Geodesics on an Ellipsoid
This page is a web resource for the papers
Charles F. F. Karney,
Algorithms for geodesics,
J. Geodesy 87(1), 43–55 (Jan. 2013);
DOI:
10.1007/s001900120578z;
pdf,
addenda.
preprint
arXiv:1109.4448,
errata.
Charles F. F. Karney,
Geodesics on an ellipsoid of revolution,
Feb. 2011,
arXiv:1102.1215,
pdf,
errata.
The implementation of the algorithms in this paper are available
as part of GeographicLib which is licensed under the
MIT/X11 License;
see LICENSE.txt for the terms.

GeographicLib home page.

GeographicLib documentation:

The C++ class
Geodesic, which solves the direct and inverse geodesic
problems.

The C++ class
GeodesicLine, which solves for points on a given geodesic.

Companion classes
GeodesicExact and
GeodesicLineExact, which implement the solution in
terms of elliptic integrals.

The C++ classes for geodesic projections:

The commandline utility
GeodSolve, for solving geodesic problems and an
online geodesic calculator.

The commandline utility
Planimeter, for measuring the area of geodesic
polygons and an
online planimeter.

The commandline utility
GeodesicProj, for performing geodesic projections.

JavaScript tools for geodesic calculations,
geodcalc, and for
displaying geodesics on Google Maps,
geodgoogle.

Transforming between geocentric and geodetic coordinates
using the method described in Appendix B of
Geodesics on an ellipsoid of revolution:

the C++ class
Geocentric,
for performing the transformation and its inverse;

the utility
CartConvert,
which is a commandline interface to this class.

GeographicLib also contains implementations of the
geodesic routines in
other languages:

Download GeographicLib
Additional material:

Supplementary documentation on
geodesics on an ellipsoid of revolution.

Test set for geodesics

Algorithms for geodesics
gives the series for geodesics accurate to 6th order.

The formulation in terms of elliptic integrals used by
GeodesicExact and
GeodesicLineExact
is given in Appendix D of
Geodesics on an ellipsoid of revolution.
Further details are given in
Geodesics in terms of elliptic integrals.

In some application it may be important to minimize roundoff
errors when taking the difference of two trigonometric sums.
This may be accomplished by using
Clenshaw evaluation of differenced sums.

Various ways that the distance along a meridian can be solved
in terms of elliptic integrals are given in
Parameters for the meridian.

Some notes on solving the inverse geodesic problem in the case of
short geodesics.

Some notes on geodesics on a triaxial ellipsoid are given
in Geodesics on a triaxial
ellipsoid. This examines the solution to this problem found
by Jacobi in 1839.

In the same paper, Jacobi gave a conformal projection for a
triaxial ellipsoid. This is expressed in terms of elliptic
integrals in these notes on
Jacobi's conformal projection.

A geodesic bibliography.
This lists many papers treating geodesics on an ellipsoid and
includes links to online versions of the papers.

Some scans of geodesic papers.

Bessel's paper on geodesics: F. W. Bessel,
The calculation of longitude and latitude
from geodesic measurements (1825),
Astron. Nachr. 331(8), 852–861 (Sept. 2010),
translated by C. F. F. Karney and R. E. Deakin; preprint:
arXiv:0908.1824
(errata).

F. R. Helmert,
Mathematical and Physical Theories of Higher Geodesy,
Vol. 1 and
Vol. 2,
English translation by Aeronautical Chart and Information Center
(St. Louis, 1964).

The Wikipedia page,
Geodesics on an ellipsoid.
Charles Karney
<charles@karney.com>
(20150514)
GeographicLib home