This paper investigates the first minors M_i,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 M_i,j = 0 passes through all of the singular points of P(t). Furthermore, the only additional points at which M_i,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 M_0.0. Previous algorithms of finding such a polynomial are less direct. We further show that M_ij = M_k,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 curve.
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