The method of resolvents: A technique for the implicitization, inversion, and intersection of nonplanar, parametric, rational cubic curves 
 
Authors:  Ronald N Goldman 
 
Institution:  Control Data Corporation, AHS  251, Arden Hills, MN 55112, U.S.A. 
 
Abstract:  Let P(t) be a nonplanar, parametric, rational cubic curve. The method of resolvents is applied to: (1) construct three quadric surfaces whose intersection is equal to P(t) (implicitization); (2) solve for the parameter t as the ratio of two linear expressions in the coordinates x, y, z (inversion). The results of these two operations are then applied to construct an optimal, robust, intersection algorithm for any two nonplanar rational cubic curves, and it is shown that two such curves can intersect in at most five points. Specializations of these results for nonplanar, integral, cubic curves are derived, and extensions of these techniques to nonplanar, rational cubic, Bézier curves are also discussed. 
 
Keywords:  Computational geometry and object modeling curve representations elimination resolvent resultant 
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