Jump to ContentJump to Main Navigation
Open Access

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

1Department of Surveying Engineering, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece

This content is open access.

Citation Information: Journal of Geodetic Science. Volume 3, Issue 3, Pages 240–249, ISSN (Online) 2081-9943, ISSN (Print) 2081-9919, DOI: 10.2478/jogs-2013-0028, October 2013

Publication History

Published Online:
2013-10-15

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.

Keywords: biaxial ellipsoid; ellipsoidal coordinates; geodesic problem; Liouville constant; numerical integration

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

  • 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.

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

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

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

  • 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. [Web of Science]

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

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

  • 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.

  • 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.

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

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

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

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

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

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

  • 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.

  • 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. [Web of Science]

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

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

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

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

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

Comments (0)

Please log in or register to comment.