Direct symbolic transformation from 3D cartesian into hyperboloidal coordinates

A direct transformation from cartesian coordinates into hyperboloidal coordinates (considered for biaxial hyperboloids) is presented in this paper. The transformation problem is reduced to the problem of finding the smallest positive root of a fourth degree polynomial. The analysis of the polynomial's roots is performed by an algebraically complete stratification, based on symbolic techniques (mainly Sturm-Habicht sequences and its properties related to real root counting), of a planar region situated in the positive quadrant. Two approaches for computing the polynomial's roots are presented, one based on the Merriman method and the other one obtained using the Computer Algebra System Maple. Our approach improves the solution presented in Feltens (2011) [1], being reduced to a few evaluations of symbolic expressions.