Conversion between Cartesian and geodetic coordinates on a rotational ellipsoid by solving a system of nonlinear equations

Journal title

Geodesy and Cartography




vol. 60


No 2

Publication authors


Cartesian and geodetic coordinates ; rotational ellipsoid ; Newton’s method ; coordinate transformation

Divisions of PAS

Nauki Techniczne


A new method to transform from Cartesian to geodetic coordinates is presented. It is based on the solution of a system of nonlinear equations with respect to the coordinates of the point projected onto the ellipsoid along the normal. Newton’s method and a modification of Newton’s method were applied to give third-order convergence. The method developed was compared to some well known iterative techniques. All methods were tested on three ellipsoidal height ranges: namely, (-10 – 10 km) (terrestrial), (20 – 1000 km), and (1000 – 36000 km) (satellite). One iteration of the presented method, implemented with the third-order convergence modified Newton’s method, is necessary to obtain a satisfactory level of accuracy for the geodetic latitude ( σ φ < 0.0004”) and height ( σ h < 10 − 6 km, i.e. less than a millimetre) for all the heights tested. The method is slightly slower than the method of Fukushima (2006) and Fukushima’s (1999) fast implementation of Bowring’s (1976) method.


Commitee on Geodesy PAS




Artykuły / Articles


ISSN 2080-7636


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