Robust Intersections under Floating Point

Our strategy for handling intersections robustly in floating point is to register degeneracies with tolerances in a hierarchical manner before registering the top level intersection primitives in order from most degenerate (vertex-vertex) to least, followed by the top level primitives (edge-edge and triangle-vertex in 2D, triangleedge and tetrahedron-vertex in 3D). Intersection pairs are stored in hashtables for efficient lookup. When processing each intersection pair, we first check these hashtables to see if a degenerate version of pair has already been registered (e.g., when checking an edge-edge pair, a vertex of one of these edges may already be registered as intersecting the other edge or one if its endpoints). If such a degeneracy is found, processing on the pair terminates and no intersection is registered. Otherwise, the pair is checked to see if it actually intersects. Handling degeneracies in this way allows us to assume that no degeneracy exists, which in turn makes it possible to guarantee that the current pair can be checked robustly. This is the key to guaranteeing robustness.