geosphere/0000755000176200001440000000000013472415406012241 5ustar liggesusersgeosphere/inst/0000755000176200001440000000000013472363553013223 5ustar liggesusersgeosphere/inst/doc/0000755000176200001440000000000013472363553013770 5ustar liggesusersgeosphere/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.R0000644000176200001440000002034413472363553016077 0ustar liggesusers### R code from vignette source 'geosphere.Rnw' ### Encoding: ISO8859-1 ################################################### ### 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.pdf0000644000176200001440000100547613472363553016461 0ustar liggesusers%PDF-1.5 % 1 0 obj << /S /GoTo /D (section.1) >> endobj 4 0 obj (Introduction) endobj 5 0 obj << /S /GoTo /D (section.2) >> endobj 8 0 obj (Great circle distance) endobj 9 0 obj << /S /GoTo /D (section.3) >> endobj 12 0 obj (Points on great circles) endobj 13 0 obj << /S /GoTo /D (section.4) >> endobj 16 0 obj (Point at distance and bearing) endobj 17 0 obj << /S /GoTo /D (section.5) >> endobj 20 0 obj (Maximum latitude on a great circle) endobj 21 0 obj << /S /GoTo /D (section.6) >> endobj 24 0 obj (Great circle intersections) endobj 25 0 obj << /S /GoTo /D (section.7) >> endobj 28 0 obj (Triangulation) endobj 29 0 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/Root 213 0 R /Info 214 0 R /ID [ ] >> startxref 260560 %%EOF geosphere/src/0000755000176200001440000000000013472363553013035 5ustar liggesusersgeosphere/src/util.c0000644000176200001440000000075313472155746014166 0ustar liggesusers#include #include #include #include #include "Rmath.h" 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/util.h0000644000176200001440000000054013472155746014165 0ustar liggesusers/* modulo */ double mod(double x, double n) ; /* Convert degrees to radians */ double toRad(double deg) ; /* Convert radians to degrees */ double toDeg(double rad) ; /* normatlize longitude between -180 .. 180 degrees*/ double normalizeLonDeg(double lon); /* normatlize longitude between -pi .. p1 radians*/ double normalizeLonRad(double lon); geosphere/src/geodesic.c0000644000176200001440000021161113472155746014770 0ustar liggesusers// Version 1.46 // Date 2016-02-15 /** * \file geodesic.c * \brief Implementation of the geodesic routines in C * * For the full documentation see geodesic.h. **********************************************************************/ /** @cond SKIP */ /* * 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.sourceforge.net/geod-addenda.html * * See the comments in geodesic.h for documentation. * * Copyright (c) Charles Karney (2012-2016) and licensed * under the MIT/X11 License. For more information, see * http://geographiclib.sourceforge.net/ */ #include "geodesic.h" #include #define GEOGRAPHICLIB_GEODESIC_ORDER 6 #define nA1 GEOGRAPHICLIB_GEODESIC_ORDER #define nC1 GEOGRAPHICLIB_GEODESIC_ORDER #define nC1p GEOGRAPHICLIB_GEODESIC_ORDER #define nA2 GEOGRAPHICLIB_GEODESIC_ORDER #define nC2 GEOGRAPHICLIB_GEODESIC_ORDER #define nA3 GEOGRAPHICLIB_GEODESIC_ORDER #define nA3x nA3 #define nC3 GEOGRAPHICLIB_GEODESIC_ORDER #define nC3x ((nC3 * (nC3 - 1)) / 2) #define nC4 GEOGRAPHICLIB_GEODESIC_ORDER #define nC4x ((nC4 * (nC4 + 1)) / 2) #define nC (GEOGRAPHICLIB_GEODESIC_ORDER + 1) typedef double real; typedef int boolx; static unsigned init = 0; static const int FALSE = 0; static const int TRUE = 1; static unsigned digits, maxit1, maxit2; static real epsilon, realmin, pi, degree, NaN, tiny, tol0, tol1, tol2, tolb, xthresh; static void Init() { if (!init) { #if defined(__DBL_MANT_DIG__) digits = __DBL_MANT_DIG__; #else digits = 53; #endif #if defined(__DBL_EPSILON__) epsilon = __DBL_EPSILON__; #else epsilon = pow(0.5, digits - 1); #endif #if defined(__DBL_MIN__) realmin = __DBL_MIN__; #else realmin = pow(0.5, 1022); #endif #if defined(M_PI) pi = M_PI; #else pi = atan2(0.0, -1.0); #endif maxit1 = 20; maxit2 = maxit1 + digits + 10; tiny = sqrt(realmin); tol0 = 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); /* Check on bisection interval */ tolb = tol0 * tol2; xthresh = 1000 * tol2; degree = pi/180; NaN = sqrt(-1.0); init = 1; } } 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, OUT_ALL = 0x7F80U }; static real sq(real x) { return x * x; } static real log1px(real x) { volatile real y = 1 + x, z = y - 1; /* Here's the explanation for this magic: y = 1 + z, exactly, and z * approx x, thus log(y)/z (which is nearly constant near z = 0) returns * a good approximation to the true log(1 + x)/x. The multiplication x * * (log(y)/z) introduces little additional error. */ return z == 0 ? x : x * log(y) / z; } static real atanhx(real x) { real y = fabs(x); /* Enforce odd parity */ y = log1px(2 * y/(1 - y))/2; return x < 0 ? -y : y; } static real copysignx(real x, real y) { return fabs(x) * (y < 0 || (y == 0 && 1/y < 0) ? -1 : 1); } static real hypotx(real x, real y) { return sqrt(x * x + y * y); } static real cbrtx(real x) { real y = pow(fabs(x), 1/(real)(3)); /* Return the real cube root */ return x < 0 ? -y : y; } static real sumx(real u, real v, real* t) { volatile real s = u + v; volatile real up = s - v; volatile real vpp = s - up; up -= u; vpp -= v; if (t) *t = -(up + vpp); /* error-free sum: * u + v = s + t * = round(u + v) + t */ return s; } static real polyval(int N, const real p[], real x) { real y = N < 0 ? 0 : *p++; while (--N >= 0) y = y * x + *p++; return y; } /* mimic C++ std::min and std::max */ static real minx(real a, real b) { return (b < a) ? b : a; } static real maxx(real a, real b) { return (a < b) ? b : a; } static void swapx(real* x, real* y) { real t = *x; *x = *y; *y = t; } static void norm2(real* sinx, real* cosx) { real r = hypotx(*sinx, *cosx); *sinx /= r; *cosx /= r; } static real AngNormalize(real x) { x = fmod(x, (real)(360)); return x < -180 ? x + 360 : (x < 180 ? x : x - 360); } static real LatFix(real x) { return fabs(x) > 90 ? NaN : x; } static real AngDiff(real x, real y, real* e) { real t, d = - AngNormalize(sumx(AngNormalize(x), AngNormalize(-y), &t)); /* Here y - x = d - t (mod 360), exactly, where d is in (-180,180] and * abs(t) <= eps (eps = 2^-45 for doubles). The only case where the * addition of t takes the result outside the range (-180,180] is d = 180 * and t < 0. The case, d = -180 + eps, t = eps, can't happen, since * sum would have returned the exact result in such a case (i.e., given t * = 0). */ return sumx(d == 180 && t < 0 ? -180 : d, -t, e); } static real AngRound(real x) { const real z = 1/(real)(16); if (x == 0) return 0; volatile real y = fabs(x); /* The compiler mustn't "simplify" z - (z - y) to y */ y = y < z ? z - (z - y) : y; return x < 0 ? -y : y; } static void sincosdx(real x, real* sinx, real* cosx) { /* In order to minimize round-off errors, this function exactly reduces * the argument to the range [-45, 45] before converting it to radians. */ real r, s, c; int q; r = fmod(x, (real)(360)); q = (int)(floor(r / 90 + (real)(0.5))); r -= 90 * q; /* now abs(r) <= 45 */ r *= degree; /* Possibly could call the gnu extension sincos */ s = sin(r); c = cos(r); switch ((unsigned)q & 3U) { case 0U: *sinx = s; *cosx = c; break; case 1U: *sinx = c; *cosx = 0 - s; break; case 2U: *sinx = 0 - s; *cosx = 0 - c; break; default: *sinx = 0 - c; *cosx = s; break; /* case 3U */ } } static real atan2dx(real y, real 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; real ang; if (fabs(y) > fabs(x)) { swapx(&x, &y); q = 2; } if (x < 0) { x = -x; ++q; } /* here x >= 0 and x >= abs(y), so angle is in [-pi/4, pi/4] */ ang = atan2(y, x) / degree; switch (q) { /* Note that atan2d(-0.0, 1.0) will return -0. However, we expect that * atan2d will not be called with y = -0. If need be, include * * case 0: ang = 0 + ang; break; */ case 1: ang = (y > 0 ? 180 : -180) - ang; break; case 2: ang = 90 - ang; break; case 3: ang = -90 + ang; break; } return ang; } static void A3coeff(struct geod_geodesic* g); static void C3coeff(struct geod_geodesic* g); static void C4coeff(struct geod_geodesic* g); static real SinCosSeries(boolx sinp, real sinx, real cosx, const real c[], int n); static void Lengths(const struct geod_geodesic* g, real eps, real sig12, real ssig1, real csig1, real dn1, real ssig2, real csig2, real dn2, real cbet1, real cbet2, real* ps12b, real* pm12b, real* pm0, real* pM12, real* pM21, /* Scratch area of the right size */ real Ca[]); static real Astroid(real x, real y); static real InverseStart(const struct geod_geodesic* g, real sbet1, real cbet1, real dn1, real sbet2, real cbet2, real dn2, real lam12, real slam12, real clam12, real* psalp1, real* pcalp1, /* Only updated if return val >= 0 */ real* psalp2, real* pcalp2, /* Only updated for short lines */ real* pdnm, /* Scratch area of the right size */ real Ca[]); static real Lambda12(const struct geod_geodesic* g, real sbet1, real cbet1, real dn1, real sbet2, real cbet2, real dn2, real salp1, real calp1, real slam120, real clam120, real* psalp2, real* pcalp2, real* psig12, real* pssig1, real* pcsig1, real* pssig2, real* pcsig2, real* peps, real* psomg12, real* pcomg12, boolx diffp, real* pdlam12, /* Scratch area of the right size */ real Ca[]); static real A3f(const struct geod_geodesic* g, real eps); static void C3f(const struct geod_geodesic* g, real eps, real c[]); static void C4f(const struct geod_geodesic* g, real eps, real c[]); 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[]); static int transit(real lon1, real lon2); static int transitdirect(real lon1, real lon2); static void accini(real s[]); static void acccopy(const real s[], real t[]); static void accadd(real s[], real y); static real accsum(const real s[], real y); static void accneg(real s[]); void geod_init(struct geod_geodesic* g, real a, real f) { if (!init) Init(); g->a = a; g->f = f; g->f1 = 1 - g->f; g->e2 = g->f * (2 - g->f); g->ep2 = g->e2 / sq(g->f1); /* e2 / (1 - e2) */ g->n = g->f / ( 2 - g->f); g->b = g->a * g->f1; g->c2 = (sq(g->a) + sq(g->b) * (g->e2 == 0 ? 1 : (g->e2 > 0 ? atanhx(sqrt(g->e2)) : atan(sqrt(-g->e2))) / sqrt(fabs(g->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. */ g->etol2 = 0.1 * tol2 / sqrt( maxx((real)(0.001), fabs(g->f)) * minx((real)(1), 1 - g->f/2) / 2 ); A3coeff(g); C3coeff(g); C4coeff(g); } static void geod_lineinit_int(struct geod_geodesicline* l, const struct geod_geodesic* g, real lat1, real lon1, real azi1, real salp1, real calp1, unsigned caps) { real cbet1, sbet1, eps; l->a = g->a; l->f = g->f; l->b = g->b; l->c2 = g->c2; l->f1 = g->f1; /* If caps is 0 assume the standard direct calculation */ l->caps = (caps ? caps : GEOD_DISTANCE_IN | GEOD_LONGITUDE) | /* always allow latitude and azimuth and unrolling of longitude */ GEOD_LATITUDE | GEOD_AZIMUTH | GEOD_LONG_UNROLL; l->lat1 = LatFix(lat1); l->lon1 = lon1; l->azi1 = azi1; l->salp1 = salp1; l->calp1 = calp1; sincosdx(AngRound(l->lat1), &sbet1, &cbet1); sbet1 *= l->f1; /* Ensure cbet1 = +epsilon at poles */ norm2(&sbet1, &cbet1); cbet1 = maxx(tiny, cbet1); l->dn1 = sqrt(1 + g->ep2 * sq(sbet1)); /* Evaluate alp0 from sin(alp1) * cos(bet1) = sin(alp0), */ l->salp0 = l->salp1 * cbet1; /* alp0 in [0, pi/2 - |bet1|] */ /* Alt: calp0 = hypot(sbet1, calp1 * cbet1). The following * is slightly better (consider the case salp1 = 0). */ l->calp0 = hypotx(l->calp1, l->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. */ l->ssig1 = sbet1; l->somg1 = l->salp0 * sbet1; l->csig1 = l->comg1 = sbet1 != 0 || l->calp1 != 0 ? cbet1 * l->calp1 : 1; norm2(&l->ssig1, &l->csig1); /* sig1 in (-pi, pi] */ /* norm2(somg1, comg1); -- don't need to normalize! */ l->k2 = sq(l->calp0) * g->ep2; eps = l->k2 / (2 * (1 + sqrt(1 + l->k2)) + l->k2); if (l->caps & CAP_C1) { real s, c; l->A1m1 = A1m1f(eps); C1f(eps, l->C1a); l->B11 = SinCosSeries(TRUE, l->ssig1, l->csig1, l->C1a, nC1); s = sin(l->B11); c = cos(l->B11); /* tau1 = sig1 + B11 */ l->stau1 = l->ssig1 * c + l->csig1 * s; l->ctau1 = l->csig1 * c - l->ssig1 * s; /* Not necessary because C1pa reverts C1a * B11 = -SinCosSeries(TRUE, stau1, ctau1, C1pa, nC1p); */ } if (l->caps & CAP_C1p) C1pf(eps, l->C1pa); if (l->caps & CAP_C2) { l->A2m1 = A2m1f(eps); C2f(eps, l->C2a); l->B21 = SinCosSeries(TRUE, l->ssig1, l->csig1, l->C2a, nC2); } if (l->caps & CAP_C3) { C3f(g, eps, l->C3a); l->A3c = -l->f * l->salp0 * A3f(g, eps); l->B31 = SinCosSeries(TRUE, l->ssig1, l->csig1, l->C3a, nC3-1); } if (l->caps & CAP_C4) { C4f(g, eps, l->C4a); /* Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0) */ l->A4 = sq(l->a) * l->calp0 * l->salp0 * g->e2; l->B41 = SinCosSeries(FALSE, l->ssig1, l->csig1, l->C4a, nC4); } l->a13 = l->s13 = NaN; } void geod_lineinit(struct geod_geodesicline* l, const struct geod_geodesic* g, real lat1, real lon1, real azi1, unsigned caps) { azi1 = AngNormalize(azi1); real salp1, calp1; /* Guard against underflow in salp0 */ sincosdx(AngRound(azi1), &salp1, &calp1); geod_lineinit_int(l, g, lat1, lon1, azi1, salp1, calp1, caps); } void geod_gendirectline(struct geod_geodesicline* l, const struct geod_geodesic* g, real lat1, real lon1, real azi1, unsigned flags, real a12_s12, unsigned caps) { geod_lineinit(l, g, lat1, lon1, azi1, caps); geod_gensetdistance(l, flags, a12_s12); } void geod_directline(struct geod_geodesicline* l, const struct geod_geodesic* g, real lat1, real lon1, real azi1, real s12, unsigned caps) { geod_gendirectline(l, g, lat1, lon1, azi1, GEOD_NOFLAGS, s12, caps); } real geod_genposition(const struct geod_geodesicline* l, unsigned flags, real s12_a12, real* plat2, real* plon2, real* pazi2, real* ps12, real* pm12, real* pM12, real* pM21, real* pS12) { real lat2 = 0, lon2 = 0, azi2 = 0, s12 = 0, m12 = 0, M12 = 0, M21 = 0, S12 = 0; /* Avoid warning about uninitialized B12. */ real sig12, ssig12, csig12, B12 = 0, AB1 = 0; real omg12, lam12, lon12; real ssig2, csig2, sbet2, cbet2, somg2, comg2, salp2, calp2, dn2; unsigned outmask = (plat2 ? GEOD_LATITUDE : 0U) | (plon2 ? GEOD_LONGITUDE : 0U) | (pazi2 ? GEOD_AZIMUTH : 0U) | (ps12 ? GEOD_DISTANCE : 0U) | (pm12 ? GEOD_REDUCEDLENGTH : 0U) | (pM12 || pM21 ? GEOD_GEODESICSCALE : 0U) | (pS12 ? GEOD_AREA : 0U); outmask &= l->caps & OUT_ALL; if (!( TRUE /*Init()*/ && (flags & GEOD_ARCMODE || (l->caps & (GEOD_DISTANCE_IN & OUT_ALL))) )) /* Uninitialized or impossible distance calculation requested */ return NaN; if (flags & GEOD_ARCMODE) { /* Interpret s12_a12 as spherical arc length */ sig12 = s12_a12 * degree; sincosdx(s12_a12, &ssig12, &csig12); } else { /* Interpret s12_a12 as distance */ real tau12 = s12_a12 / (l->b * (1 + l->A1m1)), s = sin(tau12), c = cos(tau12); /* tau2 = tau1 + tau12 */ B12 = - SinCosSeries(TRUE, l->stau1 * c + l->ctau1 * s, l->ctau1 * c - l->stau1 * s, l->C1pa, nC1p); sig12 = tau12 - (B12 - l->B11); ssig12 = sin(sig12); csig12 = cos(sig12); if (fabs(l->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 serr; ssig2 = l->ssig1 * csig12 + l->csig1 * ssig12; csig2 = l->csig1 * csig12 - l->ssig1 * ssig12; B12 = SinCosSeries(TRUE, ssig2, csig2, l->C1a, nC1); serr = (1 + l->A1m1) * (sig12 + (B12 - l->B11)) - s12_a12 / l->b; sig12 = sig12 - serr / sqrt(1 + l->k2 * sq(ssig2)); ssig12 = sin(sig12); csig12 = cos(sig12); /* Update B12 below */ } } /* sig2 = sig1 + sig12 */ ssig2 = l->ssig1 * csig12 + l->csig1 * ssig12; csig2 = l->csig1 * csig12 - l->ssig1 * ssig12; dn2 = sqrt(1 + l->k2 * sq(ssig2)); if (outmask & (GEOD_DISTANCE | GEOD_REDUCEDLENGTH | GEOD_GEODESICSCALE)) { if (flags & GEOD_ARCMODE || fabs(l->f) > 0.01) B12 = SinCosSeries(TRUE, ssig2, csig2, l->C1a, nC1); AB1 = (1 + l->A1m1) * (B12 - l->B11); } /* sin(bet2) = cos(alp0) * sin(sig2) */ sbet2 = l->calp0 * ssig2; /* Alt: cbet2 = hypot(csig2, salp0 * ssig2); */ cbet2 = hypotx(l->salp0, l->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 = l->salp0; calp2 = l->calp0 * csig2; /* No need to normalize */ if (outmask & GEOD_DISTANCE) s12 = flags & GEOD_ARCMODE ? l->b * ((1 + l->A1m1) * sig12 + AB1) : s12_a12; if (outmask & GEOD_LONGITUDE) { real E = copysignx(1, l->salp0); /* east or west going? */ /* tan(omg2) = sin(alp0) * tan(sig2) */ somg2 = l->salp0 * ssig2; comg2 = csig2; /* No need to normalize */ /* omg12 = omg2 - omg1 */ omg12 = flags & GEOD_LONG_UNROLL ? E * (sig12 - (atan2( ssig2, csig2) - atan2( l->ssig1, l->csig1)) + (atan2(E * somg2, comg2) - atan2(E * l->somg1, l->comg1))) : atan2(somg2 * l->comg1 - comg2 * l->somg1, comg2 * l->comg1 + somg2 * l->somg1); lam12 = omg12 + l->A3c * ( sig12 + (SinCosSeries(TRUE, ssig2, csig2, l->C3a, nC3-1) - l->B31)); lon12 = lam12 / degree; lon2 = flags & GEOD_LONG_UNROLL ? l->lon1 + lon12 : AngNormalize(AngNormalize(l->lon1) + AngNormalize(lon12)); } if (outmask & GEOD_LATITUDE) lat2 = atan2dx(sbet2, l->f1 * cbet2); if (outmask & GEOD_AZIMUTH) azi2 = atan2dx(salp2, calp2); if (outmask & (GEOD_REDUCEDLENGTH | GEOD_GEODESICSCALE)) { real B22 = SinCosSeries(TRUE, ssig2, csig2, l->C2a, nC2), AB2 = (1 + l->A2m1) * (B22 - l->B21), J12 = (l->A1m1 - l->A2m1) * sig12 + (AB1 - AB2); if (outmask & GEOD_REDUCEDLENGTH) /* Add parens around (csig1 * ssig2) and (ssig1 * csig2) to ensure * accurate cancellation in the case of coincident points. */ m12 = l->b * ((dn2 * (l->csig1 * ssig2) - l->dn1 * (l->ssig1 * csig2)) - l->csig1 * csig2 * J12); if (outmask & GEOD_GEODESICSCALE) { real t = l->k2 * (ssig2 - l->ssig1) * (ssig2 + l->ssig1) / (l->dn1 + dn2); M12 = csig12 + (t * ssig2 - csig2 * J12) * l->ssig1 / l->dn1; M21 = csig12 - (t * l->ssig1 - l->csig1 * J12) * ssig2 / dn2; } } if (outmask & GEOD_AREA) { real B42 = SinCosSeries(FALSE, ssig2, csig2, l->C4a, nC4); real salp12, calp12; if (l->calp0 == 0 || l->salp0 == 0) { /* alp12 = alp2 - alp1, used in atan2 so no need to normalize */ salp12 = salp2 * l->calp1 - calp2 * l->salp1; calp12 = calp2 * l->calp1 + salp2 * l->salp1; } 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 = l->calp0 * l->salp0 * (csig12 <= 0 ? l->csig1 * (1 - csig12) + ssig12 * l->ssig1 : ssig12 * (l->csig1 * ssig12 / (1 + csig12) + l->ssig1)); calp12 = sq(l->salp0) + sq(l->calp0) * l->csig1 * csig2; } S12 = l->c2 * atan2(salp12, calp12) + l->A4 * (B42 - l->B41); } if (outmask & GEOD_LATITUDE) *plat2 = lat2; if (outmask & GEOD_LONGITUDE) *plon2 = lon2; if (outmask & GEOD_AZIMUTH) *pazi2 = azi2; if (outmask & GEOD_DISTANCE) *ps12 = s12; if (outmask & GEOD_REDUCEDLENGTH) *pm12 = m12; if (outmask & GEOD_GEODESICSCALE) { if (pM12) *pM12 = M12; if (pM21) *pM21 = M21; } if (outmask & GEOD_AREA) *pS12 = S12; return flags & GEOD_ARCMODE ? s12_a12 : sig12 / degree; } void geod_setdistance(struct geod_geodesicline* l, real s13) { l->s13 = s13; l->a13 = geod_genposition(l, GEOD_NOFLAGS, l->s13, 0, 0, 0, 0, 0, 0, 0, 0); } static void geod_setarc(struct geod_geodesicline* l, real a13) { l->a13 = a13; l->s13 = NaN; geod_genposition(l, GEOD_ARCMODE, l->a13, 0, 0, 0, &l->s13, 0, 0, 0, 0); } void geod_gensetdistance(struct geod_geodesicline* l, unsigned flags, real s13_a13) { flags & GEOD_ARCMODE ? geod_setarc(l, s13_a13) : geod_setdistance(l, s13_a13); } void geod_position(const struct geod_geodesicline* l, real s12, real* plat2, real* plon2, real* pazi2) { geod_genposition(l, FALSE, s12, plat2, plon2, pazi2, 0, 0, 0, 0, 0); } real geod_gendirect(const struct geod_geodesic* g, real lat1, real lon1, real azi1, unsigned flags, real s12_a12, real* plat2, real* plon2, real* pazi2, real* ps12, real* pm12, real* pM12, real* pM21, real* pS12) { struct geod_geodesicline l; unsigned outmask = (plat2 ? GEOD_LATITUDE : 0U) | (plon2 ? GEOD_LONGITUDE : 0U) | (pazi2 ? GEOD_AZIMUTH : 0U) | (ps12 ? GEOD_DISTANCE : 0U) | (pm12 ? GEOD_REDUCEDLENGTH : 0U) | (pM12 || pM21 ? GEOD_GEODESICSCALE : 0U) | (pS12 ? GEOD_AREA : 0U); geod_lineinit(&l, g, lat1, lon1, azi1, /* Automatically supply GEOD_DISTANCE_IN if necessary */ outmask | (flags & GEOD_ARCMODE ? GEOD_NONE : GEOD_DISTANCE_IN)); return geod_genposition(&l, flags, s12_a12, plat2, plon2, pazi2, ps12, pm12, pM12, pM21, pS12); } void geod_direct(const struct geod_geodesic* g, real lat1, real lon1, real azi1, real s12, real* plat2, real* plon2, real* pazi2) { geod_gendirect(g, lat1, lon1, azi1, GEOD_NOFLAGS, s12, plat2, plon2, pazi2, 0, 0, 0, 0, 0); } static real geod_geninverse_int(const struct geod_geodesic* g, real lat1, real lon1, real lat2, real lon2, real* ps12, real* psalp1, real* pcalp1, real* psalp2, real* pcalp2, real* pm12, real* pM12, real* pM21, real* pS12) { real s12 = 0, m12 = 0, M12 = 0, M21 = 0, S12 = 0; real lon12, lon12s; int latsign, lonsign, swapp; real sbet1, cbet1, sbet2, cbet2, s12x = 0, m12x = 0; real dn1, dn2, lam12, slam12, clam12; real a12 = 0, sig12, calp1 = 0, salp1 = 0, calp2 = 0, salp2 = 0; real Ca[nC]; boolx meridian; /* somg12 > 1 marks that it needs to be calculated */ real omg12 = 0, somg12 = 2, comg12 = 0; unsigned outmask = (ps12 ? GEOD_DISTANCE : 0U) | (pm12 ? GEOD_REDUCEDLENGTH : 0U) | (pM12 || pM21 ? GEOD_GEODESICSCALE : 0U) | (pS12 ? GEOD_AREA : 0U); outmask &= OUT_ALL; /* 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. */ lon12 = AngDiff(lon1, lon2, &lon12s); /* Make longitude difference positive. */ lonsign = lon12 >= 0 ? 1 : -1; /* If very close to being on the same half-meridian, then make it so. */ lon12 = lonsign * AngRound(lon12); lon12s = AngRound((180 - lon12) - lonsign * lon12s); lam12 = lon12 * degree; if (lon12 > 90) { sincosdx(lon12s, &slam12, &clam12); clam12 = -clam12; } else sincosdx(lon12, &slam12, &clam12); /* If really close to the equator, treat as on equator. */ lat1 = AngRound(LatFix(lat1)); lat2 = AngRound(LatFix(lat2)); /* Swap points so that point with higher (abs) latitude is point 1 * If one latitude is a nan, then it becomes lat1. */ swapp = fabs(lat1) < fabs(lat2) ? -1 : 1; if (swapp < 0) { lonsign *= -1; swapx(&lat1, &lat2); } /* Make lat1 <= 0 */ latsign = lat1 < 0 ? 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. */ sincosdx(lat1, &sbet1, &cbet1); sbet1 *= g->f1; /* Ensure cbet1 = +epsilon at poles */ norm2(&sbet1, &cbet1); cbet1 = maxx(tiny, cbet1); sincosdx(lat2, &sbet2, &cbet2); sbet2 *= g->f1; /* Ensure cbet2 = +epsilon at poles */ norm2(&sbet2, &cbet2); cbet2 = maxx(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 = sbet2 < 0 ? sbet1 : -sbet1; } else { if (fabs(sbet2) == -sbet1) cbet2 = cbet1; } dn1 = sqrt(1 + g->ep2 * sq(sbet1)); dn2 = sqrt(1 + g->ep2 * sq(sbet2)); meridian = lat1 == -90 || slam12 == 0; if (meridian) { /* Endpoints are on a single full meridian, so the geodesic might lie on * a meridian. */ real ssig1, csig1, ssig2, csig2; calp1 = clam12; salp1 = slam12; /* Head to the target longitude */ calp2 = 1; salp2 = 0; /* At the target we're heading north */ /* tan(bet) = tan(sig) * cos(alp) */ ssig1 = sbet1; csig1 = calp1 * cbet1; ssig2 = sbet2; csig2 = calp2 * cbet2; /* sig12 = sig2 - sig1 */ sig12 = atan2(maxx((real)(0), csig1 * ssig2 - ssig1 * csig2), csig1 * csig2 + ssig1 * ssig2); Lengths(g, g->n, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2, &s12x, &m12x, 0, outmask & GEOD_GEODESICSCALE ? &M12 : 0, outmask & GEOD_GEODESICSCALE ? &M21 : 0, 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. */ if (sig12 < 1 || m12x >= 0) { /* Need at least 2, to handle 90 0 90 180 */ if (sig12 < 3 * tiny) sig12 = m12x = s12x = 0; m12x *= g->b; s12x *= g->b; a12 = sig12 / degree; } else /* m12 < 0, i.e., prolate and too close to anti-podal */ meridian = FALSE; } if (!meridian && sbet1 == 0 && /* and sbet2 == 0 */ /* Mimic the way Lambda12 works with calp1 = 0 */ (g->f <= 0 || lon12s >= g->f * 180)) { /* Geodesic runs along equator */ calp1 = calp2 = 0; salp1 = salp2 = 1; s12x = g->a * lam12; sig12 = omg12 = lam12 / g->f1; m12x = g->b * sin(sig12); if (outmask & GEOD_GEODESICSCALE) M12 = M21 = cos(sig12); a12 = lon12 / g->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 = 0; sig12 = InverseStart(g, 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 * g->b * dnm; m12x = sq(dnm) * g->b * sin(sig12 / dnm); if (outmask & GEOD_GEODESICSCALE) M12 = M21 = cos(sig12 / dnm); a12 = sig12 / degree; omg12 = lam12 / (g->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. */ real ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0, eps = 0; unsigned numit = 0; /* Bracketing range */ real salp1a = tiny, calp1a = 1, salp1b = tiny, calp1b = -1; boolx tripn, tripb; for (tripn = FALSE, tripb = FALSE; numit < maxit2; ++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 = 0, v = Lambda12(g, sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1, calp1, slam12, clam12, &salp2, &calp2, &sig12, &ssig1, &csig1, &ssig2, &csig2, &eps, &somg12, &comg12, numit < maxit1, &dv, Ca); /* 2 * tol0 is approximately 1 ulp for a number in [0, pi]. */ /* Reversed test to allow escape with NaNs */ if (tripb || !(fabs(v) >= (tripn ? 8 : 1) * tol0)) 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; real sdalp1 = sin(dalp1), cdalp1 = cos(dalp1), nsalp1 = salp1 * cdalp1 + calp1 * sdalp1; if (nsalp1 > 0 && fabs(dalp1) < pi) { calp1 = calp1 * cdalp1 - salp1 * sdalp1; salp1 = nsalp1; norm2(&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 postive 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; norm2(&salp1, &calp1); tripn = FALSE; tripb = (fabs(salp1a - salp1) + (calp1a - calp1) < tolb || fabs(salp1 - salp1b) + (calp1 - calp1b) < tolb); } Lengths(g, eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2, &s12x, &m12x, 0, outmask & GEOD_GEODESICSCALE ? &M12 : 0, outmask & GEOD_GEODESICSCALE ? &M21 : 0, Ca); m12x *= g->b; s12x *= g->b; a12 = sig12 / degree; } } if (outmask & GEOD_DISTANCE) s12 = 0 + s12x; /* Convert -0 to 0 */ if (outmask & GEOD_REDUCEDLENGTH) m12 = 0 + m12x; /* Convert -0 to 0 */ if (outmask & GEOD_AREA) { real /* From Lambda12: sin(alp1) * cos(bet1) = sin(alp0) */ salp0 = salp1 * cbet1, calp0 = hypotx(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 = sq(calp0) * g->ep2, eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2), /* Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0). */ A4 = sq(g->a) * calp0 * salp0 * g->e2; real B41, B42; norm2(&ssig1, &csig1); norm2(&ssig2, &csig2); C4f(g, eps, Ca); 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) { if (somg12 > 1) { somg12 = sin(omg12); comg12 = cos(omg12); } else norm2(&somg12, &comg12); } 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 += g->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) { swapx(&salp1, &salp2); swapx(&calp1, &calp2); if (outmask & GEOD_GEODESICSCALE) swapx(&M12, &M21); } salp1 *= swapp * lonsign; calp1 *= swapp * latsign; salp2 *= swapp * lonsign; calp2 *= swapp * latsign; if (psalp1) *psalp1 = salp1; if (pcalp1) *pcalp1 = calp1; if (psalp2) *psalp2 = salp2; if (pcalp2) *pcalp2 = calp2; if (outmask & GEOD_DISTANCE) *ps12 = s12; if (outmask & GEOD_REDUCEDLENGTH) *pm12 = m12; if (outmask & GEOD_GEODESICSCALE) { if (pM12) *pM12 = M12; if (pM21) *pM21 = M21; } if (outmask & GEOD_AREA) *pS12 = S12; /* Returned value in [0, 180] */ return a12; } real geod_geninverse(const struct geod_geodesic* g, real lat1, real lon1, real lat2, real lon2, real* ps12, real* pazi1, real* pazi2, real* pm12, real* pM12, real* pM21, real* pS12) { real salp1, calp1, salp2, calp2, a12 = geod_geninverse_int(g, lat1, lon1, lat2, lon2, ps12, &salp1, &calp1, &salp2, &calp2, pm12, pM12, pM21, pS12); if (pazi1) *pazi1 = atan2dx(salp1, calp1); if (pazi2) *pazi2 = atan2dx(salp2, calp2); return a12; } void geod_inverseline(struct geod_geodesicline* l, const struct geod_geodesic* g, real lat1, real lon1, real lat2, real lon2, unsigned caps) { real salp1, calp1, a12 = geod_geninverse_int(g, lat1, lon1, lat2, lon2, 0, &salp1, &calp1, 0, 0, 0, 0, 0, 0), azi1 = atan2dx(salp1, calp1); caps = caps ? caps : GEOD_DISTANCE_IN | GEOD_LONGITUDE; /* Ensure that a12 can be converted to a distance */ if (caps & (OUT_ALL & GEOD_DISTANCE_IN)) caps |= GEOD_DISTANCE; geod_lineinit_int(l, g, lat1, lon1, azi1, salp1, calp1, caps); geod_setarc(l, a12); } void geod_inverse(const struct geod_geodesic* g, real lat1, real lon1, real lat2, real lon2, real* ps12, real* pazi1, real* pazi2) { geod_geninverse(g, lat1, lon1, lat2, lon2, ps12, pazi1, pazi2, 0, 0, 0, 0); } real SinCosSeries(boolx 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 */ real ar, y0, y1; c += (n + sinp); /* Point to one beyond last element */ 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) */ } void Lengths(const struct geod_geodesic* g, real eps, real sig12, real ssig1, real csig1, real dn1, real ssig2, real csig2, real dn2, real cbet1, real cbet2, real* ps12b, real* pm12b, real* pm0, real* pM12, real* pM21, /* Scratch area of the right size */ real Ca[]) { real m0 = 0, J12 = 0, A1 = 0, A2 = 0; real Cb[nC]; /* Return m12b = (reduced length)/b; also calculate s12b = distance/b, * and m0 = coefficient of secular term in expression for reduced length. */ boolx redlp = pm12b || pm0 || pM12 || pM21; if (ps12b || redlp) { A1 = A1m1f(eps); C1f(eps, Ca); if (redlp) { A2 = A2m1f(eps); C2f(eps, Cb); m0 = A1 - A2; A2 = 1 + A2; } A1 = 1 + A1; } if (ps12b) { real B1 = SinCosSeries(TRUE, ssig2, csig2, Ca, nC1) - SinCosSeries(TRUE, ssig1, csig1, Ca, nC1); /* Missing a factor of b */ *ps12b = A1 * (sig12 + B1); if (redlp) { real B2 = SinCosSeries(TRUE, ssig2, csig2, Cb, nC2) - SinCosSeries(TRUE, ssig1, csig1, Cb, nC2); J12 = m0 * sig12 + (A1 * B1 - A2 * B2); } } else if (redlp) { /* Assume here that nC1 >= nC2 */ int l; for (l = 1; l <= nC2; ++l) Cb[l] = A1 * Ca[l] - A2 * Cb[l]; J12 = m0 * sig12 + (SinCosSeries(TRUE, ssig2, csig2, Cb, nC2) - SinCosSeries(TRUE, ssig1, csig1, Cb, nC2)); } if (pm0) *pm0 = m0; if (pm12b) /* Missing a factor of b. * Add parens around (csig1 * ssig2) and (ssig1 * csig2) to ensure * accurate cancellation in the case of coincident points. */ *pm12b = dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) - csig1 * csig2 * J12; if (pM12 || pM21) { real csig12 = csig1 * csig2 + ssig1 * ssig2; real t = g->ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2); if (pM12) *pM12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1; if (pM21) *pM21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2; } } real 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 = sq(x), q = 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 = 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; real v, uv, w; if (disc >= 0) { real T3 = S + r3, T; /* 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. cbrtx always returns the real root. cbrtx(-8) = -2. */ T = cbrtx(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); } v = sqrt(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 + 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; } real InverseStart(const struct geod_geodesic* g, real sbet1, real cbet1, real dn1, real sbet2, real cbet2, real dn2, real lam12, real slam12, real clam12, real* psalp1, real* pcalp1, /* Only updated if return val >= 0 */ real* psalp2, real* pcalp2, /* Only updated for short lines */ real* pdnm, /* Scratch area of the right size */ real Ca[]) { real salp1 = 0, calp1 = 0, salp2 = 0, calp2 = 0, dnm = 0; /* 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; boolx shortline = cbet12 >= 0 && sbet12 < (real)(0.5) && cbet2 * lam12 < (real)(0.5); real somg12, comg12, ssig12, csig12; #if defined(__GNUC__) && __GNUC__ == 4 && \ (__GNUC_MINOR__ < 6 || defined(__MINGW32__)) /* Volatile declaration needed to fix inverse cases * 88.202499451857 0 -88.202499451857 179.981022032992859592 * 89.262080389218 0 -89.262080389218 179.992207982775375662 * 89.333123580033 0 -89.333123580032997687 179.99295812360148422 * which otherwise fail with g++ 4.4.4 x86 -O3 (Linux) * and g++ 4.4.0 (mingw) and g++ 4.6.1 (tdm mingw). */ { volatile real xx1 = sbet2 * cbet1; volatile real xx2 = cbet2 * sbet1; sbet12a = xx1 + xx2; } #else sbet12a = sbet2 * cbet1 + cbet2 * sbet1; #endif if (shortline) { real sbetm2 = sq(sbet1 + sbet2), omg12; /* sin((bet1+bet2)/2)^2 * = (sbet1 + sbet2)^2 / ((sbet1 + sbet2)^2 + (cbet1 + cbet2)^2) */ sbetm2 /= sbetm2 + sq(cbet1 + cbet2); dnm = sqrt(1 + g->ep2 * sbetm2); omg12 = lam12 / (g->f1 * dnm); somg12 = sin(omg12); comg12 = cos(omg12); } else { somg12 = slam12; comg12 = clam12; } salp1 = cbet2 * somg12; calp1 = comg12 >= 0 ? sbet12 + cbet2 * sbet1 * sq(somg12) / (1 + comg12) : sbet12a - cbet2 * sbet1 * sq(somg12) / (1 - comg12); ssig12 = hypotx(salp1, calp1); csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12; if (shortline && ssig12 < g->etol2) { /* really short lines */ salp2 = cbet1 * somg12; calp2 = sbet12 - cbet1 * sbet2 * (comg12 >= 0 ? sq(somg12) / (1 + comg12) : 1 - comg12); norm2(&salp2, &calp2); /* Set return value */ sig12 = atan2(ssig12, csig12); } else if (fabs(g->n) > (real)(0.1) || /* No astroid calc if too eccentric */ csig12 >= 0 || ssig12 >= 6 * fabs(g->n) * pi * 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 y, lamscale, betscale; /* Volatile declaration needed to fix inverse case * 56.320923501171 0 -56.320923501171 179.664747671772880215 * which otherwise fails with g++ 4.4.4 x86 -O3 */ volatile real x; real lam12x = atan2(-slam12, -clam12); /* lam12 - pi */ if (g->f >= 0) { /* In fact f == 0 does not get here */ /* x = dlong, y = dlat */ { real k2 = sq(sbet1) * g->ep2, eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2); lamscale = g->f * cbet1 * A3f(g, eps) * 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; /* In the case of lon12 = 180, this repeats a calculation made in * Inverse. */ Lengths(g, g->n, pi + bet12a, sbet1, -cbet1, dn1, sbet2, cbet2, dn2, cbet1, cbet2, 0, &m12b, &m0, 0, 0, Ca); x = -1 + m12b / (cbet1 * cbet2 * m0 * pi); betscale = x < -(real)(0.01) ? sbet12a / x : -g->f * sq(cbet1) * pi; lamscale = betscale / cbet1; y = lam12x / lamscale; } if (y > -tol1 && x > -1 - xthresh) { /* strip near cut */ if (g->f >= 0) { salp1 = minx((real)(1), -(real)(x)); calp1 = - sqrt(1 - sq(salp1)); } else { calp1 = maxx((real)(x > -tol1 ? 0 : -1), (real)(x)); salp1 = sqrt(1 - 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 * ( g->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 * sq(somg12) / (1 - comg12); } } /* Sanity check on starting guess. Backwards check allows NaN through. */ if (!(salp1 <= 0)) norm2(&salp1, &calp1); else { salp1 = 1; calp1 = 0; } *psalp1 = salp1; *pcalp1 = calp1; if (shortline) *pdnm = dnm; if (sig12 >= 0) { *psalp2 = salp2; *pcalp2 = calp2; } return sig12; } real Lambda12(const struct geod_geodesic* g, real sbet1, real cbet1, real dn1, real sbet2, real cbet2, real dn2, real salp1, real calp1, real slam120, real clam120, real* psalp2, real* pcalp2, real* psig12, real* pssig1, real* pcsig1, real* pssig2, real* pcsig2, real* peps, real* psomg12, real* pcomg12, boolx diffp, real* pdlam12, /* Scratch area of the right size */ real Ca[]) { real salp2 = 0, calp2 = 0, sig12 = 0, ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0, eps = 0, somg12 = 0, comg12 = 0, dlam12 = 0; real salp0, calp0; real somg1, comg1, somg2, comg2, lam12; real B312, eta, k2; if (sbet1 == 0 && calp1 == 0) /* Break degeneracy of equatorial line. This case has already been * handled. */ calp1 = -tiny; /* sin(alp1) * cos(bet1) = sin(alp0) */ salp0 = salp1 * cbet1; calp0 = hypotx(calp1, salp1 * sbet1); /* calp0 > 0 */ /* 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; norm2(&ssig1, &csig1); /* norm2(&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(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; norm2(&ssig2, &csig2); /* norm2(&somg2, &comg2); -- don't need to normalize! */ /* sig12 = sig2 - sig1, limit to [0, pi] */ sig12 = atan2(maxx((real)(0), csig1 * ssig2 - ssig1 * csig2), csig1 * csig2 + ssig1 * ssig2); /* omg12 = omg2 - omg1, limit to [0, pi] */ somg12 = maxx((real)(0), comg1 * somg2 - somg1 * comg2); comg12 = comg1 * comg2 + somg1 * somg2; /* eta = omg12 - lam120 */ eta = atan2(somg12 * clam120 - comg12 * slam120, comg12 * clam120 + somg12 * slam120); k2 = sq(calp0) * g->ep2; eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2); C3f(g, eps, Ca); B312 = (SinCosSeries(TRUE, ssig2, csig2, Ca, nC3-1) - SinCosSeries(TRUE, ssig1, csig1, Ca, nC3-1)); lam12 = eta - g->f * A3f(g, eps) * salp0 * (sig12 + B312); if (diffp) { if (calp2 == 0) dlam12 = - 2 * g->f1 * dn1 / sbet1; else { Lengths(g, eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2, 0, &dlam12, 0, 0, 0, Ca); dlam12 *= g->f1 / (calp2 * cbet2); } } *psalp2 = salp2; *pcalp2 = calp2; *psig12 = sig12; *pssig1 = ssig1; *pcsig1 = csig1; *pssig2 = ssig2; *pcsig2 = csig2; *peps = eps; *psomg12 = somg12; *pcomg12 = comg12; if (diffp) *pdlam12 = dlam12; return lam12; } real A3f(const struct geod_geodesic* g, real eps) { /* Evaluate A3 */ return polyval(nA3 - 1, g->A3x, eps); } void C3f(const struct geod_geodesic* g, real eps, real c[]) { /* Evaluate C3 coeffs * Elements c[1] thru c[nC3 - 1] are set */ real mult = 1; int o = 0, l; for (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 * polyval(m, g->C3x + o, eps); o += m + 1; } } void C4f(const struct geod_geodesic* g, real eps, real c[]) { /* Evaluate C4 coeffs * Elements c[0] thru c[nC4 - 1] are set */ real mult = 1; int o = 0, l; for (l = 0; l < nC4; ++l) { /* l is index of C4[l] */ int m = nC4 - l - 1; /* order of polynomial in eps */ c[l] = mult * polyval(m, g->C4x + o, eps); o += m + 1; mult *= eps; } } /* The scale factor A1-1 = mean value of (d/dsigma)I1 - 1 */ real A1m1f(real eps) { static const real coeff[] = { /* (1-eps)*A1-1, polynomial in eps2 of order 3 */ 1, 4, 64, 0, 256, }; int m = nA1/2; real t = polyval(m, coeff, sq(eps)) / coeff[m + 1]; return (t + eps) / (1 - eps); } /* The coefficients C1[l] in the Fourier expansion of B1 */ void C1f(real eps, real c[]) { 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, }; real eps2 = sq(eps), d = eps; int o = 0, l; for (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 * polyval(m, coeff + o, eps2) / coeff[o + m + 1]; o += m + 2; d *= eps; } } /* The coefficients C1p[l] in the Fourier expansion of B1p */ void C1pf(real eps, real c[]) { 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, }; real eps2 = sq(eps), d = eps; int o = 0, l; for (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 * polyval(m, coeff + o, eps2) / coeff[o + m + 1]; o += m + 2; d *= eps; } } /* The scale factor A2-1 = mean value of (d/dsigma)I2 - 1 */ real A2m1f(real eps) { static const real coeff[] = { /* (eps+1)*A2-1, polynomial in eps2 of order 3 */ -11, -28, -192, 0, 256, }; int m = nA2/2; real t = polyval(m, coeff, sq(eps)) / coeff[m + 1]; return (t - eps) / (1 + eps); } /* The coefficients C2[l] in the Fourier expansion of B2 */ void C2f(real eps, real c[]) { 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, }; real eps2 = sq(eps), d = eps; int o = 0, l; for (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 * polyval(m, coeff + o, eps2) / coeff[o + m + 1]; o += m + 2; d *= eps; } } /* The scale factor A3 = mean value of (d/dsigma)I3 */ void A3coeff(struct geod_geodesic* g) { 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, }; int o = 0, k = 0, j; for (j = nA3 - 1; j >= 0; --j) { /* coeff of eps^j */ int m = nA3 - j - 1 < j ? nA3 - j - 1 : j; /* order of polynomial in n */ g->A3x[k++] = polyval(m, coeff + o, g->n) / coeff[o + m + 1]; o += m + 2; } } /* The coefficients C3[l] in the Fourier expansion of B3 */ void C3coeff(struct geod_geodesic* g) { 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, }; int o = 0, k = 0, l, j; for (l = 1; l < nC3; ++l) { /* l is index of C3[l] */ for (j = nC3 - 1; j >= l; --j) { /* coeff of eps^j */ int m = nC3 - j - 1 < j ? nC3 - j - 1 : j; /* order of polynomial in n */ g->C3x[k++] = polyval(m, coeff + o, g->n) / coeff[o + m + 1]; o += m + 2; } } } /* The coefficients C4[l] in the Fourier expansion of I4 */ void C4coeff(struct geod_geodesic* g) { 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, }; int o = 0, k = 0, l, j; for (l = 0; l < nC4; ++l) { /* l is index of C4[l] */ for (j = nC4 - 1; j >= l; --j) { /* coeff of eps^j */ int m = nC4 - j - 1; /* order of polynomial in n */ g->C4x[k++] = polyval(m, coeff + o, g->n) / coeff[o + m + 1]; o += m + 2; } } } int transit(real lon1, real lon2) { real lon12; /* Return 1 or -1 if crossing prime meridian in east or west direction. * Otherwise return zero. */ /* Compute lon12 the same way as Geodesic::Inverse. */ lon1 = AngNormalize(lon1); lon2 = AngNormalize(lon2); lon12 = AngDiff(lon1, lon2, 0); return lon1 < 0 && lon2 >= 0 && lon12 > 0 ? 1 : (lon2 < 0 && lon1 >= 0 && lon12 < 0 ? -1 : 0); } int transitdirect(real lon1, real lon2) { lon1 = fmod(lon1, (real)(720)); lon2 = fmod(lon2, (real)(720)); return ( ((lon2 >= 0 && lon2 < 360) || lon2 < -360 ? 0 : 1) - ((lon1 >= 0 && lon1 < 360) || lon1 < -360 ? 0 : 1) ); } void accini(real s[]) { /* Initialize an accumulator; this is an array with two elements. */ s[0] = s[1] = 0; } void acccopy(const real s[], real t[]) { /* Copy an accumulator; t = s. */ t[0] = s[0]; t[1] = s[1]; } void accadd(real s[], real y) { /* Add y to an accumulator. */ real u, z = sumx(y, s[1], &u); s[0] = sumx(z, s[0], &s[1]); if (s[0] == 0) s[0] = u; else s[1] = s[1] + u; } real accsum(const real s[], real y) { /* Return accumulator + y (but don't add to accumulator). */ real t[2]; acccopy(s, t); accadd(t, y); return t[0]; } void accneg(real s[]) { /* Negate an accumulator. */ s[0] = -s[0]; s[1] = -s[1]; } void geod_polygon_init(struct geod_polygon* p, boolx polylinep) { p->polyline = (polylinep != 0); geod_polygon_clear(p); } void geod_polygon_clear(struct geod_polygon* p) { p->lat0 = p->lon0 = p->lat = p->lon = NaN; accini(p->P); accini(p->A); p->num = p->crossings = 0; } void geod_polygon_addpoint(const struct geod_geodesic* g, struct geod_polygon* p, real lat, real lon) { lon = AngNormalize(lon); if (p->num == 0) { p->lat0 = p->lat = lat; p->lon0 = p->lon = lon; } else { real s12, S12; geod_geninverse(g, p->lat, p->lon, lat, lon, &s12, 0, 0, 0, 0, 0, p->polyline ? 0 : &S12); accadd(p->P, s12); if (!p->polyline) { accadd(p->A, S12); p->crossings += transit(p->lon, lon); } p->lat = lat; p->lon = lon; } ++p->num; } void geod_polygon_addedge(const struct geod_geodesic* g, struct geod_polygon* p, real azi, real s) { if (p->num) { /* Do nothing is num is zero */ real lat, lon, S12; geod_gendirect(g, p->lat, p->lon, azi, GEOD_LONG_UNROLL, s, &lat, &lon, 0, 0, 0, 0, 0, p->polyline ? 0 : &S12); accadd(p->P, s); if (!p->polyline) { accadd(p->A, S12); p->crossings += transitdirect(p->lon, lon); } p->lat = lat; p->lon = lon; ++p->num; } } unsigned geod_polygon_compute(const struct geod_geodesic* g, const struct geod_polygon* p, boolx reverse, boolx sign, real* pA, real* pP) { real s12, S12, t[2], area0; int crossings; if (p->num < 2) { if (pP) *pP = 0; if (!p->polyline && pA) *pA = 0; return p->num; } if (p->polyline) { if (pP) *pP = p->P[0]; return p->num; } geod_geninverse(g, p->lat, p->lon, p->lat0, p->lon0, &s12, 0, 0, 0, 0, 0, &S12); if (pP) *pP = accsum(p->P, s12); acccopy(p->A, t); accadd(t, S12); crossings = p->crossings + transit(p->lon, p->lon0); area0 = 4 * pi * g->c2; if (crossings & 1) accadd(t, (t[0] < 0 ? 1 : -1) * area0/2); /* area is with the clockwise sense. If !reverse convert to * counter-clockwise convention. */ if (!reverse) accneg(t); /* If sign put area in (-area0/2, area0/2], else put area in [0, area0) */ if (sign) { if (t[0] > area0/2) accadd(t, -area0); else if (t[0] <= -area0/2) accadd(t, +area0); } else { if (t[0] >= area0) accadd(t, -area0); else if (t[0] < 0) accadd(t, +area0); } if (pA) *pA = 0 + t[0]; return p->num; } unsigned geod_polygon_testpoint(const struct geod_geodesic* g, const struct geod_polygon* p, real lat, real lon, boolx reverse, boolx sign, real* pA, real* pP) { real perimeter, tempsum, area0; int crossings, i; unsigned num = p->num + 1; if (num == 1) { if (pP) *pP = 0; if (!p->polyline && pA) *pA = 0; return num; } perimeter = p->P[0]; tempsum = p->polyline ? 0 : p->A[0]; crossings = p->crossings; for (i = 0; i < (p->polyline ? 1 : 2); ++i) { real s12, S12; geod_geninverse(g, i == 0 ? p->lat : lat, i == 0 ? p->lon : lon, i != 0 ? p->lat0 : lat, i != 0 ? p->lon0 : lon, &s12, 0, 0, 0, 0, 0, p->polyline ? 0 : &S12); perimeter += s12; if (!p->polyline) { tempsum += S12; crossings += transit(i == 0 ? p->lon : lon, i != 0 ? p->lon0 : lon); } } if (pP) *pP = perimeter; if (p->polyline) return num; area0 = 4 * pi * g->c2; if (crossings & 1) tempsum += (tempsum < 0 ? 1 : -1) * area0/2; /* area is with the clockwise sense. If !reverse convert to * counter-clockwise convention. */ if (!reverse) tempsum *= -1; /* If sign put area in (-area0/2, area0/2], else put area in [0, area0) */ if (sign) { if (tempsum > area0/2) tempsum -= area0; else if (tempsum <= -area0/2) tempsum += area0; } else { if (tempsum >= area0) tempsum -= area0; else if (tempsum < 0) tempsum += area0; } if (pA) *pA = 0 + tempsum; return num; } unsigned geod_polygon_testedge(const struct geod_geodesic* g, const struct geod_polygon* p, real azi, real s, boolx reverse, boolx sign, real* pA, real* pP) { real perimeter, tempsum, area0; int crossings; unsigned num = p->num + 1; if (num == 1) { /* we don't have a starting point! */ if (pP) *pP = NaN; if (!p->polyline && pA) *pA = NaN; return 0; } perimeter = p->P[0] + s; if (p->polyline) { if (pP) *pP = perimeter; return num; } tempsum = p->A[0]; crossings = p->crossings; { real lat, lon, s12, S12; geod_gendirect(g, p->lat, p->lon, azi, GEOD_LONG_UNROLL, s, &lat, &lon, 0, 0, 0, 0, 0, &S12); tempsum += S12; crossings += transitdirect(p->lon, lon); geod_geninverse(g, lat, lon, p->lat0, p->lon0, &s12, 0, 0, 0, 0, 0, &S12); perimeter += s12; tempsum += S12; crossings += transit(lon, p->lon0); } area0 = 4 * pi * g->c2; if (crossings & 1) tempsum += (tempsum < 0 ? 1 : -1) * area0/2; /* area is with the clockwise sense. If !reverse convert to * counter-clockwise convention. */ if (!reverse) tempsum *= -1; /* If sign put area in (-area0/2, area0/2], else put area in [0, area0) */ if (sign) { if (tempsum > area0/2) tempsum -= area0; else if (tempsum <= -area0/2) tempsum += area0; } else { if (tempsum >= area0) tempsum -= area0; else if (tempsum < 0) tempsum += area0; } if (pP) *pP = perimeter; if (pA) *pA = 0 + tempsum; return num; } void geod_polygonarea(const struct geod_geodesic* g, real lats[], real lons[], int n, real* pA, real* pP) { int i; struct geod_polygon p; geod_polygon_init(&p, FALSE); for (i = 0; i < n; ++i) geod_polygon_addpoint(g, &p, lats[i], lons[i]); geod_polygon_compute(g, &p, FALSE, TRUE, pA, pP); } /** @endcond */ geosphere/src/geolib.c0000644000176200001440000000525413472155746014453 0ustar liggesusers#include "geodesic.h" #include /* Robert Hijmans, May 2015 */ SEXP _geodesic(SEXP longitude, SEXP latitude, SEXP azimuth, SEXP distance, SEXP pa, SEXP pf) { PROTECT(latitude = coerceVector(latitude, REALSXP)); PROTECT(longitude = coerceVector(longitude, REALSXP)); PROTECT(azimuth = coerceVector(azimuth, REALSXP)); PROTECT(distance = coerceVector(distance, REALSXP)); double a = REAL(pa)[0]; double f = REAL(pf)[0]; double *lat1, *lon1, *azi1, *s12, *xr; lat1 = REAL(latitude); lon1 = REAL(longitude); azi1 = REAL(azimuth); s12 = REAL(distance); double lat2, lon2, azi2; struct geod_geodesic g; geod_init(&g, a, f); int i; SEXP r; PROTECT( r = allocVector(REALSXP, 3 * length(latitude) ) ); xr = REAL(r); for (i=0; i < length(latitude); i++) { geod_direct(&g, lat1[i], lon1[i], azi1[i], s12[i], &lat2, &lon2, &azi2); xr[i*3] = lon2; xr[i*3+1] = lat2; xr[i*3+2] = azi2; } UNPROTECT(5); return r; } SEXP _inversegeodesic(SEXP longitude1, SEXP latitude1, SEXP longitude2, SEXP latitude2, SEXP pa, SEXP pf) { PROTECT(latitude1 = coerceVector(latitude1, REALSXP)); PROTECT(longitude1 = coerceVector(longitude1, REALSXP)); PROTECT(latitude2 = coerceVector(latitude2, REALSXP)); PROTECT(longitude2 = coerceVector(longitude2, REALSXP)); double a = REAL(pa)[0]; double f = REAL(pf)[0]; double *lat1, *lon1, *lat2, *lon2, *xr; lat1 = REAL(latitude1); lon1 = REAL(longitude1); lat2 = REAL(latitude2); lon2 = REAL(longitude2); double azi1, azi2, s12; struct geod_geodesic g; geod_init(&g, a, f); SEXP r; PROTECT( r = allocVector(REALSXP, 3 * length(latitude1) )); xr = REAL(r); int i; for (i=0; i < length(latitude1); i++) { geod_inverse(&g, lat1[i], lon1[i], lat2[i], lon2[i], &s12, &azi1, &azi2); xr[i*3] = s12; xr[i*3+1] = azi1; xr[i*3+2] = azi2; } UNPROTECT(5); return r; } SEXP _polygonarea(SEXP longitude, SEXP latitude, SEXP pa, SEXP pf) { PROTECT(latitude = coerceVector(latitude, REALSXP)); PROTECT(longitude = coerceVector(longitude, REALSXP)); double *lat, *lon, *xr; lat = REAL(latitude); lon = REAL(longitude); double a = REAL(pa)[0]; double f = REAL(pf)[0]; /* double a = 6378137, f = 1/298.257223563; WGS84 */ double A, P; int n, i; struct geod_geodesic g; struct geod_polygon p; geod_init(&g, a, f); geod_polygon_init(&p, 0); for (i=0; i * Algorithms for geodesics, * J. Geodesy 87, 43--55 (2013); * DOI: * 10.1007/s00190-012-0578-z; * addenda: * geod-addenda.html. * . * The principal advantages of these algorithms over previous ones (e.g., * Vincenty, 1975) are * - accurate to round off for |f| < 1/50; * - the solution of the inverse problem is always found; * - differential and integral properties of geodesics are computed. * * The shortest path between two points on the 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 forward azimuths * \e azi1 and \e azi2 at the two end points. * * Traditionally two geodesic problems are considered: * - the direct problem -- given \e lat1, \e lon1, \e s12, and \e azi1, * determine \e lat2, \e lon2, and \e azi2. This is solved by the function * geod_direct(). * - the inverse problem -- given \e lat1, \e lon1, and \e lat2, \e lon2, * determine \e s12, \e azi1, and \e azi2. This is solved by the function * geod_inverse(). * * The ellipsoid is specified by its equatorial radius \e a (typically in * meters) and flattening \e f. The routines are accurate to round off with * double precision arithmetic provided that |f| < 1/50; for the * WGS84 ellipsoid, the errors are less than 15 nanometers. (Reasonably * accurate results are obtained for |f| < 1/5.) For a prolate * ellipsoid, specify \e f < 0. * * The routines also calculate several other quantities of interest * - \e S12 is the area between the geodesic from point 1 to point 2 and the * equator; i.e., it is the area, measured counter-clockwise, of the * quadrilateral with corners (\e lat1,\e lon1), (0,\e lon1), (0,\e lon2), * and (\e lat2,\e lon2). * - \e m12, the reduced length of the geodesic is defined such that if * the initial azimuth is perturbed by \e dazi1 (radians) then the * second point is displaced by \e m12 \e dazi1 in the direction * perpendicular to the geodesic. 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. * - \e M12 and \e M21 are geodesic scales. If two geodesics are * parallel at point 1 and separated by a small distance \e dt, then * they are separated by a distance \e M12 \e dt at point 2. \e M21 * is defined similarly (with the geodesics being parallel to one * another at point 2). On a flat surface, we have \e M12 = \e M21 * = 1. * - \e a12 is the arc length on the auxiliary sphere. This is a * construct for converting the problem to one in spherical * trigonometry. \e a12 is measured in degrees. The spherical arc * length from one equator crossing to the next is always 180°. * * If points 1, 2, and 3 lie on a single geodesic, then the following * addition 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 * * 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. * * These routines are a simple transcription of the corresponding C++ classes * in GeographicLib. The * "class data" is represented by the structs geod_geodesic, geod_geodesicline, * geod_polygon and pointers to these objects are passed as initial arguments * to the member functions. Most of the internal comments have been retained. * However, in the process of transcription some documentation has been lost * and the documentation for the C++ classes, GeographicLib::Geodesic, * GeographicLib::GeodesicLine, and GeographicLib::PolygonAreaT, should be * consulted. The C++ code remains the "reference implementation". Think * twice about restructuring the internals of the C code since this may make * porting fixes from the C++ code more difficult. * * Copyright (c) Charles Karney (2012-2016) and licensed * under the MIT/X11 License. For more information, see * http://geographiclib.sourceforge.net/ * * This library was distributed with * GeographicLib 1.46. **********************************************************************/ #if !defined(GEODESIC_H) #define GEODESIC_H 1 /** * The major version of the geodesic library. (This tracks the version of * GeographicLib.) **********************************************************************/ #define GEODESIC_VERSION_MAJOR 1 /** * The minor version of the geodesic library. (This tracks the version of * GeographicLib.) **********************************************************************/ #define GEODESIC_VERSION_MINOR 46 /** * The patch level of the geodesic library. (This tracks the version of * GeographicLib.) **********************************************************************/ #define GEODESIC_VERSION_PATCH 0 /** * 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 GEODESIC_VERSION, below. **********************************************************************/ #define GEODESIC_VERSION_NUM(a,b,c) ((((a) * 10000 + (b)) * 100) + (c)) /** * The version of the geodesic library 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{.c} #if GEODESIC_VERSION >= GEODESIC_VERSION_NUM(1,40,0) ... #endif * @endcode **********************************************************************/ #define GEODESIC_VERSION \ GEODESIC_VERSION_NUM(GEODESIC_VERSION_MAJOR, \ GEODESIC_VERSION_MINOR, \ GEODESIC_VERSION_PATCH) #if defined(__cplusplus) extern "C" { #endif /** * The struct containing information about the ellipsoid. This must be * initialized by geod_init() before use. **********************************************************************/ struct geod_geodesic { double a; /**< the equatorial radius */ double f; /**< the flattening */ /**< @cond SKIP */ double f1, e2, ep2, n, b, c2, etol2; double A3x[6], C3x[15], C4x[21]; /**< @endcond */ }; /** * The struct containing information about a single geodesic. This must be * initialized by geod_lineinit(), geod_directline(), geod_gendirectline(), * or geod_inverseline() before use. **********************************************************************/ struct geod_geodesicline { double lat1; /**< the starting latitude */ double lon1; /**< the starting longitude */ double azi1; /**< the starting azimuth */ double a; /**< the equatorial radius */ double f; /**< the flattening */ double salp1; /**< sine of \e azi1 */ double calp1; /**< cosine of \e azi1 */ double a13; /**< arc length to reference point */ double s13; /**< distance to reference point */ /**< @cond SKIP */ double b, c2, f1, salp0, calp0, k2, ssig1, csig1, dn1, stau1, ctau1, somg1, comg1, A1m1, A2m1, A3c, B11, B21, B31, A4, B41; double C1a[6+1], C1pa[6+1], C2a[6+1], C3a[6], C4a[6]; /**< @endcond */ unsigned caps; /**< the capabilities */ }; /** * The struct for accumulating information about a geodesic polygon. This is * used for computing the perimeter and area of a polygon. This must be * initialized by geod_polygon_init() before use. **********************************************************************/ struct geod_polygon { double lat; /**< the current latitude */ double lon; /**< the current longitude */ /**< @cond SKIP */ double lat0; double lon0; double A[2]; double P[2]; int polyline; int crossings; /**< @endcond */ unsigned num; /**< the number of points so far */ }; /** * Initialize a geod_geodesic object. * * @param[out] g a pointer to the object to be initialized. * @param[in] a the equatorial radius (meters). * @param[in] f the flattening. **********************************************************************/ void geod_init(struct geod_geodesic* g, double a, double f); /** * Solve the direct geodesic problem. * * @param[in] g a pointer to the geod_geodesic object specifying the * ellipsoid. * @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 from point 1 to point 2 (meters); it can be * negative. * @param[out] plat2 pointer to the latitude of point 2 (degrees). * @param[out] plon2 pointer to the longitude of point 2 (degrees). * @param[out] pazi2 pointer to the (forward) azimuth at point 2 (degrees). * * \e g must have been initialized with a call to geod_init(). \e lat1 * should be in the range [−90°, 90°]. The values of \e lon2 * and \e azi2 returned are in the range [−180°, 180°). Any of * the "return" arguments \e plat2, etc., may be replaced by 0, if you do not * need some quantities computed. * * 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°.) * * Example, determine the point 10000 km NE of JFK: @code{.c} struct geod_geodesic g; double lat, lon; geod_init(&g, 6378137, 1/298.257223563); geod_direct(&g, 40.64, -73.78, 45.0, 10e6, &lat, &lon, 0); printf("%.5f %.5f\n", lat, lon); @endcode **********************************************************************/ void geod_direct(const struct geod_geodesic* g, double lat1, double lon1, double azi1, double s12, double* plat2, double* plon2, double* pazi2); /** * The general direct geodesic problem. * * @param[in] g a pointer to the geod_geodesic object specifying the * ellipsoid. * @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] flags bitor'ed combination of geod_flags(); \e flags & * GEOD_ARCMODE determines the meaning of \e s12_a12 and \e flags & * GEOD_LONG_UNROLL "unrolls" \e lon2. * @param[in] s12_a12 if \e flags & GEOD_ARCMODE is 0, this is the distance * from point 1 to point 2 (meters); otherwise it is the arc length * from point 1 to point 2 (degrees); it can be negative. * @param[out] plat2 pointer to the latitude of point 2 (degrees). * @param[out] plon2 pointer to the longitude of point 2 (degrees). * @param[out] pazi2 pointer to the (forward) azimuth at point 2 (degrees). * @param[out] ps12 pointer to the distance from point 1 to point 2 * (meters). * @param[out] pm12 pointer to the reduced length of geodesic (meters). * @param[out] pM12 pointer to the geodesic scale of point 2 relative to * point 1 (dimensionless). * @param[out] pM21 pointer to the geodesic scale of point 1 relative to * point 2 (dimensionless). * @param[out] pS12 pointer to the area under the geodesic * (meters2). * @return \e a12 arc length from point 1 to point 2 (degrees). * * \e g must have been initialized with a call to geod_init(). \e lat1 * should be in the range [−90°, 90°]. The function value \e * a12 equals \e s12_a12 if \e flags & GEOD_ARCMODE. Any of the "return" * arguments, \e plat2, etc., may be replaced by 0, if you do not need some * quantities computed. * * With \e flags & GEOD_LONG_UNROLL bit set, the longitude is "unrolled" so * that the quantity \e lon2 − \e lon1 indicates how many times and in * what sense the geodesic encircles the ellipsoid. **********************************************************************/ double geod_gendirect(const struct geod_geodesic* g, double lat1, double lon1, double azi1, unsigned flags, double s12_a12, double* plat2, double* plon2, double* pazi2, double* ps12, double* pm12, double* pM12, double* pM21, double* pS12); /** * Solve the inverse geodesic problem. * * @param[in] g a pointer to the geod_geodesic object specifying the * ellipsoid. * @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] ps12 pointer to the distance from point 1 to point 2 * (meters). * @param[out] pazi1 pointer to the azimuth at point 1 (degrees). * @param[out] pazi2 pointer to the (forward) azimuth at point 2 (degrees). * * \e g must have been initialized with a call to geod_init(). \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°). * Any of the "return" arguments, \e ps12, etc., may be replaced by 0, if you * do not need some quantities computed. * * 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. * * Example, determine the distance between JFK and Singapore Changi Airport: @code{.c} struct geod_geodesic g; double s12; geod_init(&g, 6378137, 1/298.257223563); geod_inverse(&g, 40.64, -73.78, 1.36, 103.99, &s12, 0, 0); printf("%.3f\n", s12); @endcode **********************************************************************/ void geod_inverse(const struct geod_geodesic* g, double lat1, double lon1, double lat2, double lon2, double* ps12, double* pazi1, double* pazi2); /** * The general inverse geodesic calculation. * * @param[in] g a pointer to the geod_geodesic object specifying the * ellipsoid. * @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] ps12 pointer to the distance from point 1 to point 2 * (meters). * @param[out] pazi1 pointer to the azimuth at point 1 (degrees). * @param[out] pazi2 pointer to the (forward) azimuth at point 2 (degrees). * @param[out] pm12 pointer to the reduced length of geodesic (meters). * @param[out] pM12 pointer to the geodesic scale of point 2 relative to * point 1 (dimensionless). * @param[out] pM21 pointer to the geodesic scale of point 1 relative to * point 2 (dimensionless). * @param[out] pS12 pointer to the area under the geodesic * (meters2). * @return \e a12 arc length from point 1 to point 2 (degrees). * * \e g must have been initialized with a call to geod_init(). \e lat1 and * \e lat2 should be in the range [−90°, 90°]. Any of the * "return" arguments \e ps12, etc., may be replaced by 0, if you do not need * some quantities computed. **********************************************************************/ double geod_geninverse(const struct geod_geodesic* g, double lat1, double lon1, double lat2, double lon2, double* ps12, double* pazi1, double* pazi2, double* pm12, double* pM12, double* pM21, double* pS12); /** * Initialize a geod_geodesicline object. * * @param[out] l a pointer to the object to be initialized. * @param[in] g a pointer to the geod_geodesic object specifying the * ellipsoid. * @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 geod_mask() values specifying the * capabilities the geod_geodesicline object should possess, i.e., which * quantities can be returned in calls to geod_position() and * geod_genposition(). * * \e g must have been initialized with a call to geod_init(). \e lat1 * should be in the range [−90°, 90°]. * * The geod_mask values are [see geod_mask()]: * - \e caps |= GEOD_LATITUDE for the latitude \e lat2; this is * added automatically, * - \e caps |= GEOD_LONGITUDE for the latitude \e lon2, * - \e caps |= GEOD_AZIMUTH for the latitude \e azi2; this is * added automatically, * - \e caps |= GEOD_DISTANCE for the distance \e s12, * - \e caps |= GEOD_REDUCEDLENGTH for the reduced length \e m12, * - \e caps |= GEOD_GEODESICSCALE for the geodesic scales \e M12 * and \e M21, * - \e caps |= GEOD_AREA for the area \e S12, * - \e caps |= GEOD_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. * . * A value of \e caps = 0 is treated as GEOD_LATITUDE | GEOD_LONGITUDE | * GEOD_AZIMUTH | GEOD_DISTANCE_IN (to support the solution of the "standard" * direct problem). * * When initialized by this function, point 3 is undefined (l->s13 = l->a13 = * NaN). **********************************************************************/ void geod_lineinit(struct geod_geodesicline* l, const struct geod_geodesic* g, double lat1, double lon1, double azi1, unsigned caps); /** * Initialize a geod_geodesicline object in terms of the direct geodesic * problem. * * @param[out] l a pointer to the object to be initialized. * @param[in] g a pointer to the geod_geodesic object specifying the * ellipsoid. * @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 from point 1 to point 2 (meters); it can be * negative. * @param[in] caps bitor'ed combination of geod_mask() values specifying the * capabilities the geod_geodesicline object should possess, i.e., which * quantities can be returned in calls to geod_position() and * geod_genposition(). * * This function sets point 3 of the geod_geodesicline to correspond to point * 2 of the direct geodesic problem. See geod_lineinit() for more * informaion. **********************************************************************/ void geod_directline(struct geod_geodesicline* l, const struct geod_geodesic* g, double lat1, double lon1, double azi1, double s12, unsigned caps); /** * Initialize a geod_geodesicline object in terms of the direct geodesic * problem spacified in terms of either distance or arc length. * * @param[out] l a pointer to the object to be initialized. * @param[in] g a pointer to the geod_geodesic object specifying the * ellipsoid. * @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] flags either GEOD_NOFLAGS or GEOD_ARCMODE to determining the * meaning of the \e s12_a12. * @param[in] s12_a12 if \e flags = GEOD_NOFLAGS, this is the distance * from point 1 to point 2 (meters); if \e flags = GEOD_ARCMODE, it is * the arc length from point 1 to point 2 (degrees); it can be * negative. * @param[in] caps bitor'ed combination of geod_mask() values specifying the * capabilities the geod_geodesicline object should possess, i.e., which * quantities can be returned in calls to geod_position() and * geod_genposition(). * * This function sets point 3 of the geod_geodesicline to correspond to point * 2 of the direct geodesic problem. See geod_lineinit() for more * informaion. **********************************************************************/ void geod_gendirectline(struct geod_geodesicline* l, const struct geod_geodesic* g, double lat1, double lon1, double azi1, unsigned flags, double s12_a12, unsigned caps); /** * Initialize a geod_geodesicline object in terms of the inverse geodesic * problem. * * @param[out] l a pointer to the object to be initialized. * @param[in] g a pointer to the geod_geodesic object specifying the * ellipsoid. * @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 geod_mask() values specifying the * capabilities the geod_geodesicline object should possess, i.e., which * quantities can be returned in calls to geod_position() and * geod_genposition(). * * This function sets point 3 of the geod_geodesicline to correspond to point * 2 of the inverse geodesic problem. See geod_lineinit() for more * informaion. **********************************************************************/ void geod_inverseline(struct geod_geodesicline* l, const struct geod_geodesic* g, double lat1, double lon1, double lat2, double lon2, unsigned caps); /** * Compute the position along a geod_geodesicline. * * @param[in] l a pointer to the geod_geodesicline object specifying the * geodesic line. * @param[in] s12 distance from point 1 to point 2 (meters); it can be * negative. * @param[out] plat2 pointer to the latitude of point 2 (degrees). * @param[out] plon2 pointer to the longitude of point 2 (degrees); requires * that \e l was initialized with \e caps |= GEOD_LONGITUDE. * @param[out] pazi2 pointer to the (forward) azimuth at point 2 (degrees). * * \e l must have been initialized with a call, e.g., to geod_lineinit(), * with \e caps |= GEOD_DISTANCE_IN. The values of \e lon2 and \e azi2 * returned are in the range [−180°, 180°). Any of the * "return" arguments \e plat2, etc., may be replaced by 0, if you do not * need some quantities computed. * * Example, compute way points between JFK and Singapore Changi Airport * the "obvious" way using geod_direct(): @code{.c} struct geod_geodesic g; double s12, azi1, lat[101],lon[101]; int i; geod_init(&g, 6378137, 1/298.257223563); geod_inverse(&g, 40.64, -73.78, 1.36, 103.99, &s12, &azi1, 0); for (i = 0; i < 101; ++i) { geod_direct(&g, 40.64, -73.78, azi1, i * s12 * 0.01, lat + i, lon + i, 0); printf("%.5f %.5f\n", lat[i], lon[i]); } @endcode * A faster way using geod_position(): @code{.c} struct geod_geodesic g; struct geod_geodesicline l; double lat[101],lon[101]; int i; geod_init(&g, 6378137, 1/298.257223563); geod_inverseline(&l, &g, 40.64, -73.78, 1.36, 103.99, 0); for (i = 0; i <= 100; ++i) { geod_position(&l, i * l.s13 * 0.01, lat + i, lon + i, 0); printf("%.5f %.5f\n", lat[i], lon[i]); } @endcode **********************************************************************/ void geod_position(const struct geod_geodesicline* l, double s12, double* plat2, double* plon2, double* pazi2); /** * The general position function. * * @param[in] l a pointer to the geod_geodesicline object specifying the * geodesic line. * @param[in] flags bitor'ed combination of geod_flags(); \e flags & * GEOD_ARCMODE determines the meaning of \e s12_a12 and \e flags & * GEOD_LONG_UNROLL "unrolls" \e lon2; if \e flags & GEOD_ARCMODE is 0, * then \e l must have been initialized with \e caps |= GEOD_DISTANCE_IN. * @param[in] s12_a12 if \e flags & GEOD_ARCMODE is 0, this is the * distance from point 1 to point 2 (meters); otherwise it is the * arc length from point 1 to point 2 (degrees); it can be * negative. * @param[out] plat2 pointer to the latitude of point 2 (degrees). * @param[out] plon2 pointer to the longitude of point 2 (degrees); requires * that \e l was initialized with \e caps |= GEOD_LONGITUDE. * @param[out] pazi2 pointer to the (forward) azimuth at point 2 (degrees). * @param[out] ps12 pointer to the distance from point 1 to point 2 * (meters); requires that \e l was initialized with \e caps |= * GEOD_DISTANCE. * @param[out] pm12 pointer to the reduced length of geodesic (meters); * requires that \e l was initialized with \e caps |= GEOD_REDUCEDLENGTH. * @param[out] pM12 pointer to the geodesic scale of point 2 relative to * point 1 (dimensionless); requires that \e l was initialized with \e caps * |= GEOD_GEODESICSCALE. * @param[out] pM21 pointer to the geodesic scale of point 1 relative to * point 2 (dimensionless); requires that \e l was initialized with \e caps * |= GEOD_GEODESICSCALE. * @param[out] pS12 pointer to the area under the geodesic * (meters2); requires that \e l was initialized with \e caps |= * GEOD_AREA. * @return \e a12 arc length from point 1 to point 2 (degrees). * * \e l must have been initialized with a call to geod_lineinit() with \e * caps |= GEOD_DISTANCE_IN. The value \e azi2 returned is in the range * [−180°, 180°). Any of the "return" arguments \e plat2, * etc., may be replaced by 0, if you do not need some quantities * computed. Requesting a value which \e l is not capable of computing * is not an error; the corresponding argument will not be altered. * * With \e flags & GEOD_LONG_UNROLL bit set, the longitude is "unrolled" so * that the quantity \e lon2 − \e lon1 indicates how many times and in * what sense the geodesic encircles the ellipsoid. * * Example, compute way points between JFK and Singapore Changi Airport using * geod_genposition(). In this example, the points are evenly space in arc * length (and so only approximately equally spaced in distance). This is * faster than using geod_position() and would be appropriate if drawing the * path on a map. @code{.c} struct geod_geodesic g; struct geod_geodesicline l; double lat[101], lon[101]; int i; geod_init(&g, 6378137, 1/298.257223563); geod_inverseline(&l, &g, 40.64, -73.78, 1.36, 103.99, GEOD_LATITUDE | GEOD_LONGITUDE); for (i = 0; i <= 100; ++i) { geod_genposition(&l, GEOD_ARCMODE, i * l.a13 * 0.01, lat + i, lon + i, 0, 0, 0, 0, 0, 0); printf("%.5f %.5f\n", lat[i], lon[i]); } @endcode **********************************************************************/ double geod_genposition(const struct geod_geodesicline* l, unsigned flags, double s12_a12, double* plat2, double* plon2, double* pazi2, double* ps12, double* pm12, double* pM12, double* pM21, double* pS12); /** * Specify position of point 3 in terms of distance. * * @param[inout] l a pointer to the geod_geodesicline object. * @param[in] s13 the distance from point 1 to point 3 (meters); it * can be negative. * * This is only useful if the geod_geodesicline object has been constructed * with \e caps |= GEOD_DISTANCE_IN. **********************************************************************/ void geod_setdistance(struct geod_geodesicline* l, double s13); /** * Specify position of point 3 in terms of either distance or arc length. * * @param[inout] l a pointer to the geod_geodesicline object. * @param[in] flags either GEOD_NOFLAGS or GEOD_ARCMODE to determining the * meaning of the \e s13_a13. * @param[in] s13_a13 if \e flags = GEOD_NOFLAGS, this is the distance * from point 1 to point 3 (meters); if \e flags = GEOD_ARCMODE, it is * the arc length from point 1 to point 3 (degrees); it can be * negative. * * If flags = GEOD_NOFLAGS, this calls geod_setdistance(). If flags = * GEOD_ARCMODE, the \e s13 is only set if the geod_geodesicline object has * been constructed with \e caps |= GEOD_DISTANCE. **********************************************************************/ void geod_gensetdistance(struct geod_geodesicline* l, unsigned flags, double s13_a13); /** * Initialize a geod_polygon object. * * @param[out] p a pointer to the object to be initialized. * @param[in] polylinep non-zero if a polyline instead of a polygon. * * If \e polylinep is zero, then the sequence of vertices and edges added by * geod_polygon_addpoint() and geod_polygon_addedge() define a polygon and * the perimeter and area are returned by geod_polygon_compute(). If \e * polylinep is non-zero, then the vertices and edges define a polyline and * only the perimeter is returned by geod_polygon_compute(). * * 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. * * An example of the use of this function is given in the documentation for * geod_polygon_compute(). **********************************************************************/ void geod_polygon_init(struct geod_polygon* p, int polylinep); /** * Clear the polygon, allowing a new polygon to be started. * * @param[in,out] p a pointer to the object to be cleared. **********************************************************************/ void geod_polygon_clear(struct geod_polygon* p); /** * Add a point to the polygon or polyline. * * @param[in] g a pointer to the geod_geodesic object specifying the * ellipsoid. * @param[in,out] p a pointer to the geod_polygon object specifying the * polygon. * @param[in] lat the latitude of the point (degrees). * @param[in] lon the longitude of the point (degrees). * * \e g and \e p must have been initialized with calls to geod_init() and * geod_polygon_init(), respectively. The same \e g must be used for all the * points and edges in a polygon. \e lat should be in the range * [−90°, 90°]. * * An example of the use of this function is given in the documentation for * geod_polygon_compute(). **********************************************************************/ void geod_polygon_addpoint(const struct geod_geodesic* g, struct geod_polygon* p, double lat, double lon); /** * Add an edge to the polygon or polyline. * * @param[in] g a pointer to the geod_geodesic object specifying the * ellipsoid. * @param[in,out] p a pointer to the geod_polygon object specifying the * polygon. * @param[in] azi azimuth at current point (degrees). * @param[in] s distance from current point to next point (meters). * * \e g and \e p must have been initialized with calls to geod_init() and * geod_polygon_init(), respectively. The same \e g must be used for all the * points and edges in a polygon. This does nothing if no points have been * added yet. The \e lat and \e lon fields of \e p give the location of the * new vertex. **********************************************************************/ void geod_polygon_addedge(const struct geod_geodesic* g, struct geod_polygon* p, double azi, double s); /** * Return the results for a polygon. * * @param[in] g a pointer to the geod_geodesic object specifying the * ellipsoid. * @param[in] p a pointer to the geod_polygon object specifying the polygon. * @param[in] reverse if non-zero then clockwise (instead of * counter-clockwise) traversal counts as a positive area. * @param[in] sign if non-zero 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] pA pointer to the area of the polygon (meters2); * only set if \e polyline is non-zero in the call to geod_polygon_init(). * @param[out] pP pointer to the perimeter of the polygon or length of the * polyline (meters). * @return the number of points. * * 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. Only simple polygons (which are not self-intersecting) are * allowed. There's no need to "close" the polygon by repeating the first * vertex. Set \e pA or \e pP to zero, if you do not want the corresponding * quantity returned. * * More points can be added to the polygon after this call. * * Example, compute the perimeter and area of the geodesic triangle with * vertices (0°N,0°E), (0°N,90°E), (90°N,0°E). @code{.c} double A, P; int n; struct geod_geodesic g; struct geod_polygon p; geod_init(&g, 6378137, 1/298.257223563); geod_polygon_init(&p, 0); geod_polygon_addpoint(&g, &p, 0, 0); geod_polygon_addpoint(&g, &p, 0, 90); geod_polygon_addpoint(&g, &p, 90, 0); n = geod_polygon_compute(&g, &p, 0, 1, &A, &P); printf("%d %.8f %.3f\n", n, P, A); @endcode **********************************************************************/ unsigned geod_polygon_compute(const struct geod_geodesic* g, const struct geod_polygon* p, int reverse, int sign, double* pA, double* pP); /** * 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 geod_polygon_addpoint() and geod_polygon_compute() are used. * * @param[in] g a pointer to the geod_geodesic object specifying the * ellipsoid. * @param[in] p a pointer to the geod_polygon object specifying the polygon. * @param[in] lat the latitude of the test point (degrees). * @param[in] lon the longitude of the test point (degrees). * @param[in] reverse if non-zero then clockwise (instead of * counter-clockwise) traversal counts as a positive area. * @param[in] sign if non-zero 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] pA pointer to the area of the polygon (meters2); * only set if \e polyline is non-zero in the call to geod_polygon_init(). * @param[out] pP pointer to the perimeter of the polygon or length of the * polyline (meters). * @return the number of points. * * \e lat should be in the range [−90°, 90°]. **********************************************************************/ unsigned geod_polygon_testpoint(const struct geod_geodesic* g, const struct geod_polygon* p, double lat, double lon, int reverse, int sign, double* pA, double* pP); /** * 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 geod_polygon_addedge() and * geod_polygon_compute() are used. * * @param[in] g a pointer to the geod_geodesic object specifying the * ellipsoid. * @param[in] p a pointer to the geod_polygon object specifying the polygon. * @param[in] azi azimuth at current point (degrees). * @param[in] s distance from current point to final test point (meters). * @param[in] reverse if non-zero then clockwise (instead of * counter-clockwise) traversal counts as a positive area. * @param[in] sign if non-zero 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] pA pointer to the area of the polygon (meters2); * only set if \e polyline is non-zero in the call to geod_polygon_init(). * @param[out] pP pointer to the perimeter of the polygon or length of the * polyline (meters). * @return the number of points. **********************************************************************/ unsigned geod_polygon_testedge(const struct geod_geodesic* g, const struct geod_polygon* p, double azi, double s, int reverse, int sign, double* pA, double* pP); /** * A simple interface for computing the area of a geodesic polygon. * * @param[in] g a pointer to the geod_geodesic object specifying the * ellipsoid. * @param[in] lats an array of latitudes of the polygon vertices (degrees). * @param[in] lons an array of longitudes of the polygon vertices (degrees). * @param[in] n the number of vertices. * @param[out] pA pointer to the area of the polygon (meters2). * @param[out] pP pointer to the perimeter of the polygon (meters). * * \e lats should be in the range [−90°, 90°]. * * Only simple polygons (which are not self-intersecting) are allowed. * There's no need to "close" the polygon by repeating the first vertex. The * area returned is signed with counter-clockwise traversal being treated as * positive. * * Example, compute the area of Antarctica: @code{.c} double lats[] = {-72.9, -71.9, -74.9, -74.3, -77.5, -77.4, -71.7, -65.9, -65.7, -66.6, -66.9, -69.8, -70.0, -71.0, -77.3, -77.9, -74.7}, lons[] = {-74, -102, -102, -131, -163, 163, 172, 140, 113, 88, 59, 25, -4, -14, -33, -46, -61}; struct geod_geodesic g; double A, P; geod_init(&g, 6378137, 1/298.257223563); geod_polygonarea(&g, lats, lons, (sizeof lats) / (sizeof lats[0]), &A, &P); printf("%.0f %.2f\n", A, P); @endcode **********************************************************************/ void geod_polygonarea(const struct geod_geodesic* g, double lats[], double lons[], int n, double* pA, double* pP); /** * mask values for the \e caps argument to geod_lineinit(). **********************************************************************/ enum geod_mask { GEOD_NONE = 0U, /**< Calculate nothing */ GEOD_LATITUDE = 1U<<7 | 0U, /**< Calculate latitude */ GEOD_LONGITUDE = 1U<<8 | 1U<<3, /**< Calculate longitude */ GEOD_AZIMUTH = 1U<<9 | 0U, /**< Calculate azimuth */ GEOD_DISTANCE = 1U<<10 | 1U<<0, /**< Calculate distance */ GEOD_DISTANCE_IN = 1U<<11 | 1U<<0 | 1U<<1, /**< Allow distance as input */ GEOD_REDUCEDLENGTH= 1U<<12 | 1U<<0 | 1U<<2, /**< Calculate reduced length */ GEOD_GEODESICSCALE= 1U<<13 | 1U<<0 | 1U<<2, /**< Calculate geodesic scale */ GEOD_AREA = 1U<<14 | 1U<<4, /**< Calculate reduced length */ GEOD_ALL = 0x7F80U| 0x1FU /**< Calculate everything */ }; /** * flag values for the \e flags argument to geod_gendirect() and * geod_genposition() **********************************************************************/ enum geod_flags { GEOD_NOFLAGS = 0U, /**< No flags */ GEOD_ARCMODE = 1U<<0, /**< Position given in terms of arc distance */ GEOD_LONG_UNROLL = 1U<<15 /**< Unroll the longitude */ }; #if defined(__cplusplus) } #endif #endif geosphere/src/dist.c0000644000176200001440000000772513472321466014154 0ustar liggesusers/* Robert Hijmans, June 2011 */ #include #include #include #include #include "Rmath.h" #include "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) { /* 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 based on source http://www.movable-type.co.uk/scripts/latlong-vincenty.html (c) 2002-2009 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 R_NaReal; } 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 (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 R_NaReal; // 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) { int i; if (*m > 0) { for(i=0; i < *n; i++) { dist[i] = distVinEll(lon1[i], lat1[i], lon2[i], lat2[i], a[i], b[i], f[i]); } } } geosphere/src/geosphere_init.c0000644000176200001440000000136213472155746016212 0ustar liggesusers#include #include #include // for NULL #include /* FIXME: Check these declarations against the C/Fortran source code. */ /* .Call calls */ extern SEXP _geodesic(SEXP, SEXP, SEXP, SEXP, SEXP, SEXP); extern SEXP _inversegeodesic(SEXP, SEXP, SEXP, SEXP, SEXP, SEXP); extern SEXP _polygonarea(SEXP, SEXP, SEXP, SEXP); static const R_CallMethodDef CallEntries[] = { {"_geodesic", (DL_FUNC) &_geodesic, 6}, {"_inversegeodesic", (DL_FUNC) &_inversegeodesic, 6}, {"_polygonarea", (DL_FUNC) &_polygonarea, 4}, {NULL, NULL, 0} }; void R_init_geosphere(DllInfo *dll) { R_registerRoutines(dll, NULL, CallEntries, NULL, NULL); R_useDynamicSymbols(dll, FALSE); } geosphere/NAMESPACE0000644000176200001440000000012513472155746013466 0ustar liggesusersimport(sp) useDynLib(geosphere, .registration = TRUE) exportPattern("^[^\\.\\_]") geosphere/data/0000755000176200001440000000000013472155746013162 5ustar liggesusersgeosphere/data/merc.RData0000644000176200001440000106205413472155746015035 0ustar liggesusers7zXZi"6!X~5])TW"nRʟXV%>"UZ tBk}MItrU)W*v*t6_uX鎞SH\5KRahwXA-HF*˫f\vf!0"q=1:(m7A$*˶ʕMwibj qܗa(,z)=J9(rDNrQ葹#r<\@ |Pl8Mq4hR6F 8a^|}Cby<۪EC$qŃ׉>1ЄFޅB߈- *p}j{Nk0vkI3+tG3}r:9vd^ {"e'BLYSUAqP(ܘLyOW8N5PC!].45@EG, ) n#f`'D)ZSG*0h[0e8^[ݜq>J_l s| ĠH"^Ip?}[﫟sD[`FVv94q2rPs4f# h)!PL0_F鰔ʫF76X|1(X &]tꝃ v*zo z^@{r.ɴ|tI)n5=;5̽.Sz6OoO:y[-MOcc~KG뷣>;4_!4C\oX J&ڿ}-U 5ye\ǪW ha4[\880 ׎F-|?=__5X< o 6'~l$iyQNCSbs۸ľeĹw9QRQSu$ Y'׌ôv#t>P | V?*pY5dURqR%l| CY.;*aBa8V1Xfmu80 .G86n 7SӿѐU۵b}mIUl~f,ˍk$P$KZ=$Sy=Es]}w|sqXH|ng?( J&cZf xrcG`g?kdObQw"E1ĭz4t2, ]rU{bΦLN 7>}#Qݹ;.pc{H 8Y<[4M[x# Lz ?ۯ%tfF\xGOq}]rB= 8d5,QGJ?N9i3ALRF /,aCh`d7|^}|c9^z-<0ly-  PYU8Rk㙒79PCDQ%0oS,J@=686IFmYTw-OuM 8ҷ/=U | ^ ʋ패3~a2e x1R)=7>_E&DmR.osCf}P]M$u9Hl (D ZZ^:D#A>O0kk#Hza1vDݍie{u.ҍu?|hbrUBbA<>pUG]":_v)-uѼ󕞔Vk(;OiMàuO`FH/<Q4 S@$'JkBDU|9`gb͕p3وV;G2\"iNe_^n"$\ܶNXV/ "q؆AFιqh +x E;k(BU1]`)ldM`Wڟ3,|#3%;l_8җNntcȵbYjQ ~Q(qXtb_:BO`rl3[wscDu-cݩA "cH} .$kB\wcBۄ-Ks&{K o! 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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/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/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/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/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/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/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/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/destPoint.R0000644000176200001440000000132613472155746014550 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 <- .Call("_geodesic", as.double(p[,1]), as.double(p[,2]), as.double(p[,3]), as.double(p[,4]), as.double(a), as.double(f), PACKAGE='geosphere') r <- matrix(r, ncol=3, byrow=TRUE) colnames(r) <- c('lon', 'lat', 'finalbearing') return(r[, 1:2, drop=FALSE]) } 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/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/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/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/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/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/distGeo.R0000644000176200001440000000126613472155746014200 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 <- .Call("_inversegeodesic", as.double(p[,1]), as.double(p[,2]), as.double(p[,3]), as.double(p[,4]), as.double(a), as.double(f), PACKAGE='geosphere') r <- matrix(r, ncol=3, byrow=TRUE) if (addNA){ c(r[,1], NA) } else { r[,1] } } 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/daylength.R0000644000176200001440000000325713472155746014563 0ustar liggesusers# Author: Robert J. Hijmans, r.hijmans@gmail.com # License GPL3 # Version 0.1 January 2009 daylength <- function(lat, doy) { if (class(doy) == 'Date' | class(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 (class(doy) == 'Date' | class(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/antipodal.R0000644000176200001440000000112513472155746014547 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)) (diflat < tol) & (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/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/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/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/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/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/geodesic.R0000644000176200001440000000301313472155746014354 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 <- .Call("_geodesic", as.double(p[,1]), as.double(p[,2]), as.double(p[,3]), as.double(p[,4]), as.double(a), as.double(f), PACKAGE='geosphere') 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 <- .Call("_inversegeodesic", as.double(p[,1]), as.double(p[,2]), as.double(p[,3]), as.double(p[,4]), as.double(a), as.double(f), PACKAGE='geosphere') r <- matrix(r, ncol=3, byrow=TRUE) colnames(r) <- c('distance', 'azimuth1', 'azimuth2') r } 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/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/bearing.R0000644000176200001440000000260213472155746014204 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 <- .Call("_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]), PACKAGE='geosphere') 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/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/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/gcIntermediate.R0000644000176200001440000000464313472155746015530 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) } geosphere/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/distHaversine.R0000644000176200001440000000311613472155746015406 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) * sin(dLat/2) + cos(p[,2]) * cos(p[,4]) * sin(dLon/2) * sin(dLon/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 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/geomean.R0000644000176200001440000000123513472155746014211 0ustar liggesusers# Author: Robert J. Hijmans # February 2012 # version 1 # license GPL3 geomean <- 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] xy[,1] <- xy[,1] + 180 xy <- xy * pi / 180 Sx <- mean(sin(xy[,1]) * w) Cx <- mean(cos(xy[,1]) * w) x <- atan2(Sx, Cx) x <- x %% (2 * pi) - pi Sy <- mean(sin(xy[,2]) * w) Cy <- mean(cos(xy[,2]) * w) y <- atan2(Sy, Cy) cbind(x,y) * 180 / pi } 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/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/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/perimeter.R0000644000176200001440000000340213472155746014570 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(...)$r if (!is.null(r)) { # for backwards compatibility warning('remove argument "r" to use improved method') return( .old_perimeter(x, r=r) ) } x <- .Call("_polygonarea", as.double(x[,1]), as.double(x[,2]), as.double(a), as.double(f), PACKAGE='geosphere') abs(x[2]) }) .old_perimeter <- function(x, r=6378137, ...) { x <- x[,1:2] if (isTRUE(all.equal(x[1,], x[nrow(x),]))) { x <- x[-nrow(x), ] } y <- rbind(x[-1,], x[1,]) d <- distHaversine(x, y, r=r) return(sum(d)) } geosphere/R/areaPolygon.R0000644000176200001440000000565613472155746015071 0ustar liggesusers# Robert Hijmans # April 2010 # version 1 # license GPL3 if (!isGeneric("areaPolygon")) { setGeneric("areaPolygon", function(x, ...) standardGeneric("areaPolygon")) } setMethod('areaPolygon', signature(x='data.frame'), function(x, a=6378137, f=1/298.257223563, ...) { areaPolygon(as.matrix(x), a=a, f=f, ...) } ) setMethod('areaPolygon', signature(x='SpatialPolygons'), function(x, a=6378137, f=1/298.257223563, ...) { test <- is.projected(x) if ( isTRUE (test) ) { if (is.na(test)) { warning('Coordinate reference system of SpatialPolygons object is not set. Assuming it is degrees (longitude/latitude)!') } else { stop('The coordinate reference system is not longitude/latitude. Use rgeos::gArea instead') } # or rather transform them ....? } x <- x@polygons n <- length(x) res <- vector(length=n) for (i in 1:n) { parts <- length(x[[i]]@Polygons ) sumarea <- 0 for (j in 1:parts) { crd <- x[[i]]@Polygons[[j]]@coords ar <- areaPolygon(crd, a=a, f=f, ...) if (x[[i]]@Polygons[[j]]@hole) { sumarea <- sumarea - ar } else { sumarea <- sumarea + ar } } res[i] <- sumarea } return(res) } ) setMethod('areaPolygon', signature(x='matrix'), function(x, a=6378137, f=1/298.257223563, ...) { r <- list(...)$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 <- .Call("_polygonarea", as.double(x[,1]), as.double(x[,2]), as.double(a), as.double(f), PACKAGE='geosphere') 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/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$y -dy) } .sampleAlong <- function(x, interval) { if (inherits(x, 'SpatialPolygons')) { line <- methods::as(line, 'SpatialLines') } if (inherits(x, 'SpatialLines')) { requireNamespace('raster') x <- raster::geom(x) allpts <- NULL for (p in unique(x[, 'cump'])) { y <- x[x[, 'cump']==p, c('x', 'y')] pts <- .evenspace(y, interval, direction=FALSE) allpts <- rbind(allpts, pts) } return(allpts) } else { x <- .pointsToMatrix(x) .evenspace(x, interval, direction=FALSE) } } .sampleAlongPerpendicular <- function(x, interval, pdist, np=1 ) { if (inherits(x, 'SpatialPolygons')) { line <- methods::as(line, 'SpatialLines') } if (inherits(x, 'SpatialLines')) { requireNamespace('raster') x <- raster::geom(x) allpts <- NULL for (p in unique(x[, 'cump'])) { y <- x[x[, 'cump']==p, c('x', 'y')] tspts <- .evenspace(y, interval) pts <- NULL for (i in 1:np) { pts1 <- .transect(tspts, i * pdist) pts <- cbind(pts, pts1) } allpts <- rbind(allpts, pts) } return(allpts) } else { x <- .pointsToMatrix(x) y <- .evenspace(x, interval) pts <- NULL for (i in 1:np) { pts1 <- .transect(y, i * pdist) pts <- cbind(pts, pts1) } return(pts) } } geosphere/R/geomedian.R0000644000176200001440000000310313472155746014522 0ustar liggesusers# Author: Robert J. Hijmans # March 2012 # version 1 # license GPL3 .geomedian <- function(xy, w=NULL) { xy <- .pointsToMatrix(xy) if (is.null(w)) { w <- 1 } else if (length(w) != nrow(xy)) { stop('length of weights not correct. It should be: ', nrow(xy)) } w <- w / sum(w) xyw <- cbind(xy, w) xy <- stats::na.omit(xyw) xy <- xyw[,1:2] w <- xyw[,3] est <- geomean(xy, w) fun <- function(p) { if (p[2] > 90 | p[2] < -90) { return(Inf) } else { p[1] = (p[1] + 180) %% 360 - 180 sum( distCosine(xy, p) * w) } } opt <- stats::optim(geomean(xy), fun) if (!is.null(opt$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/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/finalBearing.R0000644000176200001440000000114613472155746015160 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 <- .Call("_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]), PACKAGE='geosphere') r <- matrix(r, ncol=3, byrow=TRUE) # colnames(r) <- c('lon', 'lat', 'finalbearing') return(r[, 3]) } 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/vignettes/0000755000176200001440000000000013472363553014256 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. 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0ustar liggesusersPackage: geosphere Type: Package Title: Spherical Trigonometry Version: 1.5-10 Date: 2019-05-25 Imports: 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("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. License: GPL (>= 3) LazyLoad: yes NeedsCompilation: yes Packaged: 2019-05-26 01:10:19 UTC; rhijm Author: Robert J. Hijmans [cre, aut], Ed Williams [ctb], Chris Vennes [ctb] Maintainer: Robert J. Hijmans Repository: CRAN Date/Publication: 2019-05-26 04:50:14 UTC 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. Bug fix suggested by George Wang. geosphere/man/0000755000176200001440000000000013472155746013024 5ustar liggesusersgeosphere/man/destPoint.Rd0000644000176200001440000000355413472321670015262 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. \url{https://dx.doi.org/10.1007/s00190-012-0578-z}. Addenda: \url{http://geographiclib.sf.net/geod-addenda.html}. Also see \url{http://geographiclib.sourceforge.net/} } \examples{ p <- cbind(5,52) d <- destPoint(p,30,10000) d #final direction, when arriving at endpoint: finalBearing(d, p) } \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/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/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/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/greatCircleBearing.Rd0000644000176200001440000000151513472155746017031 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{http://www.edwilliams.org/avform.htm#Int} } \author{ Robert Hijmans based on formulae by Ed Williams } \examples{ greatCircleBearing(c(5,52), 45, n=12) } \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/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/dist2line.Rd0000644000176200001440000000345413472155746015216 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{over} functions in packages \code{sp} or \code{rgeos} for point-in-polygon queries. } \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/gcIntersectBearing.Rd0000644000176200001440000000230713472155746017057 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{http://www.edwilliams.org/avform.htm#Intersection} \url{http://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/geodesic.Rd0000644000176200001440000000664213472155746015105 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{http://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. \url{https://dx.doi.org/10.1007/s00190-012-0578-z}. Addenda: \url{http://geographiclib.sf.net/geod-addenda.html}. Also see \url{http://geographiclib.sourceforge.net/} } \seealso{ \code{\link{distGeo}} } \examples{ geodesic(cbind(0,0), 30, 1000000) geodesic_inverse(cbind(0,0), cbind(90,90)) } \keyword{ spatial } geosphere/man/distHaversine.Rd0000644000176200001440000000341713472155746016130 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{http://www.movable-type.co.uk/scripts/latlong.html} \url{http://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/gcMaxLat.Rd0000644000176200001440000000231513472155746015014 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{http://www.edwilliams.org/ftp/avsig/avform.txt} \url{http://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/distMeeus.Rd0000644000176200001440000000546513472155746015267 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{http://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/bearing.Rd0000644000176200001440000000302013472155746014715 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. \url{https://dx.doi.org/10.1007/s00190-012-0578-z}. Addenda: \url{http://geographiclib.sf.net/geod-addenda.html}. Also see \url{http://geographiclib.sourceforge.net/} } \seealso{ \code{ \link[geosphere]{bearingRhumb} } } \examples{ bearing(c(10,10),c(20,20)) } \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/distVincentySphere.Rd0000644000176200001440000000241413472155746017146 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{http://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/randomCoordinates.Rd0000644000176200001440000000233613472155746016772 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), \url{http://mathforum.org/kb/message.jspa?messageID=3985660&tstart=0} and suggstions by Michael Orion (randomCoordinates), \url{http://sci.tech-archive.net/Archive/sci.math/2005-09/msg04691.html} } \examples{ randomCoordinates(3) regularCoordinates(1) } \keyword{ spatial } geosphere/man/distGeo.Rd0000644000176200001440000000552713472155746014722 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{http://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. \url{https://dx.doi.org/10.1007/s00190-012-0578-z}. Addenda: \url{http://geographiclib.sf.net/geod-addenda.html}. Also see \url{http://geographiclib.sourceforge.net/} } \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/finalDirection.Rd0000644000176200001440000000263113472155746016247 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. \url{https://dx.doi.org/10.1007/s00190-012-0578-z}. Addenda: \url{http://geographiclib.sf.net/geod-addenda.html}. Also see \url{http://geographiclib.sourceforge.net/} } \seealso{ \code{ \link[geosphere]{bearing} } } \keyword{ spatial } geosphere/man/perimeter.Rd0000644000176200001440000000322313472322006015270 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. \url{https://dx.doi.org/10.1007/s00190-012-0578-z}. Addenda: \url{http://geographiclib.sf.net/geod-addenda.html}. Also see \url{http://geographiclib.sourceforge.net/} } \examples{ xy <- rbind(c(-180,-20), c(-140,55), c(10, 0), c(-140,-60), c(-180,-20)) perimeter(xy) } \keyword{ spatial } geosphere/man/geosphere-package.Rd0000644000176200001440000000443313472155746016671 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. \url{https://dx.doi.org/10.1007/s00190-012-0578-z}. Addenda: \url{http://geographiclib.sf.net/geod-addenda.html}. Also see \url{http://geographiclib.sourceforge.net/} \url{http://www.edwilliams.org/avform.htm} \url{http://www.movable-type.co.uk/scripts/latlong.html} \url{http://en.wikipedia.org/wiki/Great_circle_distance} \url{http://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/dist2gc.Rd0000644000176200001440000000266613472155746014664 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{http://www.movable-type.co.uk/scripts/latlong.html} \url{http://www.edwilliams.org/ftp/avsig/avform.txt} } \examples{ dist2gc(c(0,0),c(90,90),c(80,80)) } \keyword{ spatial } geosphere/man/greatCircle.Rd0000644000176200001440000000162513472155746015543 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{http://www.edwilliams.org/avform.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/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.Rd0000644000176200001440000000262613472155746015773 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{http://www.edwilliams.org/avform.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/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/bearingRhumb.Rd0000644000176200001440000000216313472155746015722 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{http://www.edwilliams.org/avform.htm#Rhumb} \url{http://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/distVincentyEllipsoid.Rd0000644000176200001440000000701313472155746017644 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{http://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{http://www.movable-type.co.uk/scripts/latlong-vincenty.html} \url{http://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/horizon.Rd0000644000176200001440000000137113472155746015005 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{http://www.edwilliams.org/avform.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/lengthLine.Rd0000644000176200001440000000114513472321755015400 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 \code{\link[rgeos:misc-gLength]{gLength}} } \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/distm.Rd0000644000176200001440000000177713472155746014447 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{http://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/daylength.Rd0000644000176200001440000000170513472155746015275 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}{Interger, day of the year (1..365) for 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/gcIntersect.Rd0000644000176200001440000000170713472155746015572 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{http://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/gcLon.Rd0000644000176200001440000000143013472155746014353 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{http://www.edwilliams.org/avform.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/distRhumb.Rd0000644000176200001440000000272013472155746015255 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{http://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/gcLat.Rd0000644000176200001440000000141513472155746014346 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{http://www.edwilliams.org/avform.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.Rd0000644000176200001440000000214013472155746016257 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{http://www.edwilliams.org/avform.htm#Rhumb} \url{http://www.movable-type.co.uk/scripts/latlong.html} \url{http://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/antipode.Rd0000644000176200001440000000222613472155746015120 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{http://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/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/distCosine.Rd0000644000176200001440000000211413472155746015415 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{http://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/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/area.Rd0000644000176200001440000000406413472155746014227 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. \url{https://dx.doi.org/10.1007/s00190-012-0578-z}. Addenda: \url{http://geographiclib.sf.net/geod-addenda.html}. Also see \url{http://geographiclib.sourceforge.net/} } \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 }