Matchmaker: Manifold Breps for Non-manifold r-sets

Abstract

Many solid modeling construction techniques produce non-manifold r-sets (solids). With each non-manifold model N we can associate a family of manifold solid models that are infinitely close to N in the geometric sense. For polyhedral solids, each non-manifold edge of N with 2k incident faces will be replicated k times in any manifold model M of that family. Furthermore, some non-manifold vertices of N must also be replicated in M, possibly several times. M can be obtained by defining, in N, a single adjacent face TA(E,F) for each pair (E,F) that combines an edge E and an incident face F. The adjacency relation satisfies TA(E,TA(E,F))=F. The choice of the map A defines which vertices of N must be replicated in M and how many times. The resulting manifold representation of a non-manifold solid may be encoded using simpler and more compact data-structures, especially for triangulated model, and leads to simpler and more efficient algorithms, when it is used instead of a non-manifold representation for a variety of tasks, such as simplification, compression, interference detection or rendering. Most choices of the map A lead to invalid (self-intersecting) boundaries and to unnecessary vertex replications for M. We propose an efficient algorithm, called Matchmaker, which computes a map A, such that there exists an infinitely small perturbation of the vertices and edges of M that produces a valid (non self-intersecting) boundary of a manifold solid. Furthermore, our approach avoids most unnecessary vertex replications.