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

This paper investigates the first miners M-i,M-j of the Bezout matrix used to implicitize a degree-n plane rational curve P(t). It is shown that the d egree n - 1 curve M-i,M-j = 0 passes through all of the singular points of P(t). Furthermore, the only additional points at which M-i,M-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(infinity). Thus, a polynomial whose roots are exactl y the parameter values of the singular points of P(t) can be obtained by in tersecting P(t) with Mg-0,Mg-0. Previous algorithms of finding such a polyn omial are less direct. We further show that M-i,M-j = M-k,M-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 Bezier c urve. (C) 2001 Elsevier Science B.V. All rights reserved.