A Short Course in Computational Geometry and Topology

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Springer Science & Business, Apr 28, 2014 - Computers - 110 pages
This monograph presents a short course in computational geometry and topology. In the first part the book covers Voronoi diagrams and Delaunay triangulations, then it presents the theory of alpha complexes which play a crucial role in biology. The central part of the book is the homology theory and their computation, including the theory of persistence which is indispensable for applications, e.g. shape reconstruction. The target audience comprises researchers and practitioners in mathematics, biology, neuroscience and computer science, but the book may also be beneficial to graduate students of these fields.
 

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Contents

1 Roots of Geometry and Topology
1
Part ITessellations
7
2 Voronoi and Delaunay Diagrams
8
3 Weighted Diagrams
17
4 Three Dimensions
23
Part ITessellations
28
Part IIComplexes
29
5 Alpha Complexes
30
8 Topological Spaces
56
9 Homology Groups
65
10 Complex Construction
72
Part IIIHomology
79
Part IVPersistence
81
11 Filtrations
83
12 PL Functions
90
13 Matrix Reduction
99

6 Holes
41
7 Area Formulas
47
Part IIComplexes
53
Part IIIHomology
55
Part IVPersistence
106
Epilogue
109
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