Automatic parsing of degenerate quadric-surface intersections

Abstract

In general, two quadric surfaces intersect in a nonsingular quartic space curve. Under special circumstances, however, this intersection may “degenerate” into a quartic with a double point, or a composite of lines, conics, and twisted cubics whose degrees, counted over the complex projective domain, sum to four. Such degenerate forms are important since they occur with surprising frequency in practice and, unlike the generic case, they admit rational parameterizations. Invoking concepts from classical algebraic geometry, we formulate the condition for a degenerate intersection in terms of the vanishing of a polynomial expression in the quadric coefficients. When this is satisfied, we apply a multivariate polynomial factorization algorithm to the projecting cone of the intersection curve. Factors of this cone which correspond to intersection components “at infinity” may be removed a priori. A careful examination of the remaining cone factors then facilitates the identification and parameterization of the various real, affine intersection elements that may arise: isolated points, lines, conics, cubics, and singular quartics. The procedure is essentially automatic (avoiding the tedium of case-by-case analyses), encompasses the full range of quadric forms, and is amenable to implementation in exact (symbolic) arithmetic.