Journal of Geodesy

, 83:1071

A symbolic analysis of Vermeille and Borkowski polynomials for transforming 3D Cartesian to geodetic coordinates

Original Article

DOI: 10.1007/s00190-009-0325-2

Cite this article as:
Gonzalez-Vega, L. & Polo-Blanco, I. J Geod (2009) 83: 1071. doi:10.1007/s00190-009-0325-2


Closed form solutions for transforming 3D Cartesian to geodetic coordinates reduce the problem to finding the real solutions of the fourth degree latitude equation or variations of it. By using symbolic tools (Sturm–Habicht coefficients and subresultants mainly) we study the methods (and polynomials) proposed by Vermeille and Borkowski to solve this problem. For Vermeille approach, the region where it cannot be applied is completely characterized. For those points it is shown how to transform 3D Cartesian to geodetic coordinates and a new method for solving Vermeille equation for those cases not yet covered is introduced. Concerning Borkowski’s approach, the symbolic analysis produces a complete characterization of the singular cases (i.e. where multiple roots appear).


Coordinate transformations3D Cartesian and geodetic coordinatesSymbolic computation

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Departamento de Matematicas, Estadistica y ComputacionUniversidad de CantabriaSantanderSpain