Using multivariate resultants to find the implicit equation of a rational surface

Abstract

Given a parametrization of a rational surface, the absence of base points is shown to be a necessary and sufficient condition for the auxiliary resultant to be a power of the implicit polynomial. The method of resultants also reveals other important properties of rational surface representations, including the coefficients of the implicit equation, the relationship between the implicit and parametric degrees, the degree of each coordinate variable of the implicit equation, and the number of correspondence of the parametrization. © 1992 Springer-Verlag.