Implicitizing Rational Tensor Product Surfaces Using the Resultant of Three Moving Planes

Abstract

Implicitizing rational surfaces is a fundamental computational task in Computer Graphics and Computer Aided Design. Ray tracing, collision detection, and solid modeling all benefit from implicitization procedures for rational surfaces. The univariate resultant of two moving lines generated by a μ-basis for a rational curve represents the implicit equation of the rational curve. But although the multivariate resultant of three moving planes corresponding to a μ-basis for a rational surface is guaranteed to contain the implicit equation of the surface as a factor, μ-bases for rational surfaces are difficult to compute. Moreover, μ-bases for a rational surface often have high degrees, so these resultants generally contain many extraneous factors. Here we develop fast algorithms to implicitize rational tensor product surfaces by computing the resultant of three moving planes corresponding to three syzygies with low degrees. These syzygies are easy to compute, and the resultants of the corresponding moving planes generally contain fewer extraneous factors than the resultants of the moving planes corresponding to μ-bases. We predict and compute all the possible extraneous factors that may appear in these resultants. Examples are provided to clarify and illuminate the theory.