Efficient transformation from Cartesian to geodetic coordinates

The derivation of algorithms for the computation of geodetic coordinates from 3D Cartesian coordinates has been a very active field of research among geodesists for more than forty years. Many authors have sought the most efficient method, i.e. the method that provides the fastest computational speed, which nevertheless yields sufficient accuracy for practical applications. The problem is a special case of a more general mathematical problem that has also been studied by researchers in other fields. This paper investigates the applicability of methods by Sampson (1982, Computer graphics and image processing, 18: 97-108) and Uteshev and Goncharova (2018, Journal of Computational and Applied Mathematics, 328: 232-251) to the computation of geodetic coordinates. Both methods have been modified to make them more suitable for this particular problem. The methods are compared to several commonly used geodetic methods in terms of accuracy and computational efficiency. It is found that a simple modification improves the accuracy of the methods by similar to 3 orders of magnitude, and the modified method of Uteshev and Goncharova (2018) achieves an accuracy of <0.1 mm anywhere on the surface of the Earth. The methods are especially efficient in the computation of ellipsoidal height. As an additional result of this study, a new formulation of the well-known method by Bowring (1976, Survey Review, 23: 323-327) is derived, and it is shown to improve the computation speed of Bowring's method by similar to 12%similar to 27% compared to the conventional formulation.