## A Short Course in Computational Geometry and TopologyThis 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 | |

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 |

### Other editions - View all

A Short Course in Computational Geometry and Topology Herbert Edelsbrunner No preview available - 2014 |

A Short Course in Computational Geometry and Topology Herbert Edelsbrunner No preview available - 2014 |

### Common terms and phrases

2-manifold abstract simplicial complex algorithm alpha shapes alternating sum Author(s Betti numbers bisector boundary matrix Brownian tree centers circle column Computational Geometry connected construct contains convex hull convex polygon corresponding Course in Computational cube cycle define Delaunay triangulation dimension disks of radius draw Edelsbrunner Euclidean distance Euler characteristic example face finite set formula four full subcomplex function values geometric realization Geometry and Topology holes homeomorphic homology groups homotopy type implies Klein bottle Lemma lower star Mathematical Methods non-empty common intersection non-zero octahedron open disk p-cycle persistence diagram persistent homology pivot PL Functions planar graph pocket points in R2 power diagram projective plane protein rank Recall set of points set of sites Short Course simplex simplices sphere SpringerBriefs in Mathematical sublevel set subset tetrahedron topological space torus union of disks vertex void Voronoi diagram Voronoi regions þ 10 points