The direct geodesic problem and an approximate analytical solution in Cartesian coordinates on a triaxial ellipsoid
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G. Panou
und
R. Korakitis
Abstract
In this work, the direct geodesic problem in Cartesian coordinates on a triaxial ellipsoid is solved by an approximate analytical method. The parametric coordinates are used and the parametric to Cartesian coordinates conversion and vice versa are presented. The geodesic equations on a triaxial ellipsoid in Cartesian coordinates are solved using a Taylor series expansion. The solution provides the Cartesian coordinates and the angle between the line of constant v and the geodesic at the end point. An extensive data set of geodesics, previously studied with a numerical method, is used in order to validate the presented analytical method in terms of stability, accuracy and execution time. We conclude that the presented method is suitable for a triaxial ellipsoid with small eccentricities and an accurate solution is obtained. At a similar accuracy level, this method is about thirty times faster than the corresponding numerical method. Finally, the presented method can also be applied in the degenerate case of an oblate spheroid, which is extensively used in geodesy.
References
[1] Bomford G. Geodesy. 2nd ed. Oxford University Press, London, 1962.Suche in Google Scholar
[2] Holmstrom JS. A new approach to the theory of geodesics on an ellipsoid. Navigation, Journal of The Institute of Navigation 23, 237–244, 1976.10.1002/j.2161-4296.1976.tb00746.xSuche in Google Scholar
[3] Karney CFF. A geodesic bibliography. GeographicLib 1.50, 2019. (Accessed November 1, 2019, at https://geographiclib.sourceforge.io/geodesic-papers/biblio.html).Suche in Google Scholar
[4] Karney CFF. Geodesics on a triaxial ellipsoid. GeographicLib 1.50, 2019. (Accessed November 1, 2019 at http://geographiclib.sourceforge.io/html/triaxial.html).Suche in Google Scholar
[5] Ligas M. Two modified algorithms to transform Cartesian to geodetic coordinates on a triaxial ellipsoid. Studia Geophysica et Geodaetica 56, 993–1006, 2012.10.1007/s11200-011-9017-5Suche in Google Scholar
[6] Mai E. A fourth order solution for geodesics on ellipsoids of revolution. Journal of Applied Geodesy 4, 145–155, 2010.10.1515/jag.2010.014Suche in Google Scholar
[7] Olver PJ. Applications of Lie groups to differential equations. 2nd ed. Springer-Verlag, New York, 2000.Suche in Google Scholar
[8] Panou G. The geodesic boundary value problem and its solution on a triaxial ellipsoid. Journal of Geodetic Science 3, 240–249, 2013.10.2478/jogs-2013-0028Suche in Google Scholar
[9] Panou G, Korakitis R. Geodesic equations and their numerical solutions in geodetic and Cartesian coordinates on an oblate spheroid. Journal of Geodetic Science 7, 31–42, 2017.10.1515/jogs-2017-0004Suche in Google Scholar
[10] Panou G, Korakitis R. Geodesic equations and their numerical solution in Cartesian coordinates on a triaxial ellipsoid. Journal of Geodetic Science 9, 1–12, 2019.10.1515/jogs-2019-0001Suche in Google Scholar
[11] Panou G, Korakitis R. Cartesian to geodetic coordinates conversion on a triaxial ellipsoid using the bisection method, 2019. Preprint, https://www.researchgate.net/publication/333904072_Cartesian_to_geodetic_coordinates_conversion_on_a_triaxial_ellipsoid_using_the_bisection_method.10.1016/j.cageo.2019.104308Suche in Google Scholar
[12] Xenophontos C. A formula for the nth derivative of the quotient of two functions. Communicated by Masanobu Taniguchi, 2007.Suche in Google Scholar
[13] Zwillinger D. Handbook of differential equations. 3rd ed. Academic Press, 1997.Suche in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Editorial
- Editorial to the special edition of the JAG on Deformation Monitoring
- Research Articles
- Determination of the tectonic plates motion parameters based on SLR, DORIS and VLBI stations positions
- Deformation detection through the realization of reference frames
- Performance of Msplit estimates in the context of vertical displacement analysis
- Progress towards a rigorous error propagation for total least-squares estimates
- Reducing multipath effect of low-cost GNSS receivers for monitoring by considering temporal correlations
- F2S3: Robustified determination of 3D displacement vector fields using deep learning
- Improved kinematic Precise Point Positioning performance with the use of map constraints
- The direct geodesic problem and an approximate analytical solution in Cartesian coordinates on a triaxial ellipsoid
- A benchmarking measurement campaign in GNSS-denied/challenged indoor/outdoor and transitional environments
- Partially error-affected point-wise weighted closed-form solution to the similarity transformation and its variants
Artikel in diesem Heft
- Frontmatter
- Editorial
- Editorial to the special edition of the JAG on Deformation Monitoring
- Research Articles
- Determination of the tectonic plates motion parameters based on SLR, DORIS and VLBI stations positions
- Deformation detection through the realization of reference frames
- Performance of Msplit estimates in the context of vertical displacement analysis
- Progress towards a rigorous error propagation for total least-squares estimates
- Reducing multipath effect of low-cost GNSS receivers for monitoring by considering temporal correlations
- F2S3: Robustified determination of 3D displacement vector fields using deep learning
- Improved kinematic Precise Point Positioning performance with the use of map constraints
- The direct geodesic problem and an approximate analytical solution in Cartesian coordinates on a triaxial ellipsoid
- A benchmarking measurement campaign in GNSS-denied/challenged indoor/outdoor and transitional environments
- Partially error-affected point-wise weighted closed-form solution to the similarity transformation and its variants
