Transformation of Cartesian to geodetic coordinates on a triaxial ellipsoid using approximations

In the last decade, several numerical methods have been presented that transform Cartesian to geodetic coordinates on a triaxial ellipsoid. In this work, a new method for this transformation is presented. The method is based on ellipsoidal coordinates. To begin with, the Cartesian coordinates of a point are transformed to the respective ellipsoidal through closed analytical formulae. Following, the ellipsoidal coordinates referred to the referenced ellipsoid, approximated by those referred to the confocal ellipsoid passed through the given point. Then, the ellipsoidal coordinates are corrected by iterative and non-iterative formulae which are derived and thoroughly analyzed. After the successive approximations, the ellipsoidal coordinates are used for the computation of the foot point and hence of the geodetic coordinates by well-known formulae. The new method is validated by numerical experiments using an extensive set of points for different ellipsoids.