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

✍ Scribed by Eng-Wee Chionh; Thomas W. Sederberg

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
127 KB
Volume
18
Category
Article
ISSN
0167-8396

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✦ Synopsis

This paper investigates the first minors M i,j of the Bézout matrix used to implicitize a degreen 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 -2ij)-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 i,j = 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 Bézier curve.

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