Combinatorial models for topology-based geometric modeling

simplicial complexes (ASCs) are one of the most known topological models [1, 89, 21, 32]. Semi-simplicial sets (SSSs) generalize them [82, 67, 68], i.e. a SSS can be associated with any ASC, but the converse is not true. In particular, SSSs can represent « curved » objects, maybe multi-incident, but it is not possible for ASCs. The definition type also differs: ASCs are sets of sets, SSSs are algebra. Their definitions are here recalled, and also basic notions and operations (operations for computing topological properties are not discussed here, though many works deal with them [35, 37, 91]). Conversion operations are mentioned, and also classical model optimizations as for simplicial quasi-manifolds. Combinatorial Models for Topology-Based Geometric Modeling 163 2.1. Abstract simplicial complexes Definition. An abstract simplicial complex K is defined upon a set V of (abstract) vertices in the following way (cf. figure 8): – a p-dimensional simplex is a set of p + 1 vertices; – K is a set of simplices, such that any non empty subset of any simplex of K is a simplex of K. The dimension of K is the highest dimension of the simplices of K. Let σ = {v0, ..., vp} be an abstract simplex. A proper face of σ is a non empty subset of vertices of σ, different from σ. The principal face of σ is σ itself. The boundary of σ is the ASC made of the proper faces of σ. The star of σ is the set of simplices for which σ is a (proper or principal) face. σ is a principal simplex if it is not the proper face of any simplex. Traversal algorithms can be defined using these notions, i.e. by traversing simplices of the boundary and / or the star of a simplex. These « neighborhood traversal » algorithms are very important for many operations. They correspond for topological models to « connectivity traversal » algorithms for graphs; they are either fundamental ones, and they can be defined in similar ways. Geometric realization. The geometric realization of an ASC is a simplicial complex (cf. figure 8). An euclidean p-dimensional simplex is the convex hull of p + 1 linearly independent points of an euclidean space. A face of an euclidean simplex is a simplex defined by a non empty subset of the points which define the simplex. A simplicial complex L is a set of euclidean simplices which satisfy the two following properties: – any face of any simplex of L is a simplex of L; – the (geometric) intersection of two simplices of L is empty, or it is a face common to the two simplices. Note that simple graphs correspond to 1-dimensional ASCs. 164 P. Lienhardt, L. Fuchs, and Y. Bertrand