Topological models for boundary representation: a comparison with n-dimensional generalized maps

In boundary representation, a geometric object is represented by the union of a ‘topological’ model, which describes the topology of the modelled object, and an ‘embedding’ model, which describes the embedding of the object, for instance in three-dimensional Euclidean space. In recent years, numerous topological models have been developed for boundary representation, and there have been important developments with respect to dimension and orientability. In the main, two types of topological models can be distinguished. ‘Incidence graphs’ are graphs or hypergraphs, where the nodes generally represent the cells of the modelled subdivision (vertex, edge, face, etc.), and the edges represent the adjacency and incidence relations between these cells. ‘Ordered’ models use a single type of basic element (more or less explicitly defined), on which ‘element functions’ act; the cells of the modelled subdivision are implicitly defined in this type of model. In this paper some of the most representative ordered topological models are compared using the concepts of the n-dimensional generalized map and the n-dimensional map. The main result is that ordered topological models are (roughly speaking) equivalent with respect to the class of objects which can be modelled (i.e. with respect to dimension and orientability).