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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01//EN"
"http://www.w3.org/TR/html4/strict.dtd">
<html>
<head>
<title>
Geodesics on an Ellipsoid
</title>
<meta name="description" content="Geodesic Problem" />
<meta name="keywords"
content="geodesics,
direct geodesic problem,
inverse geodesic problem,
geodesic projections,
geodesic scale,
reduced length,
spheroidal trigonometry,
azimuthal equidistant projection,
Cassini-Soldner projection,
spheroidal gnomonic projection,
geodesic area,
triangulation,
maritime boundaries,
median line,
WGS84 ellipsoid" />
<meta name="author" content="Charles F. F. Karney" />
<link rel="stylesheet" type="text/css" href="default.css">
<meta http-equiv="Content-Type" content="text/html; charset=ISO-8859-1">
</head>
<body topmargin=10 leftmargin=10>
<h3>Geodesics on an Ellipsoid</h3>
<p>
This page is a web resource for the papers
<blockquote>
Charles F. F. Karney,<br>
<a href="https://doi.org/10.1007/s00190-012-0578-z">
<i>Algorithms for geodesics</i></a>,<br>
J. Geodesy <b>87</b>(1), 43–55 (Jan. 2013);<br>
DOI:
<a href="https://doi.org/10.1007/s00190-012-0578-z">
10.1007/s00190-012-0578-z</a>;
<a href="https://doi.org/10.1007/s00190-012-0578-z">pdf</a>,
<a href="geod-addenda.html"><b>addenda</b></a>.<br>
preprint
<a href="https://arxiv.org/abs/1109.4448"> arXiv:1109.4448</a>,
<a href="geod-addenda.html#geodalg-errata"><b>errata</b></a>.
<br><br>
Charles F. F. Karney,<br>
<a href="https://arxiv.org/abs/1102.1215">
<i>Geodesics on an ellipsoid of revolution</i></a>,<br>
Feb. 2011,
<a href="https://arxiv.org/abs/1102.1215">arXiv:1102.1215</a>,
<a href="https://arxiv.org/pdf/1102.1215">pdf</a>,
<a href="geod-addenda.html#geod-errata"><b>errata</b></a>.
</blockquote>
The implementation of the algorithms in this paper are available
as part of GeographicLib which is licensed under the
<a href="https://opensource.org/licenses/MIT">MIT/X11 License</a>;
see <a href="html/LICENSE.txt">LICENSE.txt</a> for the terms.
</p>
<ul>
<li>
<a href="index.html">GeographicLib home page</a>.
<li>
<a href="html">GeographicLib documentation</a>:
<ul>
<li>
The C++ class
<a href="html/classGeographicLib_1_1Geodesic.html">
Geodesic</a>, which solves the direct and inverse geodesic
problems.
<li>
The C++ class
<a href="html/classGeographicLib_1_1GeodesicLine.html">
GeodesicLine</a>, which solves for points on a given geodesic.
<li>
Companion classes
<a href="html/classGeographicLib_1_1GeodesicExact.html">
GeodesicExact</a> and
<a href="html/classGeographicLib_1_1GeodesicLineExact.html">
GeodesicLineExact</a>, which implement the solution in
terms of elliptic integrals.
<li>
The C++ classes for geodesic projections:
<ul>
<li>
<a href="html/classGeographicLib_1_1AzimuthalEquidistant.html">
AzimuthalEquidistant</a>,
<li>
<a href="html/classGeographicLib_1_1CassiniSoldner.html">
CassiniSoldner</a>,
<li>
<a href="html/classGeographicLib_1_1Gnomonic.html">
Gnomonic</a>.
</ul>
<li>
The command-line utility
<a href="html/GeodSolve.1.html">
GeodSolve</a>, for solving geodesic problems and an
<a href="cgi-bin/GeodSolve">
online geodesic calculator</a>.
<li>
The command-line utility
<a href="html/Planimeter.1.html">
Planimeter</a>, for measuring the area of geodesic
polygons and an
<a href="cgi-bin/Planimeter"> online planimeter</a>.
<li>
The command-line utility
<a href="html/GeodesicProj.1.html">
GeodesicProj</a>, for performing geodesic projections.
<li>
JavaScript tools for geodesic calculations,
<a href="scripts/geod-calc.html">geod-calc</a>, and for
displaying geodesics on Google Maps,
<a href="scripts/geod-google.html">geod-google</a>.
<li>
Transforming between geocentric and geodetic coordinates
using the method described in Appendix B of
<a href="https://arxiv.org/abs/1102.1215">
<i>Geodesics on an ellipsoid of revolution</i></a>:
<ul>
<li>
the C++ class
<a href="html/classGeographicLib_1_1Geocentric.html">
Geocentric</a>,
for performing the transformation and its inverse;
<li>
the utility
<a href="html/CartConvert.1.html">
CartConvert</a>,
which is a command-line interface to this class.
</ul>
<li>
GeographicLib also contains implementations of the
geodesic routines in
<a href="html/other.html">other languages</a>:
<ul>
<li>Python:
<a href="https://pypi.python.org/pypi/geographiclib">
https://pypi.python.org/pypi/geographiclib</a>
<li>JavaScript:
<a href="https://geographiclib.sourceforge.io/scripts/geographiclib.js">
https://geographiclib.sourceforge.io/scripts/geographiclib.js</a><br>
Examples: geodesic calculations,
<a href="scripts/geod-calc.html">geod-calc</a>;
displaying geodesics on Google Maps,
<a href="scripts/geod-google.html">geod-google</a>.
<li>MATLAB/Octave:
<a href="https://www.mathworks.com/matlabcentral/fileexchange/50605">
File ID: 50605 (geodesics + other components of
GeographicLib)</a>
<li><a href="html/C/index.html">C</a>,
<a href="html/Fortran/index.html">Fortran</a>, and
<a href="html/java/index.html">Java</a>:
small self-contained libraries in
these languages are also part of the source
distribution of GeographicLib.
</ul>
</ul>
<li>
<a href="https://sourceforge.net/projects/geographiclib/files/distrib">
Download GeographicLib</a>
</ul>
<p>
Additional material:
</p>
<ul>
<li>
<a href="html/geodesic.html">Supplementary documentation</a> on
geodesics on an ellipsoid of revolution.
<li>
<a href="https://doi.org/10.5281/zenodo.32156">
<i>Test set for geodesics</i></a>
<li>
<a href="https://doi.org/10.1007/s00190-012-0578-z">
<i>Algorithms for geodesics</i></a>
gives the series for geodesics accurate to 6th order.
<ul>
<li>
<a href="html/geodesic.html#geodseries">
Series for geodesic calculations to 10th order</a>.
<li>
Series for geodesic calculations to 30th order:
<a href="html/geodseries30.html">geodseries30.html</a>.
<li>
Maxima code to generate the series for geodesics to arbitrary order:
<a href="html/geod.mac">geod.mac</a>. There is brief
documentation at the top of the file.
<li>
<a href="http://maxima.sourceforge.net/">Download maxima</a>.
</ul>
<li>
The formulation in terms of elliptic integrals used by
<a href="html/classGeographicLib_1_1GeodesicExact.html">
GeodesicExact</a> and
<a href="html/classGeographicLib_1_1GeodesicLineExact.html">
GeodesicLineExact</a>
is given in Appendix D of
<a href="https://arxiv.org/abs/1102.1215">
<i>Geodesics on an ellipsoid of revolution</i></a>.
Further details are given in
<a href="html/geodesic.html#geodellip">
Geodesics in terms of elliptic integrals</a>.
<li>
In some application it may be important to minimize round-off
errors when taking the difference of two trigonometric sums.
This may be accomplished by using
<a href="html/rhumb.html#dividedclenshaw">
Clenshaw evaluation of differenced sums</a>.
<li>
Various ways that the distance along a meridian can be solved
in terms of elliptic integrals are given in
<a href="html/geodesic.html#meridian">
Parameters for the meridian</a>.
<li>
Some notes on solving the inverse geodesic problem in the case of
<a href="html/geodesic.html#geodshort">short geodesics</a>.
<li>
Some notes on geodesics on a <i>triaxial</i> ellipsoid are given
in <a href="html/triaxial.html">Geodesics on a triaxial
ellipsoid</a>. This examines the solution to this problem found
by Jacobi in 1839.
<li>
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
<a href="html/jacobi.html">
Jacobi's conformal projection</a>.
<li>
<a href="geodesic-papers/biblio.html">
A geodesic bibliography</a>.
This lists many papers treating geodesics on an ellipsoid and
includes links to online versions of the papers.
<li>
Some <a href="geodesic-papers">scans</a> of geodesic papers.
<li>
Bessel's paper on geodesics: F. W. Bessel,
<a href="https://doi.org/10.1002/asna.201011352">
<i>The calculation of longitude and latitude
from geodesic measurements (1825)</i></a>,
Astron. Nachr. <b>331</b>(8), 852–861 (Sept. 2010),
translated by C. F. F. Karney and R. E. Deakin; preprint:
<a href="https://arxiv.org/abs/0908.1824">arXiv:0908.1824</a>
(<a href="bessel-errata.html"><b>errata</b></a>).
<li>
F. R. Helmert,
<a href="https://doi.org/10.5281/zenodo.32050">
<i>Mathematical and Physical Theories of Higher Geodesy</i>,
Vol. 1</a> and
<a href="https://doi.org/10.5281/zenodo.32051">
Vol. 2</a>,
English translation by Aeronautical Chart and Information Center
(St. Louis, 1964).
<li>
The Wikipedia page,
<a href="https://en.wikipedia.org/wiki/Geodesics_on_an_ellipsoid">
Geodesics on an ellipsoid</a>.
</ul>
<hr>
<address>Charles Karney
<a href="mailto:[email protected]"><[email protected]></a>
(2017-09-30)</address>
<br>
<a href="https://geographiclib.sourceforge.io">
GeographicLib home
</a>
</body>
</html>