Enhancing Levinʹs method for computing quadric-surface intersections Original Research Article

Levinʹs method produces a parameterization of the intersection curve of two quadrics in the form p(u)=a(u)±d(u)s(u), where a(u) and d(u) are vector valued polynomials, and s(u) is a quartic polynomial. This method, however, is incapable of classifying the morphology of the intersection curve, in terms of reducibility, singularity, and the number of connected components, which is critical structural information required by solid modeling applications. We study the theoretical foundation of Levinʹs method, as well as the parameterization p(u) it produces. The following contributions are presented in this paper: (1) It is shown how the roots of s(u) can be used to classify the morphology of an irreducible intersection curve of two quadric surfaces. (2) An enhanced version of Levinʹs method is proposed that, besides classifying the morphology of the intersection curve of two quadrics, produces a rational parameterization of the curve if the curve is singular. (3) A simple geometric proof is given for the existence of a real ruled quadric in any quadric pencil, which is the key result on which Levinʹs method is based. These results enhance the capability of Levinʹs method in processing the intersection curve of two general quadrics within its own self-contained framework.