On the minors of the implicitization Bezout matrix for a rational plane curve

Abstract

This paper investigates the first minors Mi,j of the Bezout matrix used to implicitize a degree-n plane rational curve P(t). It is shown that the degree n-1 curve Mi,j = 0 passes through all of the singular points of P(t). Furthermore, the only additional points at which Mi,j = 0 and P(t) intersect are an (i+j)-fold intersection at P(0) and a (2n-2-i-j)-fold intersection at P(∞). Thus, a polynomial whose roots are exactly the parameter values of the singular points of P(t) can be obtained by intersecting P(t) with M0,0. Previous algorithms of finding such a polynomial are less direct. We further show that Mi,j = Mk,l if i+j = k+l. The method also clarifies the applicability of inversion formulas and yields simple checks for the existence of singularities in a cubic Be´zier curve.