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The method of resolvents: A technique for the implicitization, inversion, and intersection of non-planar, 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 non-planar, 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 non-planar rational cubic curves, and it is shown that two such curves can intersect in at most five points. Specializations of these results for non-planar, integral, cubic curves are derived, and extensions of these techniques to non-planar, rational cubic, Bézier curves are also discussed.
Keywords:Computational geometry and object modeling  curve representations  elimination  resolvent  resultant
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