Various parameterizations of 'latitude' equation – Cartesian to geodetic coordinates transformation

The paper presents a solution to one of the basic problems of computational geodesy - conversion between Cartesian and geodetic coordinates on a biaxial ellipsoid. The solution is based on what is known in the literature as “latitude equation”. The equation is presented in three different parameterizations commonly used in geodesy - geodetic, parametric (reduced) and geocentric latitudes. Although the resulting equations may be derived in many ways, here, we present a very elegant one based on vectors orthogonality. As the “original latitude equations” are trigonometric ones, their representation has been changed into an irrational form after Fukushima (1999, 2006). Furthermore, in order to avoid division operations we have followed Fukushima’s strategy again and rewritten the equations in a fractional form (a pair of iterative formulas). The resulting formulas involving parametric latitude are essentially the same as those introduced by Fukushima (2006) (considered the most efficient today). All the resulting variants are solved with Newton’s second-order and Halley’s third-order formulas. It turns out that all parameterizations of the “latitude equation” show a comparable level of performance.