0),c(90,90), f=0) } \keyword{spatial} geosphere/man/alongTrackDistance.Rd0000644000176200001440000000171413472155746017056 0ustar liggesusers\name{alongTrackDistance} \Rdversion{1.1} \alias{alongTrackDistance} \title{ Along Track Distance } \description{ The "along track distance" is the distance from the start point (p1) to the closest point on the path to a third point (p3), following a great circle path defined by points p1 and p2. See \code{\link{dist2gc}} for the "cross track distance" } \usage{ alongTrackDistance(p1, p2, p3, r=6378137) } \arguments{ \item{p1}{longitude/latitude of point(s). Can be a vector of two numbers, a matrix of 2 columns (first one is longitude, second is latitude) or a SpatialPoints* object} \item{p2}{as above} \item{p3}{as above} \item{r}{radius of the earth; default = 6378137m} } \value{ A distance in units of r (default is meters) } \author{ Ed Williams and Robert Hijmans } \seealso{ \code{ \link[geosphere]{dist2gc} } } \examples{ alongTrackDistance(c(0,0),c(60,60),c(50,40)) } \keyword{ spatial } geosphere/DESCRIPTION0000644000176200001440000000172214335012510013735 0ustar liggesusersPackage: geosphere Type: Package Title: Spherical Trigonometry Version: 1.5-18 Date: 2022-11-13 LinkingTo: Rcpp Imports: Rcpp, sp Depends: R (>= 3.0.0) Suggests: methods, raster Authors@R: c( person("Robert J.", "Hijmans", role = c("cre", "aut"), email = "r.hijmans@gmail.com"), person("Charles", "Karney", role = "ctb", comment="GeographicLib"), person("Ed", "Williams", role = "ctb"), person("Chris", "Vennes", role = "ctb")) Description: Spherical trigonometry for geographic applications. That is, compute distances and related measures for angular (longitude/latitude) locations. BugReports: https://github.com/rspatial/geosphere/issues/ License: GPL (>= 3) LazyLoad: yes NeedsCompilation: yes Packaged: 2022-11-13 17:35:52 UTC; rhijm Author: Robert J. Hijmans [cre, aut], Charles Karney [ctb] (GeographicLib), Ed Williams [ctb], Chris Vennes [ctb] Maintainer: Robert J. Hijmans Repository: CRAN Date/Publication: 2022-11-15 22:40:08 UTC geosphere/build/0000755000176200001440000000000014334225370013335 5ustar liggesusersgeosphere/build/vignette.rds0000644000176200001440000000033414334225370015674 0ustar liggesusersuQ0 Аx) b 3ěO.Skf!RBmLm}nxz)Ȳ@ApQ\oV2E:it'GgE`ߥ͡D+D and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ **********************************************************************/ #include #include "Rhumb.h" #if defined(_MSC_VER) // Squelch warnings about enum-float expressions # pragma warning (disable: 5055) #endif namespace GeographicLib { using namespace std; Rhumb::Rhumb(real a, real f, bool exact) : _ell(a, f) , _exact(exact) , _c2(_ell.Area() / (2 * Math::td)) { // Generated by Maxima on 2015-05-15 08:24:04-04:00 #if GEOGRAPHICLIB_RHUMBAREA_ORDER == 4 static const real coeff[] = { // R[0]/n^0, polynomial in n of order 4 691, 7860, -20160, 18900, 0, 56700, // R[1]/n^1, polynomial in n of order 3 1772, -5340, 6930, -4725, 14175, // R[2]/n^2, polynomial in n of order 2 -1747, 1590, -630, 4725, // R[3]/n^3, polynomial in n of order 1 104, -31, 315, // R[4]/n^4, polynomial in n of order 0 -41, 420, }; // count = 20 #elif GEOGRAPHICLIB_RHUMBAREA_ORDER == 5 static const real coeff[] = { // R[0]/n^0, polynomial in n of order 5 -79036, 22803, 259380, -665280, 623700, 0, 1871100, // R[1]/n^1, polynomial in n of order 4 41662, 58476, -176220, 228690, -155925, 467775, // R[2]/n^2, polynomial in n of order 3 18118, -57651, 52470, -20790, 155925, // R[3]/n^3, polynomial in n of order 2 -23011, 17160, -5115, 51975, // R[4]/n^4, polynomial in n of order 1 5480, -1353, 13860, // R[5]/n^5, polynomial in n of order 0 -668, 5775, }; // count = 27 #elif GEOGRAPHICLIB_RHUMBAREA_ORDER == 6 static const real coeff[] = { // R[0]/n^0, polynomial in n of order 6 128346268, -107884140, 31126095, 354053700, -908107200, 851350500, 0, 2554051500LL, // R[1]/n^1, polynomial in n of order 5 -114456994, 56868630, 79819740, -240540300, 312161850, -212837625, 638512875, // R[2]/n^2, polynomial in n of order 4 51304574, 24731070, -78693615, 71621550, -28378350, 212837625, // R[3]/n^3, polynomial in n of order 3 1554472, -6282003, 4684680, -1396395, 14189175, // R[4]/n^4, polynomial in n of order 2 -4913956, 3205800, -791505, 8108100, // R[5]/n^5, polynomial in n of order 1 1092376, -234468, 2027025, // R[6]/n^6, polynomial in n of order 0 -313076, 2027025, }; // count = 35 #elif GEOGRAPHICLIB_RHUMBAREA_ORDER == 7 static const real coeff[] = { // R[0]/n^0, polynomial in n of order 7 -317195588, 385038804, -323652420, 93378285, 1062161100, -2724321600LL, 2554051500LL, 0, 7662154500LL, // R[1]/n^1, polynomial in n of order 6 258618446, -343370982, 170605890, 239459220, -721620900, 936485550, -638512875, 1915538625, // R[2]/n^2, polynomial in n of order 5 -248174686, 153913722, 74193210, -236080845, 214864650, -85135050, 638512875, // R[3]/n^3, polynomial in n of order 4 114450437, 23317080, -94230045, 70270200, -20945925, 212837625, // R[4]/n^4, polynomial in n of order 3 15445736, -103193076, 67321800, -16621605, 170270100, // R[5]/n^5, polynomial in n of order 2 -27766753, 16385640, -3517020, 30405375, // R[6]/n^6, polynomial in n of order 1 4892722, -939228, 6081075, // R[7]/n^7, polynomial in n of order 0 -3189007, 14189175, }; // count = 44 #elif GEOGRAPHICLIB_RHUMBAREA_ORDER == 8 static const real coeff[] = { // R[0]/n^0, polynomial in n of order 8 71374704821LL, -161769749880LL, 196369790040LL, -165062734200LL, 47622925350LL, 541702161000LL, -1389404016000LL, 1302566265000LL, 0, 3907698795000LL, // R[1]/n^1, polynomial in n of order 7 -13691187484LL, 65947703730LL, -87559600410LL, 43504501950LL, 61062101100LL, -184013329500LL, 238803815250LL, -162820783125LL, 488462349375LL, // R[2]/n^2, polynomial in n of order 6 30802104839LL, -63284544930LL, 39247999110LL, 18919268550LL, -60200615475LL, 54790485750LL, -21709437750LL, 162820783125LL, // R[3]/n^3, polynomial in n of order 5 -8934064508LL, 5836972287LL, 1189171080, -4805732295LL, 3583780200LL, -1068242175, 10854718875LL, // R[4]/n^4, polynomial in n of order 4 50072287748LL, 3938662680LL, -26314234380LL, 17167059000LL, -4238509275LL, 43418875500LL, // R[5]/n^5, polynomial in n of order 3 359094172, -9912730821LL, 5849673480LL, -1255576140, 10854718875LL, // R[6]/n^6, polynomial in n of order 2 -16053944387LL, 8733508770LL, -1676521980, 10854718875LL, // R[7]/n^7, polynomial in n of order 1 930092876, -162639357, 723647925, // R[8]/n^8, polynomial in n of order 0 -673429061, 1929727800, }; // count = 54 #else #error "Bad value for GEOGRAPHICLIB_RHUMBAREA_ORDER" #endif static_assert(sizeof(coeff) / sizeof(real) == ((maxpow_ + 1) * (maxpow_ + 4))/2, "Coefficient array size mismatch for Rhumb"); real d = 1; int o = 0; for (int l = 0; l <= maxpow_; ++l) { int m = maxpow_ - l; // R[0] is just an integration constant so it cancels when evaluating a // definite integral. So don't bother computing it. It won't be used // when invoking SinCosSeries. if (l) _rR[l] = d * Math::polyval(m, coeff + o, _ell._n) / coeff[o + m + 1]; o += m + 2; d *= _ell._n; } // Post condition: o == sizeof(alpcoeff) / sizeof(real) } const Rhumb& Rhumb::WGS84() { static const Rhumb wgs84(Constants::WGS84_a(), Constants::WGS84_f(), false); return wgs84; } void Rhumb::GenInverse(real lat1, real lon1, real lat2, real lon2, unsigned outmask, real& s12, real& azi12, real& S12) const { real lon12 = Math::AngDiff(lon1, lon2), psi1 = _ell.IsometricLatitude(lat1), psi2 = _ell.IsometricLatitude(lat2), psi12 = psi2 - psi1, h = hypot(lon12, psi12); if (outmask & AZIMUTH) azi12 = Math::atan2d(lon12, psi12); if (outmask & DISTANCE) { real dmudpsi = DIsometricToRectifying(psi2, psi1); s12 = h * dmudpsi * _ell.QuarterMeridian() / Math::qd; } if (outmask & AREA) S12 = _c2 * lon12 * MeanSinXi(psi2 * Math::degree(), psi1 * Math::degree()); } RhumbLine Rhumb::Line(real lat1, real lon1, real azi12) const { return RhumbLine(*this, lat1, lon1, azi12); } void Rhumb::GenDirect(real lat1, real lon1, real azi12, real s12, unsigned outmask, real& lat2, real& lon2, real& S12) const { Line(lat1, lon1, azi12).GenPosition(s12, outmask, lat2, lon2, S12); } Math::real Rhumb::DE(real x, real y) const { const EllipticFunction& ei = _ell._ell; real d = x - y; if (x * y <= 0) return d != 0 ? (ei.E(x) - ei.E(y)) / d : 1; // See DLMF: Eqs (19.11.2) and (19.11.4) letting // theta -> x, phi -> -y, psi -> z // // (E(x) - E(y)) / d = E(z)/d - k2 * sin(x) * sin(y) * sin(z)/d // // tan(z/2) = (sin(x)*Delta(y) - sin(y)*Delta(x)) / (cos(x) + cos(y)) // = d * Dsin(x,y) * (sin(x) + sin(y))/(cos(x) + cos(y)) / // (sin(x)*Delta(y) + sin(y)*Delta(x)) // = t = d * Dt // sin(z) = 2*t/(1+t^2); cos(z) = (1-t^2)/(1+t^2) // Alt (this only works for |z| <= pi/2 -- however, this conditions holds // if x*y > 0): // sin(z) = d * Dsin(x,y) * (sin(x) + sin(y))/ // (sin(x)*cos(y)*Delta(y) + sin(y)*cos(x)*Delta(x)) // cos(z) = sqrt((1-sin(z))*(1+sin(z))) real sx = sin(x), sy = sin(y), cx = cos(x), cy = cos(y); real Dt = Dsin(x, y) * (sx + sy) / ((cx + cy) * (sx * ei.Delta(sy, cy) + sy * ei.Delta(sx, cx))), t = d * Dt, Dsz = 2 * Dt / (1 + t*t), sz = d * Dsz, cz = (1 - t) * (1 + t) / (1 + t*t); return ((sz != 0 ? ei.E(sz, cz, ei.Delta(sz, cz)) / sz : 1) - ei.k2() * sx * sy) * Dsz; } Math::real Rhumb::DRectifying(real latx, real laty) const { real tbetx = _ell._f1 * Math::tand(latx), tbety = _ell._f1 * Math::tand(laty); return (Math::pi()/2) * _ell._b * _ell._f1 * DE(atan(tbetx), atan(tbety)) * Dtan(latx, laty) * Datan(tbetx, tbety) / _ell.QuarterMeridian(); } Math::real Rhumb::DIsometric(real latx, real laty) const { real phix = latx * Math::degree(), tx = Math::tand(latx), phiy = laty * Math::degree(), ty = Math::tand(laty); return Dasinh(tx, ty) * Dtan(latx, laty) - Deatanhe(sin(phix), sin(phiy)) * Dsin(phix, phiy); } Math::real Rhumb::SinCosSeries(bool sinp, real x, real y, const real c[], int n) { // N.B. n >= 0 and c[] has n+1 elements 0..n, of which c[0] is ignored. // // Use Clenshaw summation to evaluate // m = (g(x) + g(y)) / 2 -- mean value // s = (g(x) - g(y)) / (x - y) -- average slope // where // g(x) = sum(c[j]*SC(2*j*x), j = 1..n) // SC = sinp ? sin : cos // CS = sinp ? cos : sin // // This function returns only s; m is discarded. // // Write // t = [m; s] // t = sum(c[j] * f[j](x,y), j = 1..n) // where // f[j](x,y) = [ (SC(2*j*x)+SC(2*j*y))/2 ] // [ (SC(2*j*x)-SC(2*j*y))/d ] // // = [ cos(j*d)*SC(j*p) ] // [ +/-(2/d)*sin(j*d)*CS(j*p) ] // (+/- = sinp ? + : -) and // p = x+y, d = x-y // // f[j+1](x,y) = A * f[j](x,y) - f[j-1](x,y) // // A = [ 2*cos(p)*cos(d) -sin(p)*sin(d)*d] // [ -4*sin(p)*sin(d)/d 2*cos(p)*cos(d) ] // // Let b[n+1] = b[n+2] = [0 0; 0 0] // b[j] = A * b[j+1] - b[j+2] + c[j] * I for j = n..1 // t = (c[0] * I - b[2]) * f[0](x,y) + b[1] * f[1](x,y) // c[0] is not accessed for s = t[2] real p = x + y, d = x - y, cp = cos(p), cd = cos(d), sp = sin(p), sd = d != 0 ? sin(d)/d : 1, m = 2 * cp * cd, s = sp * sd; // 2x2 matrices stored in row-major order const real a[4] = {m, -s * d * d, -4 * s, m}; real ba[4] = {0, 0, 0, 0}; real bb[4] = {0, 0, 0, 0}; real* b1 = ba; real* b2 = bb; if (n > 0) b1[0] = b1[3] = c[n]; for (int j = n - 1; j > 0; --j) { // j = n-1 .. 1 swap(b1, b2); // b1 = A * b2 - b1 + c[j] * I b1[0] = a[0] * b2[0] + a[1] * b2[2] - b1[0] + c[j]; b1[1] = a[0] * b2[1] + a[1] * b2[3] - b1[1]; b1[2] = a[2] * b2[0] + a[3] * b2[2] - b1[2]; b1[3] = a[2] * b2[1] + a[3] * b2[3] - b1[3] + c[j]; } // Here are the full expressions for m and s // m = (c[0] - b2[0]) * f01 - b2[1] * f02 + b1[0] * f11 + b1[1] * f12; // s = - b2[2] * f01 + (c[0] - b2[3]) * f02 + b1[2] * f11 + b1[3] * f12; if (sinp) { // real f01 = 0, f02 = 0; real f11 = cd * sp, f12 = 2 * sd * cp; // m = b1[0] * f11 + b1[1] * f12; s = b1[2] * f11 + b1[3] * f12; } else { // real f01 = 1, f02 = 0; real f11 = cd * cp, f12 = - 2 * sd * sp; // m = c[0] - b2[0] + b1[0] * f11 + b1[1] * f12; s = - b2[2] + b1[2] * f11 + b1[3] * f12; } return s; } Math::real Rhumb::DConformalToRectifying(real chix, real chiy) const { return 1 + SinCosSeries(true, chix, chiy, _ell.ConformalToRectifyingCoeffs(), tm_maxord); } Math::real Rhumb::DRectifyingToConformal(real mux, real muy) const { return 1 - SinCosSeries(true, mux, muy, _ell.RectifyingToConformalCoeffs(), tm_maxord); } Math::real Rhumb::DIsometricToRectifying(real psix, real psiy) const { if (_exact) { real latx = _ell.InverseIsometricLatitude(psix), laty = _ell.InverseIsometricLatitude(psiy); return DRectifying(latx, laty) / DIsometric(latx, laty); } else { psix *= Math::degree(); psiy *= Math::degree(); return DConformalToRectifying(gd(psix), gd(psiy)) * Dgd(psix, psiy); } } Math::real Rhumb::DRectifyingToIsometric(real mux, real muy) const { real latx = _ell.InverseRectifyingLatitude(mux/Math::degree()), laty = _ell.InverseRectifyingLatitude(muy/Math::degree()); return _exact ? DIsometric(latx, laty) / DRectifying(latx, laty) : Dgdinv(Math::taupf(Math::tand(latx), _ell._es), Math::taupf(Math::tand(laty), _ell._es)) * DRectifyingToConformal(mux, muy); } Math::real Rhumb::MeanSinXi(real psix, real psiy) const { return Dlog(cosh(psix), cosh(psiy)) * Dcosh(psix, psiy) + SinCosSeries(false, gd(psix), gd(psiy), _rR, maxpow_) * Dgd(psix, psiy); } RhumbLine::RhumbLine(const Rhumb& rh, real lat1, real lon1, real azi12) : _rh(rh) , _lat1(Math::LatFix(lat1)) , _lon1(lon1) , _azi12(Math::AngNormalize(azi12)) { real alp12 = _azi12 * Math::degree(); _salp = _azi12 == -Math::hd ? 0 : sin(alp12); _calp = fabs(_azi12) == Math::qd ? 0 : cos(alp12); _mu1 = _rh._ell.RectifyingLatitude(lat1); _psi1 = _rh._ell.IsometricLatitude(lat1); _r1 = _rh._ell.CircleRadius(lat1); } void RhumbLine::GenPosition(real s12, unsigned outmask, real& lat2, real& lon2, real& S12) const { real mu12 = s12 * _calp * Math::qd / _rh._ell.QuarterMeridian(), mu2 = _mu1 + mu12; real psi2, lat2x, lon2x; if (fabs(mu2) <= Math::qd) { if (_calp != 0) { lat2x = _rh._ell.InverseRectifyingLatitude(mu2); real psi12 = _rh.DRectifyingToIsometric( mu2 * Math::degree(), _mu1 * Math::degree()) * mu12; lon2x = _salp * psi12 / _calp; psi2 = _psi1 + psi12; } else { lat2x = _lat1; lon2x = _salp * s12 / (_r1 * Math::degree()); psi2 = _psi1; } if (outmask & AREA) S12 = _rh._c2 * lon2x * _rh.MeanSinXi(_psi1 * Math::degree(), psi2 * Math::degree()); lon2x = outmask & LONG_UNROLL ? _lon1 + lon2x : Math::AngNormalize(Math::AngNormalize(_lon1) + lon2x); } else { // Reduce to the interval [-180, 180) mu2 = Math::AngNormalize(mu2); // Deal with points on the anti-meridian if (fabs(mu2) > Math::qd) mu2 = Math::AngNormalize(Math::hd - mu2); lat2x = _rh._ell.InverseRectifyingLatitude(mu2); lon2x = Math::NaN(); if (outmask & AREA) S12 = Math::NaN(); } if (outmask & LATITUDE) lat2 = lat2x; if (outmask & LONGITUDE) lon2 = lon2x; } } // namespace GeographicLib geosphere/src/DST.h0000644000176200001440000001355114323400722013627 0ustar liggesusers/** * \file DST.hpp * \brief Header for GeographicLib::DST class * * Copyright (c) Charles Karney (2022) and licensed under * the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ **********************************************************************/ #if !defined(GEOGRAPHICLIB_DST_HPP) #define GEOGRAPHICLIB_DST_HPP 1 #include "Constants.h" #include #include template class kissfft; namespace GeographicLib { /** * \brief Discrete sine transforms * * This decomposes periodic functions \f$ f(\sigma) \f$ (period \f$ 2\pi \f$) * which are odd about \f$ \sigma = 0 \f$ and even about \f$ \sigma = \frac12 * \pi \f$ into a Fourier series * \f[ * f(\sigma) = \sum_{l=0}^\infty F_l \sin\bigl((2l+1)\sigma\bigr). * \f] * * The first \f$ N \f$ components of \f$ F_l \f$, for \f$0 \le l < N\f$ may * be approximated by * \f[ * F_l = \frac2N \sum_{j=1}^{N} * p_j f(\sigma_j) \sin\bigl((2l+1)\sigma_j\bigr), * \f] * where \f$ \sigma_j = j\pi/(2N) \f$ and \f$ p_j = \frac12 \f$ for \f$ j = N * \f$ and \f$ 1 \f$ otherwise. \f$ F_l \f$ is a discrete sine transform of * type DST-III and may be conveniently computed using the fast Fourier * transform, FFT; this is implemented with the DST::transform method. * * Having computed \f$ F_l \f$ based on \f$ N \f$ evaluations of \f$ * f(\sigma) \f$ at \f$ \sigma_j = j\pi/(2N) \f$, it is possible to * refine these transform values and add another \f$ N \f$ coefficients by * evaluating \f$ f(\sigma) \f$ at \f$ (j-\frac12)\pi/(2N) \f$; this is * implemented with the DST::refine method. * * Here we compute FFTs using the kissfft package * https://github.com/mborgerding/kissfft by Mark Borgerding. * * Example of use: * \include example-DST.cpp * * \note The FFTW package https://www.fftw.org/ can also be used. However * this is a more complicated dependency, its CMake support is broken, and it * doesn't work with mpreals (GEOGRAPHICLIB_PRECISION = 5). **********************************************************************/ class DST { private: typedef Math::real real; int _N; typedef kissfft fft_t; std::shared_ptr _fft; // Implement DST-III (centerp = false) or DST-IV (centerp = true) void fft_transform(real data[], real F[], bool centerp) const; // Add another N terms to F void fft_transform2(real data[], real F[]) const; public: /** * Constructor specifying the number of points to use. * @param[in] N the number of points to use. **********************************************************************/ GEOGRAPHICLIB_EXPORT DST(int N = 0); /** * Reset the given number of points. * @param[in] N the number of points to use. **********************************************************************/ void GEOGRAPHICLIB_EXPORT reset(int N); /** * Return the number of points. * @return the number of points to use. **********************************************************************/ int N() const { return _N; } /** * Determine first \e N terms in the Fourier series * @param[in] f the function used for evaluation. * @param[out] F the first \e N coefficients of the Fourier series. * * The evaluates \f$ f(\sigma) \f$ at \f$ \sigma = (j + 1) \pi / (2 N) \f$ * for integer \f$ j \in [0, N) \f$. \e F should be an array of length at * least \e N. **********************************************************************/ void GEOGRAPHICLIB_EXPORT transform(std::function f, real F[]) const; /** * Refine the Fourier series by doubling the number of points sampled * @param[in] f the function used for evaluation. * @param[inout] F on input the first \e N coefficents of the Fourier * series; on output the refined transform based on 2\e N points, i.e., * the first 2\e N coefficents. * * The evaluates \f$ f(\sigma) \f$ at additional points \f$ \sigma = (j + * \frac12) \pi / (2 N) \f$ for integer \f$ j \in [0, N) \f$, computes the * DST-IV transform of these, and combines this with the input \e F to * compute the 2\e N term DST-III discrete sine transform. This is * equivalent to calling transform with twice the value of \e N but is more * efficient, given that the \e N term coefficients are already known. See * the example code above. **********************************************************************/ void GEOGRAPHICLIB_EXPORT refine(std::function f, real F[]) const; /** * Evaluate the Fourier sum given the sine and cosine of the angle * @param[in] sinx sinσ. * @param[in] cosx cosσ. * @param[in] F the array of Fourier coefficients. * @param[in] N the number of Fourier coefficients. * @return the value of the Fourier sum. **********************************************************************/ static real GEOGRAPHICLIB_EXPORT eval(real sinx, real cosx, const real F[], int N); /** * Evaluate the integral of Fourier sum given the sine and cosine of the * angle * @param[in] sinx sinσ. * @param[in] cosx cosσ. * @param[in] F the array of Fourier coefficients. * @param[in] N the number of Fourier coefficients. * @return the value of the integral. * * The constant of integration is chosen so that the integral is zero at * \f$ \sigma = \frac12\pi \f$. **********************************************************************/ static real GEOGRAPHICLIB_EXPORT integral(real sinx, real cosx, const real F[], int N); }; } // namespace GeographicLib #endif // GEOGRAPHICLIB_DST_HPP geosphere/src/a_geodesic.cpp0000644000176200001440000001054414323401121015603 0ustar liggesusers #include "Rcpp.h" //#include "geodesic.h" #include #include "Geodesic.h" #include "PolygonArea.h" //#include "Constants.hpp" #include "GeodesicLine.h" //using namespace GeographicLib; /* Robert Hijmans May 2015 December 2021 */ //SEXP _geodesic(SEXP longitude, SEXP latitude, SEXP azimuth, SEXP distance, SEXP pa, SEXP pf) { /* //[[Rcpp::export(name = ".geodesic")]] std::vector geodesic(std::vector lon1, std::vector lat1, std::vector azi1, std::vector s12, double a, double f) { struct geod_geodesic g; geod_init(&g, a, f); size_t n = lat1.size(); std::vector r(n * 3); for (size_t i=0; i < n; i++) { size_t j = i * 3; geod_direct(&g, lat1[i], lon1[i], azi1[i], s12[i], &r[j], &r[j+1], &r[j+2]); } return r; } */ //[[Rcpp::export(name = ".geodesic")]] std::vector geodesic(std::vector lon1, std::vector lat1, std::vector azi1, std::vector s12, double a, double f) { GeographicLib::Geodesic geod(a, f); size_t n = lat1.size(); std::vector r(n * 3); for (size_t i=0; i < n; i++) { size_t j = i * 3; geod.Direct(lat1[i], lon1[i], azi1[i], s12[i], r[j+1], r[j], r[j+2]); } return r; } //SEXP _inversegeodesic(SEXP longitude1, SEXP latitude1, SEXP longitude2, SEXP latitude2, SEXP pa, SEXP pf) { /* //[[Rcpp::export(name = ".inversegeodesic")]] std::vector inversegeodesic(std::vector lon1, std::vector lat1, std::vector lon2, std::vector lat2, double a, double f) { struct geod_geodesic g; geod_init(&g, a, f); size_t n = lat1.size(); std::vector r(n * 3); for (size_t i=0; i < n; i++) { size_t j = i * 3; geod_inverse(&g, lat1[i], lon1[i], lat2[i], lon2[i], &r[j], &r[j+1], &r[j+2]); } return r; } */ //[[Rcpp::export(name = ".inversegeodesic")]] std::vector inversegeodesic(std::vector lon1, std::vector lat1, std::vector lon2, std::vector lat2, double a, double f) { const GeographicLib::Geodesic& geod = GeographicLib::Geodesic::WGS84(); size_t n = lat1.size(); std::vector r(n * 3); for (size_t i=0; i < n; i++) { size_t j = i * 3; geod.Inverse(lat1[i], lon1[i], lat2[i], lon2[i], r[j], r[j+1], r[j+2]); } return r; } //SEXP _polygonarea(SEXP longitude, SEXP latitude, SEXP pa, SEXP pf) { /* //[[Rcpp::export(name = ".polygonarea")]] std::vector polygonarea(std::vector lon, std::vector lat, double a, double f) { struct geod_geodesic g; struct geod_polygon p; geod_init(&g, a, f); geod_polygon_init(&p, 0); double A, P; for (size_t i=0; i r = {n, P, A}; return(r); } */ //[[Rcpp::export(name = ".polygonarea")]] std::vector polygonarea(std::vector lon, std::vector lat, double a, double f) { std::vector out(3); // n, perimeter, area; GeographicLib::Geodesic geod(a, f); GeographicLib::PolygonArea poly(geod); try { for (size_t i=0; i> geodesic_nodes(double lon1, double lat1, double lon2, double lat2, size_t n, double distance, bool arc, double a, double f) { GeographicLib::Geodesic geod(a, f); GeographicLib::GeodesicLine line = geod.InverseLine(lat1, lon1, lat2, lon2); // number of intervals if (n == 0) { if (distance <= 0) { n = 1; } else { n = int(ceil(line.Distance() / std::abs(distance))); } } std::vector> out(2); out[0].reserve(n+1); out[1].reserve(n+1); out[0].push_back(lon1); out[1].push_back(lat1); if (arc) { // intervals of equal arc length double da = line.Arc() / n; for (size_t i=1; i #include #include template class kissfft { public: typedef std::complex cpx_t; kissfft( const std::size_t nfft, const bool inverse ) :_nfft(nfft) ,_inverse(inverse) { using std::acos; using std::cos; using std::sin; if (_nfft == 0) return; // fill twiddle factors _twiddles.resize(_nfft); { const scalar_t s = _inverse ? 1 : -1; const scalar_t d = acos( (scalar_t) -1) / (2 * _nfft); int i = 0, N = int(_nfft); // signed ints needed for subtractions // enforce trigonometric symmetries by evaluating sin and cos // with arguments in the range [-pi/4, pi/4] for (; 8*i < N; ++i) _twiddles[i] = cpx_t(+ cos((4*i )*d), +s*sin((4*i )*d)); // pi/4*[0, 1) for (; 8*i < 3*N; ++i) _twiddles[i] = cpx_t(- sin((4*i- N)*d), +s*cos((4*i- N)*d)); // pi/4*[1, 3) for (; 8*i < 5*N; ++i) _twiddles[i] = cpx_t(- cos((4*i-2*N)*d), -s*sin((4*i-2*N)*d)); // pi/4*[3, 5) for (; 8*i < 7*N; ++i) _twiddles[i] = cpx_t(+ sin((4*i-3*N)*d), -s*cos((4*i-3*N)*d)); // pi/4*[5, 7) for (; i < N; ++i) _twiddles[i] = cpx_t(+ cos((4*i-4*N)*d), +s*sin((4*i-4*N)*d)); // pi/4*[5, 8) } //factorize //start factoring out 4's, then 2's, then 3,5,7,9,... std::size_t n= _nfft; std::size_t p=4; do { while (n % p) { switch (p) { case 4: p = 2; break; case 2: p = 3; break; default: p += 2; break; } if (p*p>n) p = n;// no more factors } n /= p; _stageRadix.push_back(p); _stageRemainder.push_back(n); }while(n>1); } /// Changes the FFT-length and/or the transform direction. /// /// @post The @c kissfft object will be in the same state as if it /// had been newly constructed with the passed arguments. /// However, the implementation may be faster than constructing a /// new fft object. void assign( const std::size_t nfft, const bool inverse ) { if ( nfft != _nfft ) { kissfft tmp( nfft, inverse ); // O(n) time. std::swap( tmp, *this ); // this is O(1) in C++11, O(n) otherwise. } else if ( inverse != _inverse ) { // conjugate the twiddle factors. for ( typename std::vector::iterator it = _twiddles.begin(); it != _twiddles.end(); ++it ) it->imag( -it->imag() ); } } /// Calculates the complex Discrete Fourier Transform. /// /// The size of the passed arrays must be passed in the constructor. /// The sum of the squares of the absolute values in the @c dst /// array will be @c N times the sum of the squares of the absolute /// values in the @c src array, where @c N is the size of the array. /// In other words, the l_2 norm of the resulting array will be /// @c sqrt(N) times as big as the l_2 norm of the input array. /// This is also the case when the inverse flag is set in the /// constructor. Hence when applying the same transform twice, but with /// the inverse flag changed the second time, then the result will /// be equal to the original input times @c N. void transform(const cpx_t * fft_in, cpx_t * fft_out, const std::size_t stage = 0, const std::size_t fstride = 1, const std::size_t in_stride = 1) const { if (_nfft == 0) return; const std::size_t p = _stageRadix[stage]; const std::size_t m = _stageRemainder[stage]; cpx_t * const Fout_beg = fft_out; cpx_t * const Fout_end = fft_out + p*m; if (m==1) { do{ *fft_out = *fft_in; fft_in += fstride*in_stride; }while(++fft_out != Fout_end ); }else{ do{ // recursive call: // DFT of size m*p performed by doing // p instances of smaller DFTs of size m, // each one takes a decimated version of the input transform(fft_in, fft_out, stage+1, fstride*p,in_stride); fft_in += fstride*in_stride; }while( (fft_out += m) != Fout_end ); } fft_out=Fout_beg; // recombine the p smaller DFTs switch (p) { case 2: kf_bfly2(fft_out,fstride,m); break; case 3: kf_bfly3(fft_out,fstride,m); break; case 4: kf_bfly4(fft_out,fstride,m); break; case 5: kf_bfly5(fft_out,fstride,m); break; default: kf_bfly_generic(fft_out,fstride,m,p); break; } } /// Calculates the Discrete Fourier Transform (DFT) of a real input /// of size @c 2*N. /// /// The 0-th and N-th value of the DFT are real numbers. These are /// stored in @c dst[0].real() and @c dst[0].imag() respectively. /// The remaining DFT values up to the index N-1 are stored in /// @c dst[1] to @c dst[N-1]. /// The other half of the DFT values can be calculated from the /// symmetry relation /// @code /// DFT(src)[2*N-k] == conj( DFT(src)[k] ); /// @endcode /// The same scaling factors as in @c transform() apply. /// /// @note For this to work, the types @c scalar_t and @c cpx_t /// must fulfill the following requirements: /// /// For any object @c z of type @c cpx_t, /// @c reinterpret_cast(z)[0] is the real part of @c z and /// @c reinterpret_cast(z)[1] is the imaginary part of @c z. /// For any pointer to an element of an array of @c cpx_t named @c p /// and any valid array index @c i, @c reinterpret_cast(p)[2*i] /// is the real part of the complex number @c p[i], and /// @c reinterpret_cast(p)[2*i+1] is the imaginary part of the /// complex number @c p[i]. /// /// Since C++11, these requirements are guaranteed to be satisfied for /// @c scalar_ts being @c float, @c double or @c long @c double /// together with @c cpx_t being @c std::complex. void transform_real( const scalar_t * const src, cpx_t * const dst ) const { using std::acos; using std::exp; const std::size_t N = _nfft; if ( N == 0 ) return; // perform complex FFT transform( reinterpret_cast(src), dst ); // post processing for k = 0 and k = N dst[0] = cpx_t( dst[0].real() + dst[0].imag(), dst[0].real() - dst[0].imag() ); // post processing for all the other k = 1, 2, ..., N-1 const scalar_t pi = acos( (scalar_t) -1); const scalar_t half_phi_inc = ( _inverse ? pi : -pi ) / N; const cpx_t twiddle_mul = exp( cpx_t(0, half_phi_inc) ); for ( std::size_t k = 1; 2*k < N; ++k ) { const cpx_t w = (scalar_t)0.5 * cpx_t( dst[k].real() + dst[N-k].real(), dst[k].imag() - dst[N-k].imag() ); const cpx_t z = (scalar_t)0.5 * cpx_t( dst[k].imag() + dst[N-k].imag(), -dst[k].real() + dst[N-k].real() ); const cpx_t twiddle = k % 2 == 0 ? _twiddles[k/2] : _twiddles[k/2] * twiddle_mul; dst[ k] = w + twiddle * z; dst[N-k] = std::conj( w - twiddle * z ); } if ( N % 2 == 0 ) dst[N/2] = std::conj( dst[N/2] ); } private: void kf_bfly2( cpx_t * Fout, const size_t fstride, const std::size_t m) const { for (std::size_t k=0;k _scratchbuf.size()) _scratchbuf.resize(p); for ( std::size_t u=0; u=_nfft) twidx-=_nfft; Fout[ k ] += _scratchbuf[q] * twiddles[twidx]; } k += m; } } } std::size_t _nfft; bool _inverse; std::vector _twiddles; std::vector _stageRadix; std::vector _stageRemainder; mutable std::vector _scratchbuf; }; #endif geosphere/src/DST.cpp0000644000176200001440000001026714323403614014166 0ustar liggesusers/** * \file DST.cpp * \brief Implementation for GeographicLib::DST class * * Copyright (c) Charles Karney (2022) and licensed under * the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ **********************************************************************/ #include "DST.h" #include #include "kissfft.h" namespace GeographicLib { using namespace std; DST::DST(int N) : _N(N < 0 ? 0 : N) , _fft(make_shared(fft_t(2 * _N, false))) {} void DST::reset(int N) { N = N < 0 ? 0 : N; if (N == _N) return; _N = N; _fft->assign(2 * _N, false); } void DST::fft_transform(real data[], real F[], bool centerp) const { // Implement DST-III (centerp = false) or DST-IV (centerp = true). // Elements (0,N], resp. [0,N), of data should be set on input for centerp // = false, resp. true. F must have a size of at least N and on output // elements [0,N) of F contain the transform. if (_N == 0) return; if (centerp) { for (int i = 0; i < _N; ++i) { data[_N+i] = data[_N-1-i]; data[2*_N+i] = -data[i]; data[3*_N+i] = -data[_N-1-i]; } } else { data[0] = 0; // set [0] for (int i = 1; i < _N; ++i) data[_N+i] = data[_N-i]; // set [N+1,2*N-1] for (int i = 0; i < 2*_N; ++i) data[2*_N+i] = -data[i]; // [2*N, 4*N-1] } vector> ctemp(2*_N); _fft->transform_real(data, ctemp.data()); if (centerp) { real d = -Math::pi()/(4*_N); for (int i = 0, j = 1; i < _N; ++i, j+=2) ctemp[j] *= exp(complex(0, j*d)); } for (int i = 0, j = 1; i < _N; ++i, j+=2) { F[i] = -ctemp[j].imag() / (2*_N); } } void DST::fft_transform2(real data[], real F[]) const { // Elements [0,N), of data should be set to the N grid center values and F // should have size of at least 2*N. On input elements [0,N) of F contain // the size N transform; on output elements [0,2*N) of F contain the size // 2*N transform. fft_transform(data, F+_N, true); // Copy DST-IV order N tx to [0,N) elements of data for (int i = 0; i < _N; ++i) data[i] = F[i+_N]; for (int i = _N; i < 2*_N; ++i) // (DST-IV order N - DST-III order N) / 2 F[i] = (data[2*_N-1-i] - F[2*_N-1-i])/2; for (int i = 0; i < _N; ++i) // (DST-IV order N + DST-III order N) / 2 F[i] = (data[i] + F[i])/2; } void DST::transform(function f, real F[]) const { vector data(4 * _N); real d = Math::pi()/(2 * _N); for (int i = 1; i <= _N; ++i) data[i] = f( i * d ); fft_transform(data.data(), F, false); } void DST::refine(function f, real F[]) const { vector data(4 * _N); real d = Math::pi()/(4 * _N); for (int i = 0; i < _N; ++i) data[i] = f( (2*i + 1) * d ); fft_transform2(data.data(), F); } Math::real DST::eval(real sinx, real cosx, const real F[], int N) { // Evaluate // y = sum(F[i] * sin((2*i+1) * x), i, 0, N-1) // using Clenshaw summation. // Approx operation count = (N + 5) mult and (2 * N + 2) add real ar = 2 * (cosx - sinx) * (cosx + sinx), // 2 * cos(2 * x) y0 = N & 1 ? F[--N] : 0, y1 = 0; // accumulators for sum // Now N is even while (N > 0) { // Unroll loop x 2, so accumulators return to their original role y1 = ar * y0 - y1 + F[--N]; y0 = ar * y1 - y0 + F[--N]; } return sinx * (y0 + y1); // sin(x) * (y0 + y1) } Math::real DST::integral(real sinx, real cosx, const real F[], int N) { // Evaluate // y = -sum(F[i]/(2*i+1) * cos((2*i+1) * x), i, 0, N-1) // using Clenshaw summation. // Approx operation count = (N + 5) mult and (2 * N + 2) add int l = N; real ar = 2 * (cosx - sinx) * (cosx + sinx), // 2 * cos(2 * x) y0 = N & 1 ? F[--N]/(2*(--l)+1) : 0, y1 = 0; // accumulators for sum // Now N is even while (N > 0) { // Unroll loop x 2, so accumulators return to their original role y1 = ar * y0 - y1 + F[--N]/(2*(--l)+1); y0 = ar * y1 - y0 + F[--N]/(2*(--l)+1); } return cosx * (y1 - y0); // cos(x) * (y1 - y0) } } // namespace GeographicLib geosphere/src/a_dist.c0000644000176200001440000000744114323373006014440 0ustar liggesusers/* Robert Hijmans, June 2011 */ #include #include #include "a_util.h" double distPlane(double x1, double y1, double x2, double y2) { return( sqrt(pow((x2-x1),2) + pow((y2-y1), 2)) ); } double distCos(double lon1, double lat1, double lon2, double lat2, double r) { double cosd; lon1 = toRad(lon1); lon2 = toRad(lon2); lat1 = toRad(lat1); lat2 = toRad(lat2); cosd = sin(lat1) * sin(lat2) + cos(lat1) * cos(lat2) * cos(lon1-lon2); return acos(cosd) * r; } double distHav(double lon1, double lat1, double lon2, double lat2, double r) { double dLat, dLon, a; lon1 = toRad(lon1); lon2 = toRad(lon2); lat1 = toRad(lat1); lat2 = toRad(lat2); dLat = lat2-lat1; dLon = lon2-lon1; a = sin(dLat/2.) * sin(dLat/2.) + cos(lat1) * cos(lat2) * sin(dLon/2.) * sin(dLon/2.); return 2. * atan2(sqrt(a), sqrt(1.-a)) * r; } double distVinSph(double lon1, double lat1, double lon2, double lat2, double r) { double x, x1, x2, y; lon1 = toRad(lon1); lon2 = toRad(lon2); lat1 = toRad(lat1); lat2 = toRad(lat2); x1 = cos(lat2) * sin(lon1-lon2); x2 = cos(lat1) * sin(lat2) - sin(lat1) * cos(lat2) * cos(lon1-lon2); x = sqrt(x1*x1 + x2*x2); y = sin(lat1) * sin(lat2) + cos(lat1) * cos(lat2) * cos(lon1-lon2); return r * atan2(x, y); } double distVinEll(double lon1, double lat1, double lon2, double lat2, double a, double b, double f) { /* Calculate geodesic distance (in m) between two points specified by latitude/longitude (in numeric degrees) using Vincenty inverse formula for ellipsoids based on source http://www.movable-type.co.uk/scripts/latlong-vincenty.html by Chris Veness */ double L, U1, U2, sinU1, cosU1, sinU2, cosU2, lambda, sinLambda, cosLambda, sinSigma, cosSigma, sigma, sinAlpha, cosSqAlpha, cos2SigmaM, C, lambdaP, uSq, A, B, deltaSigma; int iterLimit, cont; if ((lon1 == lon2) & (lat1 == lat2)) { return 0.; } else if ( isnan(lon1) | isnan(lat1) | isnan(lon2) | isnan(lat2)) { return NAN; } else { lon1 = toRad(lon1); lon2 = toRad(lon2); lat1 = toRad(lat1); lat2 = toRad(lat2); L = (lon2-lon1); U1 = atan((1.-f) * tan(lat1)); U2 = atan((1.-f) * tan(lat2)); sinU1 = sin(U1); cosU1 = cos(U1); sinU2 = sin(U2); cosU2 = cos(U2); lambda = L; iterLimit = 100; cont = 1; while (cont) { sinLambda = sin(lambda); cosLambda = cos(lambda); sinSigma = sqrt((cosU2*sinLambda) * (cosU2*sinLambda) + (cosU1*sinU2-sinU1*cosU2*cosLambda) * (cosU1*sinU2-sinU1*cosU2*cosLambda)); cosSigma = sinU1*sinU2 + cosU1*cosU2*cosLambda; sigma = atan2(sinSigma, cosSigma); sinAlpha = cosU1 * cosU2 * sinLambda / sinSigma; cosSqAlpha = 1. - sinAlpha*sinAlpha; cos2SigmaM = cosSigma - 2.*sinU1*sinU2/cosSqAlpha; if (isnan(cos2SigmaM)) { cos2SigmaM = 0.; // equatorial line: cosSqAlpha=0 (par 6) } C = f/16.*cosSqAlpha*(4.+f*(4.-3.*cosSqAlpha)); lambdaP = lambda; lambda = L + (1.-C) * f * sinAlpha * (sigma + C*sinSigma*(cos2SigmaM+C*cosSigma*(-1.+2.*cos2SigmaM*cos2SigmaM))); iterLimit = iterLimit - 1; cont = (fabs(lambda-lambdaP) > 1e-12 && iterLimit > 0); } if (iterLimit==0) { return NAN; // formula failed to converge } else { uSq = cosSqAlpha * (a*a - b*b) / (b*b); A = 1. + uSq/16384.*(4096.+uSq*(-768.+uSq*(320.-175.*uSq))); B = uSq/1024. * (256.+uSq*(-128.+uSq*(74.-47.*uSq))); deltaSigma = B*sinSigma*(cos2SigmaM+B/4.*(cosSigma*(-1.+2.*cos2SigmaM*cos2SigmaM)- B/6.*cos2SigmaM*(-3.+4.*sinSigma*sinSigma)*(-3.+4.*cos2SigmaM*cos2SigmaM))); return b*A*(sigma-deltaSigma); } } } void distanceEllipsoid(int *n, double *lon1, double *lat1, double *lon2, double *lat2, double *a, double *b, double *f, int *m, double *dist) { if (*m > 0) { for(size_t i=0; i < *n; i++) { dist[i] = distVinEll(lon1[i], lat1[i], lon2[i], lat2[i], a[i], b[i], f[i]); } } } geosphere/src/Rhumb.h0000644000176200001440000006260714323377037014274 0ustar liggesusers/** * \file Rhumb.hpp * \brief Header for GeographicLib::Rhumb and GeographicLib::RhumbLine classes * * Copyright (c) Charles Karney (2014-2022) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ **********************************************************************/ #if !defined(GEOGRAPHICLIB_RHUMB_HPP) #define GEOGRAPHICLIB_RHUMB_HPP 1 #include "Constants.h" #include "Ellipsoid.h" #if !defined(GEOGRAPHICLIB_RHUMBAREA_ORDER) /** * The order of the series approximation used in rhumb area calculations. * GEOGRAPHICLIB_RHUMBAREA_ORDER can be set to any integer in [4, 8]. **********************************************************************/ # define GEOGRAPHICLIB_RHUMBAREA_ORDER \ (GEOGRAPHICLIB_PRECISION == 2 ? 6 : \ (GEOGRAPHICLIB_PRECISION == 1 ? 4 : 8)) #endif namespace GeographicLib { class RhumbLine; template class PolygonAreaT; /** * \brief Solve of the direct and inverse rhumb problems. * * The path of constant azimuth between two points on an ellipsoid at (\e * lat1, \e lon1) and (\e lat2, \e lon2) is called the rhumb line (also * called the loxodrome). Its length is \e s12 and its azimuth is \e azi12. * (The azimuth is the heading measured clockwise from north.) * * Given \e lat1, \e lon1, \e azi12, and \e s12, we can determine \e lat2, * and \e lon2. This is the \e direct rhumb problem and its solution is * given by the function Rhumb::Direct. * * Given \e lat1, \e lon1, \e lat2, and \e lon2, we can determine \e azi12 * and \e s12. This is the \e inverse rhumb problem, whose solution is given * by Rhumb::Inverse. This finds the shortest such rhumb line, i.e., the one * that wraps no more than half way around the earth. If the end points are * on opposite meridians, there are two shortest rhumb lines and the * east-going one is chosen. * * These routines also optionally calculate the area under the rhumb line, \e * S12. This is the area, measured counter-clockwise, of the rhumb line * quadrilateral with corners (lat1,lon1), (0,lon1), * (0,lon2), and (lat2,lon2). * * Note that rhumb lines may be appreciably longer (up to 50%) than the * corresponding Geodesic. For example the distance between London Heathrow * and Tokyo Narita via the rhumb line is 11400 km which is 18% longer than * the geodesic distance 9600 km. * * For more information on rhumb lines see \ref rhumb. * * Example of use: * \include example-Rhumb.cpp **********************************************************************/ class GEOGRAPHICLIB_EXPORT Rhumb { private: typedef Math::real real; friend class RhumbLine; template friend class PolygonAreaT; Ellipsoid _ell; bool _exact; real _c2; static const int tm_maxord = GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER; static const int maxpow_ = GEOGRAPHICLIB_RHUMBAREA_ORDER; // _rR[0] unused real _rR[maxpow_ + 1]; static real gd(real x) { using std::atan; using std::sinh; return atan(sinh(x)); } // Use divided differences to determine (mu2 - mu1) / (psi2 - psi1) // accurately // // Definition: Df(x,y,d) = (f(x) - f(y)) / (x - y) // See: // W. M. Kahan and R. J. Fateman, // Symbolic computation of divided differences, // SIGSAM Bull. 33(2), 7-28 (1999) // https://doi.org/10.1145/334714.334716 // http://www.cs.berkeley.edu/~fateman/papers/divdiff.pdf static real Dlog(real x, real y) { using std::sqrt; using std::asinh; real t = x - y; // Change // // atanh(t / (x + y)) // // to // // asinh(t / (2 * sqrt(x*y))) // // to avoid taking atanh(1) when x is large and y is 1. N.B., this // routine is invoked with positive x and y, so no need to guard against // taking the sqrt of a negative quantity. This fixes bogus results for // the area being returning when an endpoint is at a pole. return t != 0 ? 2 * asinh(t / (2 * sqrt(x*y))) / t : 1 / x; } // N.B., x and y are in degrees static real Dtan(real x, real y) { real d = x - y, tx = Math::tand(x), ty = Math::tand(y), txy = tx * ty; return d != 0 ? (2 * txy > -1 ? (1 + txy) * Math::tand(d) : tx - ty) / (d * Math::degree()) : 1 + txy; } static real Datan(real x, real y) { using std::atan; real d = x - y, xy = x * y; return d != 0 ? (2 * xy > -1 ? atan( d / (1 + xy) ) : atan(x) - atan(y)) / d : 1 / (1 + xy); } static real Dsin(real x, real y) { using std::sin; using std::cos; real d = (x - y) / 2; return cos((x + y)/2) * (d != 0 ? sin(d) / d : 1); } static real Dsinh(real x, real y) { using std::sinh; using std::cosh; real d = (x - y) / 2; return cosh((x + y) / 2) * (d != 0 ? sinh(d) / d : 1); } static real Dcosh(real x, real y) { using std::sinh; real d = (x - y) / 2; return sinh((x + y) / 2) * (d != 0 ? sinh(d) / d : 1); } static real Dasinh(real x, real y) { using std::asinh; using std::hypot; real d = x - y, hx = hypot(real(1), x), hy = hypot(real(1), y); return d != 0 ? asinh(x*y > 0 ? d * (x + y) / (x*hy + y*hx) : x*hy - y*hx) / d : 1 / hx; } static real Dgd(real x, real y) { using std::sinh; return Datan(sinh(x), sinh(y)) * Dsinh(x, y); } // N.B., x and y are the tangents of the angles static real Dgdinv(real x, real y) { return Dasinh(x, y) / Datan(x, y); } // Copied from LambertConformalConic... // Deatanhe(x,y) = eatanhe((x-y)/(1-e^2*x*y))/(x-y) real Deatanhe(real x, real y) const { real t = x - y, d = 1 - _ell._e2 * x * y; return t != 0 ? Math::eatanhe(t / d, _ell._es) / t : _ell._e2 / d; } // (E(x) - E(y)) / (x - y) -- E = incomplete elliptic integral of 2nd kind real DE(real x, real y) const; // (mux - muy) / (phix - phiy) using elliptic integrals real DRectifying(real latx, real laty) const; // (psix - psiy) / (phix - phiy) real DIsometric(real latx, real laty) const; // (sum(c[j]*sin(2*j*x),j=1..n) - sum(c[j]*sin(2*j*x),j=1..n)) / (x - y) static real SinCosSeries(bool sinp, real x, real y, const real c[], int n); // (mux - muy) / (chix - chiy) using Krueger's series real DConformalToRectifying(real chix, real chiy) const; // (chix - chiy) / (mux - muy) using Krueger's series real DRectifyingToConformal(real mux, real muy) const; // (mux - muy) / (psix - psiy) // N.B., psix and psiy are in degrees real DIsometricToRectifying(real psix, real psiy) const; // (psix - psiy) / (mux - muy) real DRectifyingToIsometric(real mux, real muy) const; real MeanSinXi(real psi1, real psi2) const; // The following two functions (with lots of ignored arguments) mimic the // interface to the corresponding Geodesic function. These are needed by // PolygonAreaT. void GenDirect(real lat1, real lon1, real azi12, bool, real s12, unsigned outmask, real& lat2, real& lon2, real&, real&, real&, real&, real&, real& S12) const { GenDirect(lat1, lon1, azi12, s12, outmask, lat2, lon2, S12); } void GenInverse(real lat1, real lon1, real lat2, real lon2, unsigned outmask, real& s12, real& azi12, real&, real& , real& , real& , real& S12) const { GenInverse(lat1, lon1, lat2, lon2, outmask, s12, azi12, S12); } public: /** * Bit masks for what calculations to do. They specify which results to * return in the general routines Rhumb::GenDirect and Rhumb::GenInverse * routines. RhumbLine::mask is a duplication of this enum. **********************************************************************/ enum mask { /** * No output. * @hideinitializer **********************************************************************/ NONE = 0U, /** * Calculate latitude \e lat2. * @hideinitializer **********************************************************************/ LATITUDE = 1U<<7, /** * Calculate longitude \e lon2. * @hideinitializer **********************************************************************/ LONGITUDE = 1U<<8, /** * Calculate azimuth \e azi12. * @hideinitializer **********************************************************************/ AZIMUTH = 1U<<9, /** * Calculate distance \e s12. * @hideinitializer **********************************************************************/ DISTANCE = 1U<<10, /** * Calculate area \e S12. * @hideinitializer **********************************************************************/ AREA = 1U<<14, /** * Unroll \e lon2 in the direct calculation. * @hideinitializer **********************************************************************/ LONG_UNROLL = 1U<<15, /** * Calculate everything. (LONG_UNROLL is not included in this mask.) * @hideinitializer **********************************************************************/ ALL = 0x7F80U, }; /** * Constructor for an ellipsoid with * * @param[in] a equatorial radius (meters). * @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere. * Negative \e f gives a prolate ellipsoid. * @param[in] exact if true (the default) use an addition theorem for * elliptic integrals to compute divided differences; otherwise use * series expansion (accurate for |f| < 0.01). * @exception GeographicErr if \e a or (1 − \e f) \e a is not * positive. * * See \ref rhumb, for a detailed description of the \e exact parameter. **********************************************************************/ Rhumb(real a, real f, bool exact = true); /** * Solve the direct rhumb problem returning also the area. * * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] azi12 azimuth of the rhumb line (degrees). * @param[in] s12 distance between point 1 and point 2 (meters); it can be * negative. * @param[out] lat2 latitude of point 2 (degrees). * @param[out] lon2 longitude of point 2 (degrees). * @param[out] S12 area under the rhumb line (meters2). * * \e lat1 should be in the range [−90°, 90°]. The value of * \e lon2 returned is in the range [−180°, 180°]. * * If point 1 is a pole, the cosine of its latitude is taken to be * 1/ε2 (where ε is 2-52). This * position, which is extremely close to the actual pole, allows the * calculation to be carried out in finite terms. If \e s12 is large * enough that the rhumb line crosses a pole, the longitude of point 2 * is indeterminate (a NaN is returned for \e lon2 and \e S12). **********************************************************************/ void Direct(real lat1, real lon1, real azi12, real s12, real& lat2, real& lon2, real& S12) const { GenDirect(lat1, lon1, azi12, s12, LATITUDE | LONGITUDE | AREA, lat2, lon2, S12); } /** * Solve the direct rhumb problem without the area. **********************************************************************/ void Direct(real lat1, real lon1, real azi12, real s12, real& lat2, real& lon2) const { real t; GenDirect(lat1, lon1, azi12, s12, LATITUDE | LONGITUDE, lat2, lon2, t); } /** * The general direct rhumb problem. Rhumb::Direct is defined in terms * of this function. * * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] azi12 azimuth of the rhumb line (degrees). * @param[in] s12 distance between point 1 and point 2 (meters); it can be * negative. * @param[in] outmask a bitor'ed combination of Rhumb::mask values * specifying which of the following parameters should be set. * @param[out] lat2 latitude of point 2 (degrees). * @param[out] lon2 longitude of point 2 (degrees). * @param[out] S12 area under the rhumb line (meters2). * * The Rhumb::mask values possible for \e outmask are * - \e outmask |= Rhumb::LATITUDE for the latitude \e lat2; * - \e outmask |= Rhumb::LONGITUDE for the latitude \e lon2; * - \e outmask |= Rhumb::AREA for the area \e S12; * - \e outmask |= Rhumb::ALL for all of the above; * - \e outmask |= Rhumb::LONG_UNROLL to unroll \e lon2 instead of wrapping * it into the range [−180°, 180°]. * . * With the Rhumb::LONG_UNROLL bit set, the quantity \e lon2 − * \e lon1 indicates how many times and in what sense the rhumb line * encircles the ellipsoid. **********************************************************************/ void GenDirect(real lat1, real lon1, real azi12, real s12, unsigned outmask, real& lat2, real& lon2, real& S12) const; /** * Solve the inverse rhumb problem returning also the area. * * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] lat2 latitude of point 2 (degrees). * @param[in] lon2 longitude of point 2 (degrees). * @param[out] s12 rhumb distance between point 1 and point 2 (meters). * @param[out] azi12 azimuth of the rhumb line (degrees). * @param[out] S12 area under the rhumb line (meters2). * * The shortest rhumb line is found. If the end points are on opposite * meridians, there are two shortest rhumb lines and the east-going one is * chosen. \e lat1 and \e lat2 should be in the range [−90°, * 90°]. The value of \e azi12 returned is in the range * [−180°, 180°]. * * If either point is a pole, the cosine of its latitude is taken to be * 1/ε2 (where ε is 2-52). This * position, which is extremely close to the actual pole, allows the * calculation to be carried out in finite terms. **********************************************************************/ void Inverse(real lat1, real lon1, real lat2, real lon2, real& s12, real& azi12, real& S12) const { GenInverse(lat1, lon1, lat2, lon2, DISTANCE | AZIMUTH | AREA, s12, azi12, S12); } /** * Solve the inverse rhumb problem without the area. **********************************************************************/ void Inverse(real lat1, real lon1, real lat2, real lon2, real& s12, real& azi12) const { real t; GenInverse(lat1, lon1, lat2, lon2, DISTANCE | AZIMUTH, s12, azi12, t); } /** * The general inverse rhumb problem. Rhumb::Inverse is defined in terms * of this function. * * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] lat2 latitude of point 2 (degrees). * @param[in] lon2 longitude of point 2 (degrees). * @param[in] outmask a bitor'ed combination of Rhumb::mask values * specifying which of the following parameters should be set. * @param[out] s12 rhumb distance between point 1 and point 2 (meters). * @param[out] azi12 azimuth of the rhumb line (degrees). * @param[out] S12 area under the rhumb line (meters2). * * The Rhumb::mask values possible for \e outmask are * - \e outmask |= Rhumb::DISTANCE for the latitude \e s12; * - \e outmask |= Rhumb::AZIMUTH for the latitude \e azi12; * - \e outmask |= Rhumb::AREA for the area \e S12; * - \e outmask |= Rhumb::ALL for all of the above; **********************************************************************/ void GenInverse(real lat1, real lon1, real lat2, real lon2, unsigned outmask, real& s12, real& azi12, real& S12) const; /** * Set up to compute several points on a single rhumb line. * * @param[in] lat1 latitude of point 1 (degrees). * @param[in] lon1 longitude of point 1 (degrees). * @param[in] azi12 azimuth of the rhumb line (degrees). * @return a RhumbLine object. * * \e lat1 should be in the range [−90°, 90°]. * * If point 1 is a pole, the cosine of its latitude is taken to be * 1/ε2 (where ε is 2-52). This * position, which is extremely close to the actual pole, allows the * calculation to be carried out in finite terms. **********************************************************************/ RhumbLine Line(real lat1, real lon1, real azi12) const; /** \name Inspector functions. **********************************************************************/ ///@{ /** * @return \e a the equatorial radius of the ellipsoid (meters). This is * the value used in the constructor. **********************************************************************/ Math::real EquatorialRadius() const { return _ell.EquatorialRadius(); } /** * @return \e f the flattening of the ellipsoid. This is the * value used in the constructor. **********************************************************************/ Math::real Flattening() const { return _ell.Flattening(); } /** * @return total area of ellipsoid in meters2. The area of a * polygon encircling a pole can be found by adding * Geodesic::EllipsoidArea()/2 to the sum of \e S12 for each side of the * polygon. **********************************************************************/ Math::real EllipsoidArea() const { return _ell.Area(); } ///@} /** * A global instantiation of Rhumb with the parameters for the WGS84 * ellipsoid. **********************************************************************/ static const Rhumb& WGS84(); }; /** * \brief Find a sequence of points on a single rhumb line. * * RhumbLine facilitates the determination of a series of points on a single * rhumb line. The starting point (\e lat1, \e lon1) and the azimuth \e * azi12 are specified in the call to Rhumb::Line which returns a RhumbLine * object. RhumbLine.Position returns the location of point 2 (and, * optionally, the corresponding area, \e S12) a distance \e s12 along the * rhumb line. * * There is no public constructor for this class. (Use Rhumb::Line to create * an instance.) The Rhumb object used to create a RhumbLine must stay in * scope as long as the RhumbLine. * * Example of use: * \include example-RhumbLine.cpp **********************************************************************/ class GEOGRAPHICLIB_EXPORT RhumbLine { private: typedef Math::real real; friend class Rhumb; const Rhumb& _rh; real _lat1, _lon1, _azi12, _salp, _calp, _mu1, _psi1, _r1; // copy assignment not allowed RhumbLine& operator=(const RhumbLine&) = delete; RhumbLine(const Rhumb& rh, real lat1, real lon1, real azi12); public: /** * Construction is via default copy constructor. **********************************************************************/ RhumbLine(const RhumbLine&) = default; /** * This is a duplication of Rhumb::mask. **********************************************************************/ enum mask { /** * No output. * @hideinitializer **********************************************************************/ NONE = Rhumb::NONE, /** * Calculate latitude \e lat2. * @hideinitializer **********************************************************************/ LATITUDE = Rhumb::LATITUDE, /** * Calculate longitude \e lon2. * @hideinitializer **********************************************************************/ LONGITUDE = Rhumb::LONGITUDE, /** * Calculate azimuth \e azi12. * @hideinitializer **********************************************************************/ AZIMUTH = Rhumb::AZIMUTH, /** * Calculate distance \e s12. * @hideinitializer **********************************************************************/ DISTANCE = Rhumb::DISTANCE, /** * Calculate area \e S12. * @hideinitializer **********************************************************************/ AREA = Rhumb::AREA, /** * Unroll \e lon2 in the direct calculation. * @hideinitializer **********************************************************************/ LONG_UNROLL = Rhumb::LONG_UNROLL, /** * Calculate everything. (LONG_UNROLL is not included in this mask.) * @hideinitializer **********************************************************************/ ALL = Rhumb::ALL, }; /** * Compute the position of point 2 which is a distance \e s12 (meters) from * point 1. The area is also computed. * * @param[in] s12 distance between point 1 and point 2 (meters); it can be * negative. * @param[out] lat2 latitude of point 2 (degrees). * @param[out] lon2 longitude of point 2 (degrees). * @param[out] S12 area under the rhumb line (meters2). * * The value of \e lon2 returned is in the range [−180°, * 180°]. * * If \e s12 is large enough that the rhumb line crosses a pole, the * longitude of point 2 is indeterminate (a NaN is returned for \e lon2 and * \e S12). **********************************************************************/ void Position(real s12, real& lat2, real& lon2, real& S12) const { GenPosition(s12, LATITUDE | LONGITUDE | AREA, lat2, lon2, S12); } /** * Compute the position of point 2 which is a distance \e s12 (meters) from * point 1. The area is not computed. **********************************************************************/ void Position(real s12, real& lat2, real& lon2) const { real t; GenPosition(s12, LATITUDE | LONGITUDE, lat2, lon2, t); } /** * The general position routine. RhumbLine::Position is defined in term so * this function. * * @param[in] s12 distance between point 1 and point 2 (meters); it can be * negative. * @param[in] outmask a bitor'ed combination of RhumbLine::mask values * specifying which of the following parameters should be set. * @param[out] lat2 latitude of point 2 (degrees). * @param[out] lon2 longitude of point 2 (degrees). * @param[out] S12 area under the rhumb line (meters2). * * The RhumbLine::mask values possible for \e outmask are * - \e outmask |= RhumbLine::LATITUDE for the latitude \e lat2; * - \e outmask |= RhumbLine::LONGITUDE for the latitude \e lon2; * - \e outmask |= RhumbLine::AREA for the area \e S12; * - \e outmask |= RhumbLine::ALL for all of the above; * - \e outmask |= RhumbLine::LONG_UNROLL to unroll \e lon2 instead of * wrapping it into the range [−180°, 180°]. * . * With the RhumbLine::LONG_UNROLL bit set, the quantity \e lon2 − \e * lon1 indicates how many times and in what sense the rhumb line encircles * the ellipsoid. * * If \e s12 is large enough that the rhumb line crosses a pole, the * longitude of point 2 is indeterminate (a NaN is returned for \e lon2 and * \e S12). **********************************************************************/ void GenPosition(real s12, unsigned outmask, real& lat2, real& lon2, real& S12) const; /** \name Inspector functions **********************************************************************/ ///@{ /** * @return \e lat1 the latitude of point 1 (degrees). **********************************************************************/ Math::real Latitude() const { return _lat1; } /** * @return \e lon1 the longitude of point 1 (degrees). **********************************************************************/ Math::real Longitude() const { return _lon1; } /** * @return \e azi12 the azimuth of the rhumb line (degrees). **********************************************************************/ Math::real Azimuth() const { return _azi12; } /** * @return \e a the equatorial radius of the ellipsoid (meters). This is * the value inherited from the Rhumb object used in the constructor. **********************************************************************/ Math::real EquatorialRadius() const { return _rh.EquatorialRadius(); } /** * @return \e f the flattening of the ellipsoid. This is the value * inherited from the Rhumb object used in the constructor. **********************************************************************/ Math::real Flattening() const { return _rh.Flattening(); } }; } // namespace GeographicLib #endif // GEOGRAPHICLIB_RHUMB_HPP geosphere/src/Geodesic.h0000644000176200001440000012770414323377037014741 0ustar liggesusers/** * \file Geodesic.hpp * \brief Header for GeographicLib::Geodesic class * * Copyright (c) Charles Karney (2009-2022) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ **********************************************************************/ #if !defined(GEOGRAPHICLIB_GEODESIC_HPP) #define GEOGRAPHICLIB_GEODESIC_HPP 1 #include "Constants.h" #if !defined(GEOGRAPHICLIB_GEODESIC_ORDER) /** * The order of the expansions used by Geodesic. * GEOGRAPHICLIB_GEODESIC_ORDER can be set to any integer in [3, 8]. **********************************************************************/ # define GEOGRAPHICLIB_GEODESIC_ORDER \ (GEOGRAPHICLIB_PRECISION == 2 ? 6 : \ (GEOGRAPHICLIB_PRECISION == 1 ? 3 : \ (GEOGRAPHICLIB_PRECISION == 3 ? 7 : 8))) #endif namespace GeographicLib { class GeodesicLine; /** * \brief %Geodesic calculations * * The shortest path between two points on an ellipsoid at (\e lat1, \e lon1) * and (\e lat2, \e lon2) is called the geodesic. Its length is \e s12 and * the geodesic from point 1 to point 2 has azimuths \e azi1 and \e azi2 at * the two end points. (The azimuth is the heading measured clockwise from * north. \e azi2 is the "forward" azimuth, i.e., the heading that takes you * beyond point 2 not back to point 1.) In the figure below, latitude is * labeled φ, longitude λ (with λ12 = * λ2 − λ1), and azimuth α. * * spheroidal triangle * * Given \e lat1, \e lon1, \e azi1, and \e s12, we can determine \e lat2, \e * lon2, and \e azi2. This is the \e direct geodesic problem and its * solution is given by the function Geodesic::Direct. (If \e s12 is * sufficiently large that the geodesic wraps more than halfway around the * earth, there will be another geodesic between the points with a smaller \e * s12.) * * Given \e lat1, \e lon1, \e lat2, and \e lon2, we can determine \e azi1, \e * azi2, and \e s12. This is the \e inverse geodesic problem, whose solution * is given by Geodesic::Inverse. Usually, the solution to the inverse * problem is unique. In cases where there are multiple solutions (all with * the same \e s12, of course), all the solutions can be easily generated * once a particular solution is provided. * * The standard way of specifying the direct problem is the specify the * distance \e s12 to the second point. However it is sometimes useful * instead to specify the arc length \e a12 (in degrees) on the auxiliary * sphere. This is a mathematical construct used in solving the geodesic * problems. The solution of the direct problem in this form is provided by * Geodesic::ArcDirect. An arc length in excess of 180° indicates that * the geodesic is not a shortest path. In addition, the arc length between * an equatorial crossing and the next extremum of latitude for a geodesic is * 90°. * * This class can also calculate several other quantities related to * geodesics. These are: * - reduced length. If we fix the first point and increase \e azi1 * by \e dazi1 (radians), the second point is displaced \e m12 \e dazi1 in * the direction \e azi2 + 90°. The quantity \e m12 is called * the "reduced length" and is symmetric under interchange of the two * points. On a curved surface the reduced length obeys a symmetry * relation, \e m12 + \e m21 = 0. On a flat surface, we have \e m12 = \e * s12. The ratio s12/\e m12 gives the azimuthal scale for an * azimuthal equidistant projection. * - geodesic scale. Consider a reference geodesic and a second * geodesic parallel to this one at point 1 and separated by a small * distance \e dt. The separation of the two geodesics at point 2 is \e * M12 \e dt where \e M12 is called the "geodesic scale". \e M21 is * defined similarly (with the geodesics being parallel at point 2). On a * flat surface, we have \e M12 = \e M21 = 1. The quantity 1/\e M12 gives * the scale of the Cassini-Soldner projection. * - area. The area between the geodesic from point 1 to point 2 and * the equation is represented by \e S12; it is the area, measured * counter-clockwise, of the geodesic quadrilateral with corners * (lat1,lon1), (0,lon1), (0,lon2), and * (lat2,lon2). It can be used to compute the area of any * geodesic polygon. * * Overloaded versions of Geodesic::Direct, Geodesic::ArcDirect, and * Geodesic::Inverse allow these quantities to be returned. In addition * there are general functions Geodesic::GenDirect, and Geodesic::GenInverse * which allow an arbitrary set of results to be computed. The quantities \e * m12, \e M12, \e M21 which all specify the behavior of nearby geodesics * obey addition rules. If points 1, 2, and 3 all lie on a single geodesic, * then the following rules hold: * - \e s13 = \e s12 + \e s23 * - \e a13 = \e a12 + \e a23 * - \e S13 = \e S12 + \e S23 * - \e m13 = \e m12 \e M23 + \e m23 \e M21 * - \e M13 = \e M12 \e M23 − (1 − \e M12 \e M21) \e m23 / \e m12 * - \e M31 = \e M32 \e M21 − (1 − \e M23 \e M32) \e m12 / \e m23 * * Additional functionality is provided by the GeodesicLine class, which * allows a sequence of points along a geodesic to be computed. * * The shortest distance returned by the solution of the inverse problem is * (obviously) uniquely defined. However, in a few special cases there are * multiple azimuths which yield the same shortest distance. Here is a * catalog of those cases: * - \e lat1 = −\e lat2 (with neither point at a pole). If \e azi1 = * \e azi2, the geodesic is unique. Otherwise there are two geodesics and * the second one is obtained by setting [\e azi1, \e azi2] → [\e * azi2, \e azi1], [\e M12, \e M21] → [\e M21, \e M12], \e S12 → * −\e S12. (This occurs when the longitude difference is near * ±180° for oblate ellipsoids.) * - \e lon2 = \e lon1 ± 180° (with neither point at a pole). If * \e azi1 = 0° or ±180°, the geodesic is unique. Otherwise * there are two geodesics and the second one is obtained by setting [\e * azi1, \e azi2] → [−\e azi1, −\e azi2], \e S12 → * −\e S12. (This occurs when \e lat2 is near −\e lat1 for * prolate ellipsoids.) * - Points 1 and 2 at opposite poles. There are infinitely many geodesics * which can be generated by setting [\e azi1, \e azi2] → [\e azi1, \e * azi2] + [\e d, −\e d], for arbitrary \e d. (For spheres, this * prescription applies when points 1 and 2 are antipodal.) * - \e s12 = 0 (coincident points). There are infinitely many geodesics * which can be generated by setting [\e azi1, \e azi2] → * [\e azi1, \e azi2] + [\e d, \e d], for arbitrary \e d. * * The calculations are accurate to better than 15 nm (15 nanometers) for the * WGS84 ellipsoid. See Sec. 9 of * arXiv:1102.1215v1 for * details. The algorithms used by this class are based on series expansions * using the flattening \e f as a small parameter. These are only accurate * for |f| < 0.02; however reasonably accurate results will be * obtained for |f| < 0.2. Here is a table of the approximate * maximum error (expressed as a distance) for an ellipsoid with the same * equatorial radius as the WGS84 ellipsoid and different values of the * flattening.