Computing the intersection of two ellipsoids based on a fast algebraic topology determination strategy

Abstract

Ellipsoids serve as the most commonly used geometric primitives and bounding volumes in computer-aided design and computer graphics, where an efficient and topologically stable intersection algorithm between two ellipsoids is highly required. Although there has been extensive research on intersections of two general quadrics, ellipsoids have their own specialty in both algebra and geometry which guides to new possibilities to break the bottleneck in intersection computation. In this paper, we use a topology-determination-based strategy in computing the intersection of ellipsoids. Firstly, the topology of the intersection curve is quickly determined using some algebraic discriminants without computing any point on the intersection curve; then an octree strategy is applied to efficiently compute at least one point on each intersection branch; finally, by tracing the branch, we get the complete intersection loci. Plenty of examples show that our algorithm is topologically stable when facing challenging cases including multi-branches, small loops, singular or tangent intersections, and is more efficient compared with existing algorithms.