Polyhedral perturbations that preserve topological form

Abstract

The idea, that we are willing to accept variation in an object but that we insist it should retain its original topological form, has powerful intuitive appeal, and the concept appears in many applied fields. Some of the most important of these are tolerancing and metrology, solid modeling, engineering design, finite element analysis, surface reconstruction, computer graphics, path planning in robotics, fairing procedures, image analysis, and medical imaging. In this paper we focus on the field of tolerancing and metrology. The requirement that two objects or sets should have the same topological form requires a precise definition. We specify “same topological form” to mean that there exists a “space homeomorphism” from R 3 onto R 3 that carries a nominal object S onto another design object. In general, establishing the existence of such space homeomorphisms can be considerably more difficult than demonstrating classical topological equivalence by a homeomorphism. In the special case when the boundary of S is a polyhedral two-sphere in R 3, one of the authors has previously given a simple sufficient condition for the existence of a space homeomorphism mapping S onto another design object. This paper presents an analogous sufficient condition for the case when S is a finite polyhedron in R 3. The result relies upon a triangulation of the boundary and upon a dependent parameter that specifies the maximum size of permissible perturbations of the vertices of the polyhedron.