A univariate resultant-based implicitization algorithm for surfaces

Abstract

In this paper, we present a new algorithm for computing the implicit equation of a rational surface $\mathcal{V}$ from a rational parametrization $\mathcal{P}\left(\overline{t}\right)$. The algorithm is valid independent of the existence of base points, and is based on the computation of polynomial gcds and univariate resultants. Moreover, we prove that the resultant-based formula provides a power of the implicit equation. In addition, performing a suitable linear change of parameters, we prove that this power is indeed the degree of the rational map induced by the parametrization. We also present formulas for computing the partial degrees of the implicit equation.