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Journal of Geodetic Science

Editor-in-Chief: Eshagh, Mehdi

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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


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


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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, Volume 7, Issue 1, Pages 31–42, 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

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