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Combining algebraic rigor with geometric robustness for the detection and calculation of conic sections in the intersection of two natural quadric surfaces

One of the most challenging aspects of the surface-surface intersection problem is the proper disposition of degenerate configurations. The topology of the intersection as well as the basic geometric representation of the curve itself is usually at stake. In the boundary evaluation algorithm for solid modeling, detecting such conditions is often the difference between a successful and an unsuccessful implementation. Even in the domain of quadric surfaces, this problem has proven to be quite difficult. 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 real affine space. The presence of such degeneracies can, in theory, be detected using classical algebraic geometry. Unfortunately it has proven to be extremely difficult, in practice, to make computer implementations of such methods reliable numerically. In contrast, methods based on case-by-case geometric analysis and construction have proven to be extremely reliable, but there has been no way to guarantee that all degenerate situations are detected. In this paper we begin by demonstrating the use of algebraic methods to characterize precisely those situations in which two natural quadric surfaces intersect in planar curves. We then re-interpret the resulting algebraic conditions geometricrdly. The result is a definitive list of all possible configurations of two natural quadrics that intersect in planar curves, not all of which are intuitive. We go on to demonstrate how to use this list to develop robust geometric schemes in the actual computer implementation to detect and calculate these planar interseetions. Thus we secure the guaranteed completeness of algebraic analysis, but retain the numerical robustness of geometric schemes since only the latter are actually implemented in the code.