A Taxonomy on Geometric and Topological Models

Mathematical fundamentals of point set topology and combinatorial topology are reviewed. All geometric models of practical interest are found to be finite collections of manifold elements of different dimensionalities, with boundary incidence as the most important relation among the elements. Based on this observation a classification scheme is proposed, characterizing a model topologically according to the compositional structure, boundary relations, dimensionality and connectivity of its elements. Geometric shapes are characterized as embeddings of manifolds into the Euclidian space, giving emphasis to homotopy groups. Accidental inconsistencies between embeddings and topological data are discussed, along with surgery operations on them. In focus of criticism is the inexact use of common terms like “boundary representation”, “regularization” and “non-manifold model”, which are here reconsidered mathematically.