Efficient computation of moving planes for rational parametric surfaces with base points using Dixon resultants

Abstract

Moving planes have been widely recognized as a potent algebraic tool in various fundamental problems of geometric modeling, including implicitization, intersection computation, singularity calculation, and point inversion of parametric surfaces. For instance, a matrix representation that inherits the key properties of a parametric surface is constructed from a set of moving planes. In this paper, we present an efficient approach to computing such a set of moving planes that follow the given rational parametric surface. Our method is based on the calculation of Dixon resultant matrices, which allows for the computation of moving planes with simpler coefficients, improved efficiency and superior numerical stability when compared to the direct way of solving a linear system of equations for the same purpose. We also demonstrate the performance of our algorithm through experimental examples when applied to implicitization, surface intersection, singularity computation as well as inversion formula computation.