Hyperboloidal coordinates: transformations and applications in special constructions

The vector-based algorithms for biaxial and triaxial ellipsoidal coordinates presented by Feltens (J Geod 82:493–504, 2008; 83:129–137, 2009) have been extended to hyperboloids of one sheet. For the backward transformation from Cartesian to hyperboloidal coordinates, of two iterative process candidates one was identified to be well suited. It turned out that a careful selection of the center of curvature is essential for the establishment of a stable and reliable iteration process. In addition, for zero hyperboloidal heights a closed solution is presented. The hyperboloid algorithms are again based on simple formulae and have been successfully tested for various theoretical hyperboloids. The paper concludes with a practical application example on a cooling tower construction.