An optimized method to transform the Cartesian to geodetic coordinates on a triaxial ellipsoid

A general triaxial ellipsoid is suitable to represent the reference surface of the celestial bodies. The transformation from the Cartesian to geodetic coordinates on the triaxial ellipsoid becomes an important issue in geodesy. In the literature, the vector iterative method and the Newton’s iterative method for solving the nonlinear system of equations or an algebraic fraction equation is applied to compute the geodetic coordinates, but may lead to the non-convergence regions. In this work, the universal algorithm including the Newton’s iterative solutions of an algebraic sextic equation for the points outside the equatorial plane and the analytic solutions for the points inside the equatorial plane are used to compute the geodetic coordinates. The numerical experiments show the algorithm is fast, highly accurate and well convergent. The algorithm is valid at any point inside and outside the celestial bodies including the points near the celestial bodies’ center and in the singular elliptical disc.