Direct transformation from Cartesian into geodetic coordinates on a triaxial ellipsoid

Abstract

This paper presents two new direct symbolic-numerical algorithms for the transformation of Cartesian coordinates into geodetic coordinates considering the general case of a triaxial reference ellipsoid. The problem in both algorithms is reduced to finding a real positive root of a sixth degree polynomial. The first approach consists of algebraic manipulations of the equations describing the geometry of the problem and the second one uses Gröbner bases. In order to perform numerical tests and accurately compare efficiency and reliability, our algorithms together with the iterative methods presented by M. Ligas (2012) and J. Feltens ( 2009) have been implemented in C++. The numerical tests have been accomplished by considering 10 celestial bodies, referenced in the available literature. The obtained results clearly show that our algorithms improve the aforementioned iterative methods, in terms of both efficiency and accuracy.

Determination of the parameters of the triaxial earth ellipsoid as derived from present-day geospatial techniques

“…The final listed triaxial ellipsoid determined by Soler and Han, also based on EGM2008, is presented herein for the first time. It should be mentioned here that triaxial ellipsoids are often used in planetology to represent mathematical models of celestial bodies; e.g., Drummond and Christou (2008), Diaz-Toca et al (2019) where a few examples identifying the corresponding sources are tabulated.…”