Singularity Computation for Rational Parametric Surfaces Using Moving Planes

Abstract

Singularity computation is a fundamental problem in Computer Graphics and Computer Aided Geometric Design, since it is closely related to topology determination, intersection, mesh generation, rendering, simulation, and modeling of curves and surfaces. In this article, we present an efficient and robust algorithm for computing all the singularities (including their orders) of rational parametric surfaces using the technique of moving planes. The main approach is first to construct a representation matrix whose columns correspond to moving planes following the parametric surface. Then, by substituting the parametric equation of the rational surface into this representation matrix, one can extract the singularity information from the corresponding matrix and return all the singular loci including self-intersection curves, cusp curves, and isolated singular points of the rational surface, together with the order of each singular locus. We present some examples to compare our algorithm with state-of-the-art methods from different perspectives including robustness, efficiency, order computation, and numerical stability, and the experimental results show that our method outperforms existing methods in all these aspects. Furthermore, applications of our algorithm in surface rendering, mesh generation and surface/surface intersections are provided to demonstrate that correctly computing the self-intersection curves of a surface is essential to generate high quality results for these applications.