A symbolic analysis of Vermeille and Borkowski polynomials for transforming 3D Cartesian to geodetic coordinates
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- Gonzalez-Vega, L. & Polo-Blanco, I. J Geod (2009) 83: 1071. doi:10.1007/s00190-009-0325-2
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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).