Filtrations and Signatures Simplicial Complexes
2.5 Manifold Sweeps
Alpha-shapes allow us to explore the shape of finite point sets and unions of balls. In addition to such spaces, we are interested in exploring manifolds with height functions. In Example 1.10, we saw how the geometry of a manifold dictates the topology of its iso-lines. We use this example to motivate another geometrically ordered filtration in this section, postponing theoretical justifi- cation for it until we have been introduced to Morse Theory in Chapter 5. Let K be a triangulation of a compact 2-manifold without boundary M. Let h : M → R be a function that is linear on every triangle. The function is de- fined, consequently, by its values at the vertices of K. We will assume that h u = hv for all vertices u = v ∈ K. Again, simulation of simplicity is the computational justification for this assumption Edelsbrunner and Mücke, 1990. It is common to refer to h as the height function, because it matches our intuition of a geographic landscape. One needs to be careful, however, not to allow the intuition to limit one’s imagination, as h can be any continuous function. In a simplicial complex, the natural concept of a neighborhood of a vertexParts
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» Geometric Definition Simplicial Complexes
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» Manifold Sweeps Topology for Computing
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» Surface Reconstruction Topology for Computing
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