[R-sig-Geo] LL from bearing and distance
Eric Archer
Eric.Archer at noaa.gov
Fri Sep 28 00:59:27 CEST 2007
All,
I have ported JavaScript code from a couple of websites to R for three
functions that solve my problem.
'calc.destination.sphere' assumes a perfect spheroid with a given radius
'calc.destination.ellipsoid' assumes an ellipsoid
'calc.destination.Vincenty' uses the Vincenty model which appears to
have the most accuracy
I'm unfamiliar with the steps involved in including these functions into
a package, but will be happy to do it if someone can point me in the
right direction.
Cheers,
eric.
---
as.radians <- function(degrees) degrees * pi / 180
as.degrees <- function(radians) radians * 180 / pi
as.bearing <- function(radians) (as.degrees(radians) + 360) %% 360
ellipsoid <- function(model = "WGS84") {
switch(model,
WGS84 = c(a = 6378137, b = 6356752.3142, f = 1 / 298.257223563),
GRS80 = c(a = 6378137, b = 6356752.3141, f = 1 / 298.257222101),
Airy = c(a = 6377563.396, b = 6356256.909, f = 1 / 299.3249646),
International = c(a = 6378888, b = 6356911.946, f = 1 / 297),
Clarke = c(a = 6378249.145, b = 6356514.86955, f = 1 / 293.465),
GRS67 = c(a = 6378160, b = 6356774.719, f = 1 / 298.25),
c(a = NA, b = NA, f = NA)
)
}
calc.destination.sphere <- function(lat, lon, brng, dist, dist.units =
"km", radius = 6371) {
# lat, lon : lattitude and longitude in decimal degrees
# brng : bearing from 0 to 360 degrees
# dist : distance travelled
# dist.units : units of distance "km" (kilometers), "nm" (nautical
miles), "mi" (statute miles)
# radius : radius of sphere in kilometers
#
# Code adapted from JavaScript by Chris Veness
(scripts at movable-type.co.uk) at
http://www.movable-type.co.uk/scripts/latlong.html#ellipsoid
# originally from Ed Williams' Aviation Formulary,
http://williams.best.vwh.net/avform.htm#LL
#
#
dist <- switch(dist.units,
km = dist,
nm = dist * 1.852,
mi = dist * 1.609344
)
lat <- as.radians(lat)
lon <- as.radians(lon)
brng <- as.radians(brng)
psi <- dist / radius
lat2 <- asin(sin(lat) * cos(psi) + cos(lat) * sin(psi) * cos(brng))
lon2 <- lon + atan2(sin(brng) * sin(psi) * cos(lat), cos(psi) -
sin(lat) * sin(lat2))
if (is.nan(lat2) || is.nan(lon2)) return(c(lat = NA, lon = NA))
c(lat = as.degrees(lat2), lon = as.degrees(lon2))
}
calc.destination.ellipsoid <- function(lat, lon, brng, dist, dist.units
= "km", model = "WGS84") {
# lat, lon : lattitude and longitude in decimal degrees
# brng : bearing from 0 to 360 degrees
# dist : distance travelled
# dist.units : units of distance "km" (kilometers), "nm" (nautical
miles), "mi" (statute miles)
# model : choice of ellipsoid model ("WGS84", "GRS80", "Airy",
"International", "Clarke", "GRS67")
#
# Code adapted from JavaScript by Larry Bogan (larry at go.ednet.ns.ca)
at http://www.go.ednet.ns.ca/~larry/bsc/jslatlng.html
#
#
dist <- switch(dist.units,
km = dist,
nm = dist * 1.852,
mi = dist * 1.609344
)
lat <- as.radians(lat)
lon <- as.radians(lon)
brng <- as.radians(brng)
e <- 0.08181922
ellips <- ellipsoid(model)
radius <- (ellips["a"] / 1000) * (1 - e ^ 2) / ((1 - e ^ 2 * sin(lat)
^ 2) ^ 1.5)
psi <- dist / radius
phi <- pi / 2 - lat
arc.cos <- cos(psi) * cos(phi) + sin(psi) * sin(phi) * cos(brng)
lat2 <- as.degrees((pi / 2) - acos(arc.cos))
arc.sin <- sin(brng) * sin(psi) / sin(phi)
lon2 <- as.degrees(lon + asin(arc.sin))
c(lat = lat2, lon = lon2)
}
calc.destination.Vincenty <- function(lat, lon, brng, dist, dist.units =
"km", model = "WGS84") {
# lat, lon : lattitude and longitude in decimal degrees
# brng : bearing from 0 to 360 degrees
# dist : distance travelled
# dist.units : units of distance "km" (kilometers), "nm" (nautical
miles), "mi" (statute miles)
# model : choice of ellipsoid model ("WGS84", "GRS80", "Airy",
"International", "Clarke", "GRS67")
#
# Code adapted from JavaScript by Chris Veness
(scripts at movable-type.co.uk) at
http://www.movable-type.co.uk/scripts/latlong-vincenty-direct.html
# Original reference (http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf):
# Vincenty, T. 1975. Direct and inverse solutions of geodesics on
the ellipsoid with application of nested equations.
# Survey Review 22(176):88-93
#
#
dist <- switch(dist.units,
km = dist,
nm = dist * 1.852,
mi = dist * 1.609344
)
dist <- dist * 1000
lat <- as.radians(lat)
lon <- as.radians(lon)
brng <- as.radians(brng)
ellips <- ellipsoid(model)
sin.alpha1 <- sin(brng)
cos.alpha1 <- cos(brng)
tan.u1 <- (1 - ellips["f"]) * tan(lat)
cos.u1 <- 1 / sqrt(1 + (tan.u1 ^ 2))
sin.u1 <- tan.u1 * cos.u1
sigma1 <- atan2(tan.u1, cos.alpha1)
sin.alpha <- cos.u1 * sin.alpha1
cos.sq.alpha <- 1 - (sin.alpha ^ 2)
u.sq <- cos.sq.alpha * ((ellips["a"] ^ 2) - (ellips["b"] ^ 2)) /
(ellips["b"] ^ 2)
cap.A <- 1 + u.sq / 16384 * (4096 + u.sq * (-768 + u.sq * (320 - 175 *
u.sq)))
cap.B <- u.sq / 1024 * (256 + u.sq * (-128 + u.sq * (74 - 47 * u.sq)))
sigma <- dist / (ellips["b"] * cap.A)
sigma.p <- 2 * pi
cos.2.sigma.m <- cos(2 * sigma1 + sigma)
while(abs(sigma - sigma.p) > 1e-12) {
cos.2.sigma.m <- cos(2 * sigma1 + sigma)
sin.sigma <- sin(sigma)
cos.sigma <- cos(sigma)
delta.sigma <- cap.B * sin.sigma * (cos.2.sigma.m + cap.B / 4 *
(cos.sigma *
(-1 + 2 * cos.2.sigma.m ^ 2) - cap.B / 6 * cos.2.sigma.m *
(-3 + 4 * sin.sigma ^ 2) * (-3 + 4 * cos.2.sigma.m ^ 2)))
sigma.p <- sigma
sigma <- dist / (ellips["a"] * cap.A) + delta.sigma
}
tmp <- sin.u1 * sin.sigma - cos.u1 * cos.sigma * cos.alpha1
lat2 <- atan2(sin.u1 * cos.sigma + cos.u1 * sin.sigma * cos.alpha1,
(1 - ellips["f"]) * sqrt(sin.alpha ^ 2 + tmp ^ 2))
lambda <- atan2(sin.sigma * sin.alpha1, cos.u1 * cos.sigma - sin.u1 *
sin.sigma * cos.alpha1)
cap.C <- ellips["f"] / 16 * cos.sq.alpha * (4 + ellips["f"] *
(ellips["f"] - 3 * cos.sq.alpha))
cap.L <- lambda - (1 - cap.C) * ellips["f"] * sin.alpha *
(sigma + cap.C * sin.sigma * (cos.2.sigma.m + cap.C * cos.sigma *
(-1 + 2 * cos.2.sigma.m ^ 2)))
c(lat = as.degrees(lat2), lon = as.degrees(lon + cap.L))
}
--
Eric Archer, Ph.D.
NOAA-SWFSC
8604 La Jolla Shores Dr.
La Jolla, CA 92037
858-546-7121,7003(FAX)
eric.archer at noaa.gov
http://swfsc.noaa.gov/prd-etp.aspx
"Innocence about Science is the worst crime today."
- Sir Charles Percy Snow
"Lighthouses are more helpful than churches."
- Benjamin Franklin
"...but I'll take a GPS over either one."
- John C. "Craig" George
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