Geometric Algorithms for Detecting and Calculating All Conic Sections in the Intersection of Any 2 Natural Quadric Surfaces

Abstract One of the most challenging aspects of the surface-surface intersection problem is the proper disposition of degenerate configurations. Even in the domain of quadric surfaces, this problem has proven to be quite difficult. The topology of the intersection as well as the basic geometric representation of the curve itself is often at stake. By Bezout′s Theorem, two quadric surfaces always intersect in a degree four curve in complex projective space. This degree four curve is degenerate if it splits into two (possibly degenerate) conic sections. In theory the presence of such degeneracies can be detected using classical algebraic geometry. Unfortunately in practice it has proven to be extremely difficult to make computer implementations of such methods reliable numerically. Here, we present geometric algorithms that detect the presence of these degeneracies and compute the resulting planar intersections. The theoretical basis of these algorithms-in particular, proofs of correctness and completeness-are extremely long and tedious. We briefly outline the approach, but present only the results of the analysis as embodied in the geometric algorithms. Interested readers are referred to R. N. Goldman and J. R. Miller (Detecting and calculating conic sections in the intersection of two natural quadric surfaces, part I: Theoretical analysis; and Detecting and calculating conic sections in the intersection of two natural quadric surfaces, part II: Geometric constructions for detection and calculation, Technical Reports TR-93-1 and TR-93-2, Department of Computer Science, University of Kansas, January 1993) for details.