Direct transformation from Cartesian into geodetic coordinates on a triaxial ellipsoid

TL;DR

This paper introduces two novel algorithms for converting Cartesian coordinates to geodetic coordinates on a triaxial ellipsoid, improving efficiency and accuracy over existing iterative methods through symbolic-numerical approaches.

Contribution

The paper presents the first direct symbolic-numerical algorithms for this transformation on a triaxial ellipsoid, reducing the problem to solving a sixth degree polynomial.

Findings

Algorithms outperform iterative methods in efficiency.

Algorithms achieve higher accuracy in numerical tests.

Validated on 10 celestial bodies.

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 Gr\"obner 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.