Decomposition and Modeling in the Non-Manifold domain

Abstract

The problem of decomposing non-manifold object has already been studied in solid modeling. However, the few proposed solutions are limited to the problem of decomposing solids described through their boundaries. In this thesis we study the problem of decomposing an arbitrary non-manifold simplicial complex into more regular components. A formal notion of decomposition is developed using combinatorial topology. The proposed decomposition is unique, for a given complex, and is computable for complexes of any dimension. A decomposition algorithm is proposed that is linear w.r.t. the size of the input. In three or higher dimensions a decomposition into manifold parts is not always possible. Thus, in higher dimensions, we decompose a non-manifold into a decidable super class of manifolds, that we call, Initial-Quasi-Manifolds. We also defined a two-layered data structure, the Extended Winged data structure. This data structure is a dimension independent data structure conceived to model non-manifolds through their decomposition into initial-quasi-manifold parts. Our two layered data structure describes the structure of the decomposition and each component separately. In the second layer we encode the connectivity structure of the decomposition. We analyze the space requirements of the Extended Winged data structure and give algorithms to build and navigate it. Finally, we discuss time requirements for the computation of topological relations and show that, for surfaces and tetrahedralizations, embedded in real 3D space, all topological relations can be extracted in optimal time. This approach offers a compact, dimension independent, representation for non-manifolds that can be useful whenever the modeled object has few non-manifold singularities.