Sum-of-Squares Collision Detection for Curved Shapes and Paths

Sum-of-Squares Programming (SOSP) has recently been introduced to graphics as a unified way to address a large set of difficult problems involving higher order primitives. Unfortunately, a challenging aspect of this approach is the computational cost—especially for problems involving multiple geometries like collision detection. In this paper, we present techniques to reduce the cost of SOSP significantly. We use these improvements to speed up difficult problems like collision detection between Bézier triangles by as much as 300 ×. In addition, motivated by hair bundle simulation, we present SOSP based collision detection on the tapered cubic cylinder. We also present an algebraic formulation of rigid body motion enabling SOSP based collision detection for curved geometries and trajectories simultaneously. While these new formulations are complex, our speedups make them feasible. These advances improve the applicability of SOSP based collision detection and enable the continued progress of higher-order geometry processing.