Elimination and resultants. 1. Elimination and bivariate resultants

摘要

We discuss the relevance of elimination theory and resultants in computing, especially in computer graphics and CAGD. We list resultant properties to enhance overall understanding of resultants. For bivariate resultants, we present two explicit expressions: the Sylvester and the Bezout determinants. The Sylvester matrix is easier to construct, but the symmetrical Bezout matrix is structurally richer and thus sometimes more revealing. It let Kajiya (1982) observe directly that a line and a bicubic patch could intersect in at most 18 points, not 36 points, as a naive analysis would presume. For Bezier curves, there is an interesting algebraic and geometric relationship between the implicit equation in Bezout determinant form and the properties of end point interpolation and de Casteljau subdivision. When the two polynomials are of different degrees, the Bezout resultant suffers from extraneous factors. Fortunately, we can easily discard these factors. For problems related to surfaces, we need multivariate resultants: in particular, multivariate resultants for three homogeneous polynomials in three variables.