A Short Course in Computational Geometry and Topology

Description

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 Front Matter....Pages i-ix Roots of Geometry and Topology....Pages 1-6 Front Matter....Pages 7-7 Voronoi and Delaunay Diagrams....Pages 9-15 Weighted Diagrams....Pages 17-22 Three Dimensions....Pages 23-27 Back Matter....Pages 29-29 Front Matter....Pages 31-31 Alpha Complexes....Pages 33-39 Holes....Pages 41-46 Area Formulas....Pages 47-52 Back Matter....Pages 53-54 Front Matter....Pages 55-55 Topological Spaces....Pages 57-63 Homology Groups....Pages 65-71 Complex Construction....Pages 73-78 Back Matter....Pages 79-80 Front Matter....Pages 81-81 Filtrations....Pages 83-89 PL Functions....Pages 91-97 Matrix Reduction....Pages 99-105 Back Matter....Pages 107-108 Back Matter....Pages 109-110