Surface intersection by switching from recursive subdivision to iterative refinement

Abstract

This paper discusses an attempt to devise an efficient (involving minimal computations), accurate (numerically high precision), exhaustive (detecting all possible solutions), and robust (working without failures) method for detecting intersection of two parametric surfaces. The method starts with subdivision to ensure that all solutions are detected. Later it switches over to numerical iterative refinement for efficient and accurate evaluation of the intersection curve. The switching takes place only when the convergence of the refinement method is guaranteed. The necessary theory to arrive at a computable condition leading to this guarantee has been developed using fixed-point and contractionmapping theorems from topology and mathematical analysis. The implementation is discussed elaborating the data structures and the algorithms used for (1) detecting segments of the intersection curve, (2) generating points on these segments using refinement, and (3) tracing a continuous curve by identifying neighboring segments and joining them in order.