Transforming Geocentric Cartesian Coordinates to Geodetic Coordinates by a New Initial Value Calculation Paradigm

Transforming geocentric Cartesian coordinates (X, Y, Z) to geodetic curvilinear coordinates (φ, λ, h) on a biaxial ellipsoid is one of the problems used in satellite positioning, coordinates conversion between reference systems, astronomy and geodetic calculations. For this purpose, various methods including Closed-form, Vector method and Fixed-point method have been developed. In this paper, a new paradigm for calculation of initial values is presented. According to the new initial values, two state of the art iterative methods are modified to calculate the geodetic height and the geodetic latitude accurately and without iteration. The results show that for those points with height values between -10 to 1,000,000 km (30-fold more than the altitude of GPS satellites), the maximum error of the calculated height and geodetic latitude is less than 1.5×10-8 m and 1×10-14 rad (error lower than 0.001 mm in horizontal), respectively.