Analytical and numerical methods of converting Cartesian to ellipsoidal coordinates

Abstract

Abstract In this work, two analytical and two numerical methods of converting Cartesian to ellipsoidal coordinates of a point in space are presented. After slightly modifying a well-known exact analytical method, a new exact analytical method is developed. Also, two well-known numerical methods, which were developed for points exactly on the surface of a triaxial ellipsoid, are generalized for points in space. The four methods are validated with numerical experiments using an extensive set of points for the case of the Earth. Then, a theoretical and a numerical comparative assessment of the four methods is made. Furthermore, the new exact analytical method is applied for an almost oblate spheroid and for the case of the Moon and the results are compared. We conclude that, the generalized Panou and Korakitis’ numerical method, starting with approximate values from the new exact analytical method, is the best choice in terms of accuracy of the resulting ellipsoidal coordinates.