Geometric and Solid Modeling

Simplicial Complexes and Geometric Realization Since we de ned simplices as convex combinations of points, it is conceivable that this de nition is too narrow. That is, when constructing a simplicial complex, can we obtain more complicated structures using simplices that are only homeomorphic to convex combinations? From a topological point of view, the answer is no, and is justi ed as follows. We de ne an abstract simplex S as a nite set of points, called the vertices of S. Every proper subset of S is a face of S. If S consists of d + 1 points, then we say that it has the dimension d. An abstract simplicial complex C is de ned as follows: 1. There is a nite set of vertices V. 2. C is a set of subsets S of V with the property that all subsets of S are in C. Intuitively, the subsets S are the simplices in C. It can be proved that every abstract simplicial complex C has a geometric realization jCj in Euclidian space as a complex of simplices that are convex combinations. That is, given an abstract complex C with vertices fv1; :::; vmg, we can nd m points in Euclidian n-dimensional space E such that, for every abstract simplex S = hp0; :::; pdi in C, the points in E corresponding to the pk are linearly independent and hence de ne a simplex jSj in E that is a convex combination of those points. Theorem If C is an abstract simplicial complex of dimension n, then C can be realized by a corresponding concrete simplicial complex jCj in E, where the vertices are points and the simplices are convex combinations of them. In other words, the abstract complex C has a \nice" piecewise linear realization in a Euclidian space of suÆciently high dimension. Thus, we do not lose generality by using the concrete de nition of simplices as convex combinations. 2.4 Topological Validity of B-rep Solids 53 Figure 2.43 Opposite Faces in a 3-Simplex Manifold Triangulations We return to the problem of characterizing manifolds as topological polyhedra, and describe the local structure of a simplicial complex triangulating the manifold. Because of the above, we may assume that the simplicial complexes are piecewise linear in a suitable Euclidian space. Let S be a d-simplex with vertices p0; :::; pd. A proper subset q0; :::; qr of the vertices of S de nes an r-simplex that is a face S1 of S. Let qr+1; :::; qd be the remaining vertices of S. Then these vertices de ne another face S2. We say that S1 and S2 are opposite faces of S. Figure 2.43 shows examples in the case of d = 3. Let S and S 0 be two simplices in a simplicial complex. Then S and S 0 are adjacent if they have a common face. If S 00 is a face in which S and S 0 are adjacent, then S and S 0 are incident to S 00. Finally, a simplicial complex C is connected, if for all pairs of simplices S and S 0 in C, we can nd a sequence of simplices S1; :::; Sr in C such that, for 1 k < r, we have S = S1 and S 0 = Sr; and Sk is incident to Sk+1, or vice versa. Let S be a simplex in some simplicial complex. S will be incident to a nite set of simplices S1; :::; Sr in C. For each simplex Si of which S is a face, let Ti be the face of Si opposite S. The set of all such opposite faces is the link of S in C; see Figure 2.44 for an example. We are now in a position to characterize 2and 3-manifolds in terms of simplicial complexes. Although stated as de nitions, these characterizations can be proved formally. A 2-manifold without boundary is homeomorphic to a simplicial complex C of dimension 2 satisfying the following restrictions: 1. Every 1-simplex in C is incident to exactly two 2-simplices. 2. The link of every 0-simplex in C is a triangulation of the circle. See also Figure 2.45 for an illustration of the vertex structure in a 2-manifold without boundary. Similarly, a 3-manifold without boundary is homeomorphic to a simplicial complex C of dimension 3 satisfying the following restrictions: