Guaranteed consistency of surface intersections and trimmed surfaces using a coupled topology resolution and domain decomposition scheme

Abstract

We describe a method that serves to simultaneously determine the topological configuration of the intersection curve of two parametric surfaces and generate compatible decompositions of their parameter domains, that are amenable to the application of existing perturbation schemes ensuring exact topological consistency of the trimmed surface representations. To illustrate this method, we begin with the simpler problem of topology resolution for a planar algebraic curve F(x,y)=0 in a given domain, and then extend concepts developed in this context to address the intersection of two tensor-product parametric surfaces p(s,t) and q(u,v) defined on (s,t)∈[0,1]2 and (u,v)∈[0,1]2. The algorithms assume the ability to compute, to any specified precision, the real solutions of systems of polynomial equations in at most four variables within rectangular domains, and proofs for the correctness of the algorithms under this assumption are given.