Cartesian to geodetic coordinates conversion on a triaxial ellipsoid using the bisection method

Abstract

A new method to convert Cartesian to geodetic coordinates on a triaxial ellipsoid is presented. The geodetic latitude and longitude are determined by an analytical and a numerical method and the geodetic height by the Euclidean distance, after the computation of the foot point. The algorithm for computing the foot point, which uses the bisection method, always converges, especially important for a region in the interior of the ellipsoid where some published methods are not applicable. The new numerical method is validated with numerical experiments using an extensive test set of points, for several ellipsoids with different eccentricities, and then is compared to the methods of Ligas (2012b), Chen et al. (2019) and Diaz-Toca et al. (2020). The method also gives accurate results for an oblate spheroid, which is obtained as a degenerate case. We conclude that a complete, stable, accurate and universal solution of the problem is accomplished.

Appendix

Pseudocode for robustly computing the foot point $\left( {x,y,z} \right)$.

The preconditions are $0 < b \le a_{y} \le a_{x}$, $X \ge 0$, $Y \ge 0$ and $Z \ge 0$.