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Multiple Covers with Balls

Contents:
Abstract
Acknowledgments
List of publications
List of tables
List of figures
List of abbreviations
1 Introduction
2 Inclusion-Exclusion
3 Weighted Averages
4 Relaxed packing
5 Applications
6 Conclusion
Bibliography
A Appendix - Maximizing the relaxed covering quality
B Appendix - Analyzing the 3D Voroni Cell
C Appendix - 2D Proofs
D Appendix - 3D Case Analysis
Summary: We describe arrangements of three-dimensional spheres from a geometrical and topological point of view. Real data (fitting this setup) often consist of soft spheres which show certain degree of deformation while strongly packing against each other. In this context, we answer the following questions: If we model a soft packing of spheres by hard spheres that are allowed to overlap, can we measure the volume in the overlapped areas? Can we be more specific about the overlap volume, i.e. quantify how much volume is there covered exactly twice, three times, or $k$ times? What would be a good optimization criteria that rule the arrangement of soft spheres while making a \emph{good} use of the available space? Fixing a particular criterion, what would be the optimal sphere configuration? The first result of this thesis are short formulas for the computation of volumes covered by at least $k$ of the balls. The formulas exploit information contained in the order-$k$ Voronoi diagrams and its closely related Level-$k$ complex. The used complexes lead to a natural generalization into \emph{poset diagrams}, a theoretical formalism that contains the order-$k$ and degree-$k$ diagrams as special cases. In parallel, we define different criteria to determine what could be considered an optimal arrangement from a geometrical point of view. Fixing a criterion, we find optimal soft packing configurations in 2D and 3D where the ball centers lie on a lattice. As a last step, we use tools from computational topology on real physical data, to show the potentials of higher-order diagrams in the description of melting crystals. The results of the experiments leaves us with an open window to apply the theories developed in this thesis in real applications.
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Thesis

Abstract

Acknowledgments

List of publications

List of tables

List of figures

List of abbreviations

1 Introduction

2 Inclusion-Exclusion

3 Weighted Averages

4 Relaxed packing

5 Applications

6 Conclusion

Bibliography

A Appendix - Maximizing the relaxed covering quality

B Appendix - Analyzing the 3D Voroni Cell

C Appendix - 2D Proofs

D Appendix - 3D Case Analysis

We describe arrangements of three-dimensional spheres from a geometrical and topological point of view. Real data (fitting this setup) often consist of soft spheres which show certain degree of deformation while strongly packing against each other. In this context, we answer the following questions: If we model a soft packing of spheres by hard spheres that are allowed to overlap, can we measure the volume in the overlapped areas? Can we be more specific about the overlap volume, i.e. quantify how much volume is there covered exactly twice, three times, or $k$ times? What would be a good optimization criteria that rule the arrangement of soft spheres while making a \emph{good} use of the available space? Fixing a particular criterion, what would be the optimal sphere configuration? The first result of this thesis are short formulas for the computation of volumes covered by at least $k$ of the balls. The formulas exploit information contained in the order-$k$ Voronoi diagrams and its closely related Level-$k$ complex. The used complexes lead to a natural generalization into \emph{poset diagrams}, a theoretical formalism that contains the order-$k$ and degree-$k$ diagrams as special cases. In parallel, we define different criteria to determine what could be considered an optimal arrangement from a geometrical point of view. Fixing a criterion, we find optimal soft packing configurations in 2D and 3D where the ball centers lie on a lattice. As a last step, we use tools from computational topology on real physical data, to show the potentials of higher-order diagrams in the description of melting crystals. The results of the experiments leaves us with an open window to apply the theories developed in this thesis in real applications.

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