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Journal of Geodetic Science

Editor-in-Chief: Sjöberg, Lars

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2081-9943
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Research Article. Geodesic equations and their numerical solutions in geodetic and Cartesian coordinates on an oblate spheroid

G. Panou
  • Corresponding author
  • Department of Surveying Engineering, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece
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/ R. Korakitis
  • Department of Surveying Engineering, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2017-05-11 | DOI: https://doi.org/10.1515/jogs-2017-0004

Abstract

The direct geodesic problem on an oblate spheroid is described as an initial value problem and is solved numerically using both geodetic and Cartesian coordinates. The geodesic equations are formulated by means of the theory of differential geometry. The initial value problem under consideration is reduced to a system of first-order ordinary differential equations, which is solved using a numerical method. The solution provides the coordinates and the azimuths at any point along the geodesic. The Clairaut constant is not used for the solution but it is computed, allowing to check the precision of the method. An extensive data set of geodesics is used, in order to evaluate the performance of the method in each coordinate system. The results for the direct geodesic problem are validated by comparison to Karney’s method. We conclude that a complete, stable, precise, accurate and fast solution of the problem in Cartesian coordinates is accomplished.

Keywords: Clairaut’s constant; Direct geodesic problem; Geometrical geodesy; Karney’s method; Numerical method

References

  • Bessel F.W., 1826. Über die Berechnung der geographischen Längen und Breiten aus geodätischen Vermessungen. Astronomische Nachrichten 4, 241-254. doi:CrossrefGoogle Scholar

  • Bowring B.R., 1983. The geodesic inverse problem. Bulletin Géodésique 57, 109-120. doi:CrossrefGoogle Scholar

  • Butcher J.C., 1987. The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods. Wiley, New York.Google Scholar

  • Deakin R.E., Hunter M.N., 2010. Geometric geodesy, Part B. Lecture Notes, School of Mathematical & Geospatial Sciences, RMIT University, Melbourne, Australia.Google Scholar

  • Felski A., 2011.Computation of the azimuth of the great circle in Cartesian coordinates. Annual of Navigation 18, 45-53.Google Scholar

  • Fox C., 1987. An introduction to the calculus of variations. Dover, New York.Google Scholar

  • Hildebrand F.B., 1974. Introduction to numerical analysis, 2nd ed. Dover, New York.Google Scholar

  • Holmstrom J.S., 1976. A new approach to the theory of geodesics on an ellipsoid. Navigation, Journal of The Institute of Navigation 23, 237-244. doi:CrossrefGoogle Scholar

  • Jank W., Kivioja L.A., 1980. Solution of the direct and inverse problems on reference ellipsoids by point-by-point integration using programmable pocket calculators. Surveying and Mapping 40, 325-337.Google Scholar

  • Karney C.F.F., 2010. Test set for geodesics. https://doi.org/10.5281/zenodo.32156. Accessed 01 November 2016.CrossrefGoogle Scholar

  • Karney C.F.F., 2013. Algorithms for geodesics. Journal of Geodesy 87, 43-55. doi:CrossrefGoogle Scholar

  • Karney C.F.F., 2016. GeographicLib. http://geographiclib.sourceforge.net/html/. Accessed 01 November 2016.Google Scholar

  • Kivioja L.A., 1971. Computation of geodetic direct and indirect problems by computers accumulating increments from geodetic line elements. Bulletin Géodésique 99, 55-63. doi:CrossrefGoogle Scholar

  • Mai E., 2010. A fourth order solution for geodesics on ellipsoids of revolution. Journal of Applied Geodesy 4, 145-155. doi:CrossrefGoogle Scholar

  • Panou G., 2013. The geodesic boundary value problem and its solution on a triaxial ellipsoid. Journal of Geodetic Science 3, 240-249. doi:CrossrefGoogle Scholar

  • Panou G., Delikaraoglou D., Korakitis R., 2013. Solving the geodesics on the ellipsoid as a boundary value problem. Journal of Geodetic Science 3, 40-47. doi:CrossrefGoogle Scholar

  • Pittman M.E., 1986. Precision direct and inverse solutions of the geodesic. Surveying and Mapping 46, 47-54.Google Scholar

  • Rainsford H.F., 1955. Long geodesics on the ellipsoid. Bulletin Géodésique 37, 12-22. doi:CrossrefGoogle Scholar

  • Rapp R.H., 1993. Geometric geodesy, Part II. Department of Geodetic Science and Surveying, The Ohio State University, Columbus, Ohio, USA.Google Scholar

  • Robbins A.R., 1962. Long lines on the spheroid. Survey Review 16, 301-309. doi:CrossrefGoogle Scholar

  • Rollins C.M., 2010. An integral for geodesic length. Survey Review 42, 20-26. doi:CrossrefGoogle Scholar

  • Saito T., 1970. The computation of long geodesics on the ellipsoid by non-series expanding procedure. Bulletin Géodésique 98, 341-373. doi:CrossrefGoogle Scholar

  • Saito T., 1979. The computation of long geodesics on the ellipsoid through Gaussian quadrature. Bulletin Géodésique 53, 165-177. doi:CrossrefGoogle Scholar

  • Sjöberg L.E., 2012. Solutions to the ellipsoidal Clairaut constant and the inverse geodetic problem by numerical integration. Journal of Geodetic Science 2, 162-171. doi:CrossrefGoogle Scholar

  • Sjöberg L.E., Shirazian M., 2012. Solving the direct and inverse geodetic problems on the ellipsoid by numerical integration. Journal of Surveying Engineering 138, 9-16. doi:CrossrefGoogle Scholar

  • Sodano E.M., 1965. General non-iterative solution of the inverse and direct geodetic problems. Bulletin Géodésique 75, 69-89. doi:CrossrefGoogle Scholar

  • Struik D.J., 1961. Lectures on classical differential geometry, 2nd ed. Dover, New York.Google Scholar

  • Thomas C.M., Featherstone W.E., 2005. Validation of Vincenty’s formulas for the geodesic using a new fourth-order extension of Kivioja’s formula. Journal of Surveying Engineering 131, 20-26. doi:CrossrefGoogle Scholar

  • Tseng W.K., 2014. An algorithm for the inverse solution of geodesic sailing without auxiliary sphere. Journal of Navigation 67, 825-844. doi:CrossrefWeb of ScienceGoogle Scholar

  • Vermeer M., 2015. Mathematical geodesy. Lecture Notes, School of Engineering, Aalto University, Espoo, Finland.Google Scholar

  • Vincenty T., 1975. Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations. Survey Review 23, 88-93. doi:CrossrefGoogle Scholar

About the article

Received: 2016-12-05

Accepted: 2017-02-17

Published Online: 2017-05-11

Published in Print: 2017-02-23


Citation Information: Journal of Geodetic Science, ISSN (Online) 2081-9943, DOI: https://doi.org/10.1515/jogs-2017-0004.

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© by G. Panou. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Comments (1)

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  • The authors hold up their method as an efficient way of tracing a geodesic. I would like to remind readers that the "traditional" methods for solving geodesics pioneered by Bessel (1825), Helmert (1880), Rainsford (1995), and Vincenty (1975) also allow tracing a geodesic. For example, a user of my software library GeographicLib can calculate waypoints on a geodesic at a rate of 0.37 us per point (Karney 2013). This is slightly longer than the 0.3 us quoted by the authors for a single integration step with their method. However a crucial distinction is that with GeographicLib, the user can arbitrarily choose the spacing of the waypoints without affecting the accuracy of the calculation, while with the authors' approach the spacing is dictated by the needs of accuracy. Thus if the user wishes desires 1 um accuracy, for example, she is forced to compute roughly 1000 waypoints which may be far more than is needed.

    posted by: Charles Karney on 2017-05-10 11:53 PM (Europe/Berlin)