Algorithms for geodesics

Charles F. F. Karney

doi:10.1007/s00190-012-0578-z

Download PDF

Journal of Geodesy

January 2013, Volume 87, Issue 1, pp 43–55

Algorithms for geodesics

Authors
Authors and affiliations

Charles F. F. KarneyEmail author

Open Access

Original Article

First Online:: 26 June 2012

Received:: 21 September 2011
Accepted:: 30 May 2012

24 Citations
16 Shares
11k Downloads

Abstract

Algorithms for the computation of geodesics on an ellipsoid of revolution are given. These provide accurate, robust, and fast solutions to the direct and inverse geodesic problems and they allow differential and integral properties of geodesics to be computed.

Keywords

Geometrical geodesy Geodesics Polygonal areas Gnomonic projection Numerical methods

Download to read the full article text

References

Beltrami E (1865) Risoluzione del problema: Riportare i punti di una superficie sopra un piano in modo che le linee geodetiche vengano rappresentate da linee rette. Annali Mat Pura Appl 7:185–204. http://books.google.com/books?id=dfgEAAAAYAAJ&pg=PA185
Bessel FW (1825) Über die Berechnung der geographischen Längen und Breiten aus geodätischen Vermessungen. Astron Nachr 4(86):241–254. http://adsabs.harvard.edu/abs/1825AN......4..241B [translated into English by Karney CFF and Deakin RE as The calculation of longitude and latitude from geodesic measurements. Astron Nachr 331(8):852–861 (2010)]Google Scholar
Bowring BR (1997) The central projection of the spheroid and surface lines. Surv Rev 34(265): 163–173CrossRefGoogle Scholar
Bugayevskiy LM, Snyder JP (1995) Map projections: a reference manual. Taylor & Francis, LondonGoogle Scholar
Christoffel EB (1868) Allgemeine Theorie der geodätischen Dreiecke. Math Abhand König Akad der Wiss zu Berlin 8:119–176. http://books.google.com/books?id=EEtFAAAAcAAJ&pg=PA119
Clairaut AC (1735) Détermination géometrique de la perpendiculaire à la méridienne tracée par M Cassini. Mém de l’Acad Roy des Sciences de Paris 1733:406–416. http://books.google.com/books?id=GOAEAAAAQAAJ&pg=PA406
Clenshaw CW (1955) A note on the summation of Chebyshev series. Math Tables Other Aids Comput 9(51): 118–120Google Scholar
Danielsen JS (1989) The area under the geodesic. Surv Rev 30(232): 61–66CrossRefGoogle Scholar
Gauss CF (1828) Disquisitiones Generales circa Superficies Curvas. Dieterich, Göttingen. http://books.google.com/books?id=bX0AAAAAMAAJ [translated into English by Morehead JC and Hiltebeitel AM as General investigations of curved surfaces of 1827 and 1825. Princeton Univ Lib (1902) http://books.google.com/books?id=a1wTJR3kHwUC]
Helmert FR (1880) Die Mathematischen und Physikalischen Theorieen der Höheren Geodäsie, vol 1. Teubner, Leipzig. http://books.google.com/books?id=qt2CAAAAIAAJ [translated into English by Aeronautical chart and information center (St Louis, 1964) as Mathematical and physical theories of higher geodesy, part 1. http://geographiclib.sf.net/geodesic-papers/helmert80-en.html]
Jacobi CGJ (1891) Über die Curve, welche alle von einem Punkte ausgehenden geodätischen Linien eines Rotationsellipsoides berührt. In: Weierstrass KTW (ed) Gesammelte Werke, vol 7. Reimer, Berlin, pp 72–87. http://books.google.com/books?id=_09tAAAAMAAJ&pg=PA72 [op post, completed by Wangerin FHA]
Karney CFF (2011) Geodesics on an ellipsoid of revolution. Tech rep, SRI International. http://arxiv.org/abs/1102.1215v1
Karney CFF (2012) GeographicLib, version 1.22. http://geographiclib.sf.net
Legendre AM (1806) Analyse des triangles tracés sur la surface d’un sphéroïde. Mém de l’Inst Nat de France, 1st sem, pp 130–161. http://books.google.com/books?id=-d0EAAAAQAAJ&pg=PA130-IA4
Letoval’tsev IG (1963) Generalization of the gnomonic projection for a spheroid and the principal geodetic problems involved in the alignment of surface routes. Geod Aerophotogr 5:271–274 [translation of Geodeziya i Aerofotos’emka 5:61–68 (1963)]Google Scholar
Maxima (2009) A computer algebra system, version 5.20.1. http://maxima.sf.net
Olver FWJ, Lozier DW, Boisvert RF, Clark CW (eds) (2010) NIST handbook of mathematical functions. Cambridge University Press, London. http://dlmf.nist.gov
Oriani B (1806) Elementi di trigonometria sferoidica, Pt 1. Mem dell’Ist Naz Ital 1(1):118–198. http://books.google.com/books?id=SydFAAAAcAAJ&pg=PA118
Oriani B (1808) Elementi di trigonometria sferoidica, Pt 2. Mem dell’Ist Naz Ital 2(1):1–58. http://www.archive.org/stream/memoriedellistit21isti#page/1
Oriani B (1810) Elementi di trigonometria sferoidica, Pt 3. Mem dell’Ist Naz Ital 2(2):1–58. http://www.archive.org/stream/memoriedellistit22isti#page/1
Rapp RH (1993) Geometric geodesy, part II. Tech rep, Ohio State University. http://hdl.handle.net/1811/24409
Snyder JP (1987) Map projection—a working manual. Professional Paper 1395, US Geological Survey. http://pubs.er.usgs.gov/publication/pp1395
Vermeille H (2002) Direct transformation from geocentric coordinates to geodetic coordinates. J Geod 76(9): 451–454CrossRefGoogle Scholar
Vincenty T (1975a) Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations. Surv Rev 23(176):88–93 [addendum: Surv Rev 23(180):294 (1976)]Google Scholar
Vincenty T (1975b) Geodetic inverse solution between antipodal points. http://geographiclib.sf.net/geodesic-papers/vincenty75b.pdf (unpublished report dated Aug 28)
Wessel P, Smith WHF (2010) Generic mapping tools, 4.5.5. http://gmt.soest.hawaii.edu/
Williams R (1997) Gnomonic projection of the surface of an ellipsoid. J Navig 50(2): 314–320CrossRefGoogle Scholar

Algorithms for geodesics

Abstract

Keywords

Copyright information

Authors and Affiliations

Personalised recommendations

Export citation

Cookies