Surface-to-surface intersection with complete and guaranteed results

We present an algorithm for the robust computation of the complete intersection curve of two general parametric surfaces based on interval arithmetic. The subdivision algorithm we introduce follows a divide-and-conquer-approach. It avoids loss of any parts of the intersection curve by using safe bounding volumes for all parts (patches) of the surfaces. For each pair of patches, it first checks for intersection of the bounding volumes. If two bounding volumes intersect, it splits one patch, and treats both new pairs recursively until a predefined termination condition is satisfied. We use parallelepipeds as tight bounding volumes. Each parallelepiped considers the shape and orientation of its patch, overestimating it only by second order terms. With the help of interval inclusions for the partial derivatives, we compute parts of the parameter domain, where one patch cannot reach the enclosure of the other one. Cutting off such dispensable regions, and corresponding parts in object space result in a faster convergence of the algorithm. Interval arithmetic is used for all critical operations to achieve robustness.