Universal parametrization and interpolation on cubic surfaces Original Research Article

Universal parametrizations make it possible to find all rational parametrizations and all rational curves on an implicit surface. The rational curves and patches can be described with optimal degree as images under such a universal parametrization. Hence, the rational curves on the surface can be classified in a systematic way. Interpolation problems on the surface are reduced to ordinary interpolation problems. Other than a usual parametrization, a universal parametrizations allows to control the degree of the interpolating curve or patch on the surface. To avoid nonlinear equation systems, simpler parametrizations can be derived from a universal parametrization. We present universal parametrizations for three different classes of cubic surfaces. According to a conjecture by Krasauskas, this covers all cubic surfaces, which possess a universal parametrization.