Trilateration algorithm to transform Cartesian coordinates into geodetic coordinates

The transformation of Cartesian coordinates (x(p), y(p), z(p)) of a point (P) into their geodetic equivalent (phi, lambda, h) in reference to the geodetic ellipsoid, is an essential requirement in geodesy. There are many well-known algorithms solving this transformation in closed-form, approximate or iterative approaches. This paper presents a new algorithm named Trilateration Algorithm for this transformation. It is based on the new Seta-Point Theorem in the meridian plan, which defines a new deterministic Twin-Point (P-0) for the point (P). From the Twin-Point (P-0), a single iteration solution is processed to achieve highly-accurate values for (phi, h) in a relatively simple and deterministic computation algorithm which is valid and stable for all values of (phi, h). The proposed solution was tested on a sample of 4277 points that cover all possible cases of point (P). The produced maximum absolute error in latitude is (0.000 0026 '') and (0.000 476 mm) in height, computed from first iteration.