Torus/Sphere Intersection Based on a Configuration Space Approach

This paper presents an efficient and robust geometric algorithm that classifies and detects all possible types of torus/sphere intersections, including all degenerate conic sections (circles) and singular intersections. Given a torus and a sphere, we treat one surface as an obstacle and the other surface as the envelope surface of a moving ball. In this case, the Configuration space (C-space) obstacle is the same as the constant radius offset of the original obstacle, where the radius of the moving ball is taken as the offset distance. Based on the intersection between the C-space obstacle and the trajectory of the center of the moving ball, we detect all the intersection loops and singular contact point/circle of the original torus and sphere. Moreover, we generate exactly one starting point (for numerical curve tracing) on each connected component of the intersection curve. All required computations involve vector/distance computations and circle/circle intersections, which can be implemented efficiently and robustly. All degenerate conic sections (circles) can also be detected using a few additional simple geometric tests. The intersection curve itself (a quartic space curve, in general) is then approximated with a sequence of cubic curve segments.