Universal Rational Parametrizations and Spline Curves on Toric Surfaces

Abstract

Recently a constructive description of all rational parametriza-tions for toric surfaces was described in terms of the universal rational parametrizations (URP). We give an elementary introduction to this theory from the Geometric Modelling point of view: toric surfaces are defined via homogeneous coordinates; projections, singular cases, and non-canonical real structures are described; the URP theorem is explained. A theory of rational C 1 spline curves with certain interpolation properties on toric surfaces is developed. Applications for smooth blending of natural quadrics are sketched.