Calculate geodesics (great circle paths) on a flattened sphere (ellipsoid of rotation), using Vincenty's  formulae.
Geodesics is compatible with Julia version 0.7 upwards. Install using:
julia> import Pkg julia> Pkg.pkg"add https://github.com/anowacki/Geodesics.jl"
(Geodesics versions below 0.2 install on Julia v0.6 upwards, using
A standard forward Vincenty computation finds the end point on a flattened sphere,
given a starting location at longitude-latitude
(lon,lat), a forward azimuth
and a distance
dist travelled along the surface. You also need to specify the
semimajor (equatorial) radius of the ellipsoid
a and the flattening
Let's travel 30 km northeast of Trafalgar Square, london, and see where we end up:
julia> using Geodesics julia> lon, lat, az, dist, a, f = deg2rad(0.1281), deg2rad(51.5080), deg2rad(45), 30_000, Geodesics.EARTH_R_MAJOR_WGS84, Geodesics.F_WGS84 julia> Geodesics.forward(lon, lat, az, dist, a, f) (0.007590801193327456, 0.9023049202104327, 3.9311877139141966) julia> lon1, lat1, backazimuth = rad2deg.(ans) (0.4349208715005321, 51.69826376194629, 225.24046448096595)
So we get to here, which is most of the way to Chelmsford in Essex. Exciting!
Note that here, we've use the semimajor Earth radius from the WGS84 ellipsoid, and WGS84's flattening, avaialable in the geodesics package. If we'd assumed a perfectly spherical Earth, then we end up very slightly elsewhere.
A standard inverse Vincenty calculation finds the distance, azimuth and backazimuth
between two known points on the ellipsoid, (
lon1,lat1) and (
we need to specify the semimajor radius and flattening.
julia> lon1, lat1 = -1.4951547, 54.967618; # Jarrow julia> lon2, lat2 = -0.1270032, 51.4994794; # Palace of Westminster julia> a, f = Geodesics.EARTH_R_MAJOR_WGS84, Geodesics.F_WGS84; julia> dist, az, baz = Geodesics.inverse(deg2rad.((lon1, lat1, lon2, lat2))..., a, f) (396614.87733361026, 2.899562465488462, 6.06029305842987)
So the Jarrow Marchers had at the very least (ignoring topography) 396 km to walk!
Geodesics.jl comes with convenience functions which use angles in either degrees
or radians, as you prefer. It is also often true that you are only interested in
one quantity at the time, such only wanting to find out the forward azimuth between
two known points. In this case, use
Geodesics.azimuth to find which azimuth you should
sail along (on a perfectly calm day) to reach St Malo from Jersey:
julia> lon0, lat0 = -2.117641, 49.176924; # St. Helier, Jersey julia> lon1, lat1 = -2.032614, 48.641570; # St. Malo, France julia> Geodesics.azimuth(lon0, lat0, lon1, lat1) # Uses degrees and WGS84 ellipsoid by default 173.99132840869288
Note that I didn't need to specify the flattening (which defaults to that of
the WGS84 ellipsoid), but this can be overridden using the
f keyword argument.
The full list of convenience functions:
angular_distance: Find the surface distance between two points in terms of an angle measured from the centre of the ellipsoid.
surface_distance: The distance between two points.
angular_step: Find the end point and backazimuth from one point when travelling a set angular distance along a defined azimuth.
azimuth: The forward azimuth between two points.
Choice of geodesic calculation
This package so far only implements Vincenty's methods, but others are available. Pull requests to add these are welcome.
Thaddeus Vincenty published the forward and inverse methods used in this package in the following paper:
- Vincenty, T. (1975). "Direct and Inverse Solutions of Geodesics on the Ellipsoid with application of nested equations" (PDF). Survey Review. XXIII (176): 88–93. doi:10.1179/sre.19184.108.40.206.