Topology for Computing (Cambridge Monographs on Applied and Computational Mathematics)

Topology for Computing (Cambridge Monographs on Applied and Computational Mathematics)

Afra J. Zomorodian

Language: English

Pages: 258

ISBN: 0521136091

Format: PDF / Kindle (mobi) / ePub

Written by a computer scientist for computer scientists, this book teaches topology from a computational point of view, and shows how to solve real problems that have topological aspects involving computers. Such problems arise in many areas, such as computer graphics, robotics, structural biology, and chemistry. The author starts from the basics of topology, assuming no prior exposure to the subject, and moves rapidly up to recent advances in the area, including topological persistence and hierarchical Morse complexes. Algorithms and data structures are presented when appropriate.

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fundamentally related, as they are both properties of the same space. Geometric modifications, such as decimation in Example 1.5, could alter the topology. Is the simplified dragon in Figure 1.2(c) connected the same way as the original? In this case, the answer is yes, because the decimation algorithm excludes geometric modifications that have topological impact. We have changed the geometry of the surface without changing its topology. When creating photo-realistic images, however, appearance

Resolution: We eliminate conflicts by refining the complex. (ii) Diminution: We allow conflicts to exist and minimize their effects through appropriate reordering algorithms. In approach (i), we realize our goal of a reordered filtration with Betti numbers equivalent to the p-persistent Betti numbers of the original filtration. The reordered filtration, however, is refined. In approach (ii), we approximate our goal of the reordered filtration but maintain the same complex. 8.3.2 Resolution We

edges until it hits (a) a minimum or a maximum, (b) a previously traced path at a regular point, or (c) another saddle, at which point the path ends. Case (a) corresponds to the generic case for smooth height functions, Case (b) corresponds to a merging or forking, and Case (c) is the PL counterpart of a nontransversal intersection between a stable and an unstable 1-manifold. In Case (b), the regular point is special, so we call it a junction. Definition 9.1 (junction) A junction is a regular

surface. This approach mimics the construction of regular neighborhoods for complexes (Giblin, 1981). For a vertex u ∈ s p , the edge property guaranteed by subdivision enables us to mark each edge uv ∈ St u, v ∈ s p as positive or negative, depending on the location of v with respect to s p . Figure 10.5 illustrates this marking for a vertex. After marking the edges, we walk once around zq , starting at a vertex not on s p . If such a vertex does not exist, then λ(z p , zq ) = 0. Otherwise, we

subsets (cells) such that every element of the set is in one and only one of the subsets. Definition 2.16 (equivalence) Let S be a nonempty set and let ∼ be a relation between elements of S that satisfies the following properties for all a, b, c ∈ S: (a) (Reflexive) a ∼ a. (b) (Symmetric) If a ∼ b, then b ∼ a. 2.2 Manifolds 19 (c) (Transitive) If a ∼ b and b ∼ c, then a ∼ c. Then, the relation ∼ is an equivalence relation on S. The following theorem allows us to derive a partition from an

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