# Geodesic Calculations

Geodesic calculations are calculations along lines (great circle) on the surface of the earth. They can answer questions like:

- What is the distance between these two points?
- If I travel X meters from point A at bearing phi, where will I be.

They are done in native lat-long coordinates, rather than in projected coordinates.

# Mailing list threads

- http://thread.gmane.org/gmane.comp.gis.proj-4.devel/3361
- http://thread.gmane.org/gmane.comp.gis.proj-4.devel/3375
- http://thread.gmane.org/gmane.comp.gis.proj-4.devel/3435
- http://thread.gmane.org/gmane.comp.gis.proj-4.devel/3588
- http://thread.gmane.org/gmane.comp.gis.proj-4.devel/3925
- http://thread.gmane.org/gmane.comp.gis.proj-4.devel/4047
- http://thread.gmane.org/gmane.comp.gis.proj-4.devel/4083

# Terminology

The shortest distance on the surface of a solid is generally termed a geodesic, be it an ellipsoid of revolution, aposphere, etc. On a sphere, the geodesic is termed a Great Circle.

HOWEVER, when computing the distance between two points using a projected coordinate system, that is a conformal projection such as Transverse Mercator, Oblique Mercator, Normal Mercator, Stereographic, or Lambert Conformal Conic - that then is a GRID distance which can be converted to an equivalent GEODETIC distance using the function for "Scale Factor at a Point." The conversion is then termed "Grid Distance to Geodetic Distance," even though it will not be as exactly correct as a true ellipsoidal geodesic. Closer to the truth with a TM than with a Lambert or other conformal projection, but still not exactly "on."

So, it can be termed "geodetic distance" or a "geodesic distance," depending on just how you got there ...

# The Math

## Spherical Approximation

The simplest way to compute geodesics is using a sphere as an approximation for the earth. This from Mikael Rittri on the Proj mailing list:

If 1 percent accuracy is enough, I think you can use spherical formulas with a fixed Earth radius. You can find good formulas in the Aviation Formulary of Ed Williams, http://williams.best.vwh.net/avform.htm.

For the fixed Earth radius, I would choose the average of the:

c = radius of curvature at the poles,

b^{2}/ a = radius of curvature in a meridian plane at the equator,

since these are the extreme values for the local radius of curvature of the earth ellipsoid.

If your coordinates are given in WGS84, then

c = 6 399 593.626 m,

b^{2}/ a = 6 335 439.327 m,

(see http://home.online.no/~sigurdhu/WGS84_Eng.html) so their average is 6,367,516.477 m. The maximal error for distance calculation should then be less than 0.51 percent.

When computing the azimuth between two points by the spherical formulas, I think the maximal error on WGS84 will be 0.2 degrees, at least if the points are not too far away (less than 1000 km apart, say). The error should be maximal near the equator, for azimuths near northeast etc.

I am not sure about the spherical errors for the forward geodetic problem: point positioning given initial point, distance and azimuth.

## Ellipsoidal Approximation

For more accuracy, the earth can be approximated with an ellipsoid, complicating the math somewhat. See the wikipedia page, Geodesics on an ellipsoid, for more information.

### Thaddeus Vincenty's method, April 1975

For a very good procedure to calculate inter point distances see:

http://www.ngs.noaa.gov/PC_PROD/Inv_Fwd/ (Fortan code, DOS executables, and an online app)

and algorithm details published in:

#### Javascript code

Chris Veness has coded Vincenty's formulas as JavaScript.

distance: http://www.movable-type.co.uk/scripts/latlong-vincenty.html

direct: http://www.movable-type.co.uk/scripts/latlong-vincenty-direct.html

#### C code

From Gerald Evenden: a library of the converted NGS Vincenty geodesic procedure and an application program, 'geodesic'. In the case of a spherical earth Snyder's preferred equations are used.

The link in this message is broken. The correct URL is http://home.comcast.net/~gevenden56/proj/

Earlier Mr. Evenden had posted to the PROJ.4 mailing list this code for determination of true distance and respective forward and back azimuths between two points on the ellipsoid. Good for any pair of points that are not antipodal. Later he posted that this was not in fact the translation of NGS FORTRAN code, but something else. But, for what it's worth, here is the posted code (source unknown):

### PROJ.4 - geod program

The PROJ.4 geod program can be used for great circle distances on an ellipsoid. Currently the underlying geodesic calculation API is not exposed as part of the PROJ.4 library. Gerald writes that geod is based upon a poorer algorithm (than the NGS procedure above) and he no longer supports it. But it is probably good enough for most applications. The method is documented here:

Paul D. Thomas, 1970

"Spheroidal Geodesics, Reference Systems, and
Local Geometry"

U.S. Naval Oceanographic Office, p. 162

Engineering Library 526.3 T36s

http://handle.dtic.mil/100.2/AD0703541

### GeographicLib::Geodesic

Charles Karney has written a C++ class to do geodesic calculations and a utility GeodSolve to call it. See

An online version of GeodSolve is available at

This is an attempt to do geodesic calculations "right", i.e.,

- accurate to round-off (i.e., about 15 nm);
- inverse solution always succeeds (even for near anti-podal points);
- reasonably fast (comparable in speed to Vincenty);
- differential properties of geodesics are computed (these give the scales of geodesic projections);
- the area between a geodesic and the equator is computed (allowing the area of geodesic polygons to be found);
- included also is an implementation in terms of elliptic integrals which can deal with ellipsoids with 0.01 < b/a < 100.

A Javascript implementation is included, see

- geod-calc, a text interface to geodesic calculations;
- geod-google, a tool for drawing geodesics on Google Maps.

Implementations in Python, Matlab, C, Fortran, and Java are also available.

The algorithms are described in

- C. F. F. Karney,
*Algorithms for geodesics*, J. Geodesy**87**(1), 43-55 (2013),

DOI: 10.1007/s00190-012-0578-z; addenda geod-addenda.html.

## Triaxial Ellipsoid

A triaxial ellipsoid is a marginally better approximation to the shape of the earth than an ellipsoid of revolution. The problem of geodesics on a triaxial ellipsoid was solved by Jacobi in 1838. For a discussion of this problem see

- http://geographiclib.sourceforge.net/html/triaxial.html
- the wikipedia entry: Geodesics on a triaxial ellipsoid

# The History

The bibliography of papers on the geodesic problem for an ellipsoid is available at

this includes links to online copies of the papers.