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Journal of Geodesy

, Volume 87, Issue 1, pp 43–55 | Cite as

Algorithms for geodesics

  • Charles F. F. KarneyEmail author
Open Access
Original Article
First Online:

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 
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Notes

Acknowledgments

I would like to thank Rod Deakin, John Nolton, Peter Osborne, and the referees of this paper for their helpful comments.

Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

References

  1. 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
  2. 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
  3. Bowring BR (1997) The central projection of the spheroid and surface lines. Surv Rev 34(265): 163–173CrossRefGoogle Scholar
  4. Bugayevskiy LM, Snyder JP (1995) Map projections: a reference manual. Taylor & Francis, LondonGoogle Scholar
  5. 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
  6. 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
  7. Clenshaw CW (1955) A note on the summation of Chebyshev series. Math Tables Other Aids Comput 9(51): 118–120Google Scholar
  8. Danielsen JS (1989) The area under the geodesic. Surv Rev 30(232): 61–66CrossRefGoogle Scholar
  9. 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]
  10. 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]
  11. 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]
  12. Karney CFF (2011) Geodesics on an ellipsoid of revolution. Tech rep, SRI International. http://arxiv.org/abs/1102.1215v1
  13. Karney CFF (2012) GeographicLib, version 1.22. http://geographiclib.sf.net
  14. 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
  15. 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
  16. Maxima (2009) A computer algebra system, version 5.20.1. http://maxima.sf.net
  17. Olver FWJ, Lozier DW, Boisvert RF, Clark CW (eds) (2010) NIST handbook of mathematical functions. Cambridge University Press, London. http://dlmf.nist.gov
  18. 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
  19. 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
  20. 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
  21. Rapp RH (1993) Geometric geodesy, part II. Tech rep, Ohio State University. http://hdl.handle.net/1811/24409
  22. Snyder JP (1987) Map projection—a working manual. Professional Paper 1395, US Geological Survey. http://pubs.er.usgs.gov/publication/pp1395
  23. Vermeille H (2002) Direct transformation from geocentric coordinates to geodetic coordinates. J Geod 76(9): 451–454CrossRefGoogle Scholar
  24. 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
  25. Vincenty T (1975b) Geodetic inverse solution between antipodal points. http://geographiclib.sf.net/geodesic-papers/vincenty75b.pdf (unpublished report dated Aug 28)
  26. Wessel P, Smith WHF (2010) Generic mapping tools, 4.5.5. http://gmt.soest.hawaii.edu/
  27. Williams R (1997) Gnomonic projection of the surface of an ellipsoid. J Navig 50(2): 314–320CrossRefGoogle Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.SRI InternationalPrincetonUSA

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