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/s00190-012-0578-z; pdf, addenda.
preprint arXiv:1109.4448, errata.

Charles F. F. Karney,
Geodesics on an ellipsoid of revolution,
Feb. 2011, arXiv:1102.1215, pdf, errata.

• 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:
    • AzimuthalEquidistant,
    • CassiniSoldner,
    • Gnomonic.
  • The command-line utility GeodSolve, for solving geodesic problems and an online geodesic calculator.
  • The command-line utility Planimeter, for measuring the area of geodesic polygons and an online planimeter.
  • The command-line utility GeodesicProj, for performing geodesic projections.
  • JavaScript tools for geodesic calculations, geod-calc, and for displaying geodesics on Google Maps, geod-google.
  • 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 command-line interface to this class.
  • GeographicLib also contains implementations of the geodesic routines in C, Fortran, Java, JavaScript, Python, and Octave/MATLAB (in addition to C++). For details see this link.
• 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.
  • Series for geodesic calculations to 10th order.
  • Series for geodesic calculations to 30th order: geodseries30.html.
  • Maxima code to generate the series for geodesics to arbitrary order: geod.mac. There is brief documentation at the top of the file.
  • Download maxima.
• 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 round-off 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.