Implicitization and parametrization of nonsingular cubic surfaces

Abstract

In this paper we unify the two subjects of implicitization and parametrization of nonsingular cubic surfaces. Beginning with a cubic parametrization with six basepoints, we first form a three by four Hilbert–Burch matrix, and then a three by three matrix of linear forms whose determinant is the implicit equation. Beginning with an implicit equation, we show how to construct a three by three matrix of linear forms whose determinant is the implicit equation, and from it construct the Hilbert–Burch matrix and a parametrization. The intermediate three by three matrix is shown to contain information about lines and cubic curves that lie on the surface, as well as to aid in the construction of inversion formulas.