Modeling bounded and unbounded space with polyhedra: Topology and operators for manifold cell complexes

Abstract

This paper proposes polyhedral space partitioning as an alternative to component assembly of digital models of objects with complex linear shapes. A partition is specified with a path-connected user model, where each object is bounded by n-manifolds. Faces and cells can be non-convex, multiply-connected and unbounded. The user interacts with the user model and specifies work steps. Each work step splits one edge, face or cell of the partition, or merges two neighboring objects of equal dimension. As a consequence, only a small subset of the model objects, consisting of the user specified objects and their neighbors, are affected by a work step. The user model is automatically mapped to a core model containing methods for topological relations and navigation. The topological structure is described by bundles of twin arrows of opposite direction arranged in polygons, twin facets with normal vectors of opposite direction and dihedral facet cycles at the edges. Imaginary topological objects are introduced to define unbounded cells, faces and edges. The approach guarantees that there is no overlap or gap between any pair of neighboring objects. It supports modelling of non-convex and multiply connected bounded and unbounded objects. For verification, several example models are presented and visualized. The paper ends with conclusions and an outlook to ongoing and planned further research in this field.