On the lower degree intersections of two natural quadrics

Abstract

In general, two quadric surface intersect in a space quartic curve. However, the intersection frequently degenerates to a collection of plane curves. Degenerate cases are frequent in geometric/solid modeling because degeneracies are often required by design. Their detection is important because degenerate intersections can be computed more easily and allow simpler treatment of important problems. In this paper, we investigate this problem for natural quadrics. Algorithms are presented to detect and compute conic intersections and linear intersections. These methods reveal the relationship between the planes of the degenerate intersections and the quadrics. Using the theory developed in the paper, we present a new and simplified proof of a necessary and sufficient condition for conic intersection. Finally, we present a simple method for determining the types of conic in a degenerate intersection without actually computing the intersection, and an enumeration of all possible conic types. Since only elementary geometric routines such as line intersection are used, all of the above algorithms are intuitive and easily implementable.