On the minors of the implicitization Bézout matrix for a rational plane curve Original Research Article

This paper investigates the first minors Mi,j of the Bézout 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 Bézier curve.