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The method of moving planes and moving quadrics can express the implicit equation of a parametric surface as the determinant of a matrix M. The rows of M correspond to moving planes or moving quadrics that follow the parametric surface. Previous papers on the method of moving surfaces have shown that a simple base point has the effect of converting one(More)
Given a parametrization of a rational surface, the absence of base points is shown to be a necessary and sufficient condition for the auxiliary resultant to be a power of the implicit polynomial. The method of resultants also reveals other important properties of rational surface representations, including the coefficients of the implicit equation, the(More)
Dixon 1908] introduces three distinct determinant formulations for the resultant of three bivariate polynomials of bidegree (m; n). The rst technique applies Sylvester's dialytic method to construct the resultant as the determinant of a matrix of order 6mn. The second approach uses Cayley's determinant device to form a more compact representation for the(More)
An inversion equation takes the Cartesian coordinates of a point on a parametric curve or surface and returns the parameter value(s) of that point. A 2-D curve inversion equation has the form t = f (x, y)/g(x, y). This paper shows that practical insight into inversion can be obtained by studying the geometry of the implicit curves f (x, y) = 0 and g(x, y) =(More)
Macaulay's concise but explicit expression for nmltivariate resultants has many potential applications in computer-aided geometric design. Here we describe its use in solid modeling for finding the intersections of three implicit quadric surfaces. By B6zout's theorem, three quadric surfaces have either at most eight or intlnitely many intersections. Our(More)
EEcient algorithms are derived for computing the entries of the Bezout resultant matrix for two univariate polynomials of degree n and for calculating the entries of the Dixon-Cayley resultant matrix for three bivariate polynomials of bidegree (m; n). Standard methods based on explicit formulas require O(n 3) additions and multiplications to compute all the(More)