Show Summary Details
More options …

# Journal of Geodetic Science

Editor-in-Chief: Eshagh, Mehdi

Open Access
Online
ISSN
2081-9943
See all formats and pricing
More options …
Volume 3, Issue 3

# The geodesic boundary value problem and its solution on a triaxial ellipsoid

G. Panou
• Corresponding author
• Department of Surveying Engineering, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece
• Email
• Other articles by this author:
• De Gruyter OnlineGoogle Scholar
Published Online: 2013-10-15 | DOI: https://doi.org/10.2478/jogs-2013-0028

## Abstract

The geodesic problem on a triaxial ellipsoid is solved as a boundary value problem, using the calculus of variations. The boundary value problem consists of solving a non-linear second order ordinary differential equation, subject to the Dirichlet conditions. Subsequently, this problem is reduced to an initial value problem with Dirichlet and Neumann conditions. The Neumann condition is determined iteratively by solving a system of four first-order ordinary differential equations with numerical integration. The last iteration yields the solution of the boundary value problem. From the solution, the ellipsoidal coordinates and the angle between the line of constant longitude and the geodesic, at any point along the geodesic, are determined. Also, the constant in Liouville’s equation is determined and the geodesic distance between the two points, as an integral, is computed by numerical integration. To demonstrate the validity of the method presented here, numerical examples are given. The geodesic boundary value problem and its solution on a biaxial ellipsoid are obtained as a degenerate case.

• Burša M. and Šíma Z., 1980, Tri-axiality of the Earth, the Moon and Mars, Stud. Geoph. Geod., 24, 211-217.Google Scholar

• Chen W.-H. and Chen S.-G., 2011, A note of boundary geodesic problem on regular surfaces, Proceedings of the European Computing Conference, 105-109.Google Scholar

• Dassios G., 2012, Ellipsoidal harmonics: theory and applications, Cambridge University Press, Cambridge.Google Scholar

• Deakin R. E. and Hunter M. N., 2008, Geometric Geodesy - Part A, Lecture Notes, School of Mathematical & Geospatial Sciences, RMIT University, Melbourne, Australia.Google Scholar

• Featherstone W. E. and Claessens S. J., 2008, Closed-form transformation between geodetic and ellipsoidal coordinates, Stud. Geoph. Geod., 52, 1-18.

• Feltens J., 2009, Vector method to compute the Cartesian (X, Y , Z) to geodetic (ϕ, _, h) transformation on a triaxial ellipsoid, J. Geod., 83, 129-137.

• Guggenheimer H. W., 1977, Differential geometry, Dover, New York.Google Scholar

• Heiskanen W. A. and Moritz H., 1967, Physical geodesy, W. H. Freeman and Co., San Francisco and London.Google Scholar

• İz H. B., Ding X. L., Dai C. L. and Shum C. K., 2011, Polyaxial figures of the Moon, J. Geod. Sci., 1, 348-354.Google Scholar

• Jacobi C. G. J., 1839, Note von der geodätischen linie auf einem ellipsoid und den verschiedenen anwendungen einer merkwürdigen analytischen substitution, J. Crelle, 19, 309-313.Google Scholar

• Karney C. F. F., 2013, Algorithms for geodesics, J. Geod., 87, 43-55.Google Scholar

• Klingenberg W., 1982, Riemannian geometry, Walter de Gruyter, Berlin, New York.Google Scholar

• Ligas M., 2012a, Cartesian to geodetic coordinates conversion on a triaxial ellipsoid, J. Geod., 86, 249-256.

• Ligas M., 2012b, Two modified algorithms to transform Cartesian to geodetic coordinates on a triaxial ellipsoid, Stud. Geoph. Geod., 56, 993-1006.Google Scholar

• Maekawa T., 1996, Computation of shortest paths on free-form parametric surfaces, J. Mechanical Design, ASME Transactions, 118, 499-508.Google Scholar

• Moritz H., 1980, Geodetic Reference System 1980, Bull. Geod., 54, 395-405.Google Scholar

• Panou G., Delikaraoglou D. and Korakitis R., 2013, Solving the geodesics on the ellipsoid as a boundary value problem, J. Geod. Sci., 3, 40-47.Google Scholar

• Sjöberg L. E. and Shirazian M., 2012a, Solving the direct and inverse geodetic problems on the ellipsoid by numerical integration, J. Surv. Eng., 138, 9-16.

• Sjöberg L. E., 2012b, Solutions to the ellipsoidal Clairaut constant and the inverse geodetic problem by numerical integration, J. Geod. Sci., 2, 162-171.Google Scholar

• Shebl S. A. and Farag A. M., 2007, An inverse conformal projection of the spherical and ellipsoidal geodetic elements, Surv. Rev., 39, 116-123.

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

• Tabanov M. B., 1999, Normal forms of equations of wave functions in new natural ellipsoidal coordinates, American Mathematical Society, Translations, 193, 225-238. Google Scholar

• van Brunt B., 2004, The calculus of variations, Springer- Verlag, New York. Google Scholar

## About the article

Published Online: 2013-10-15

Published in Print: 2013-09-01

Citation Information: Journal of Geodetic Science, Volume 3, Issue 3, Pages 240–249, ISSN (Online) 2081-9943, ISSN (Print) 2081-9919,

Export Citation

This content is open access.