Multiple covers with balls Dissertation Thesis

Author(s): Iglesias-Ham, Mabel
Advisor(s): Edelsbrunner, Herbert
Committee Chair(s): Wojtan, Chris
Committee Member(s): Wagner, Uli
Title: Multiple covers with balls
Affiliation IST Austria
Abstract: 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 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 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.
Publication Title: IST Dissertation
Degree Granting Institution: IST Austria  
Degree: PhD
Degree Date: 2018-06-11
Start Page: 1
Total Pages: 171
DOI: 10.15479/AT:ISTA:th_1026
Notes: I would like to thank IST Austria and its Grad School for providing the conditions for this thesis to be developed. Special thanks to my supervisor Herbert Edelsbrunner, for trusting me and accepting me in his group, for the many hours of motivating research discussions and guidance that contributed to my professional development, for the opportunity to see at close distance his talent and learn from his academic example, and for the pleasure to meet him as an extraordinary person. Likewise, I am very grateful to his well selected group members, old and new, which contributed to a pleasant working environment and fruitful seminars. Furthermore, I thank my mother, Jacqueline, for taking care of my sweet daughter Sophia while I finish my thesis, and to my loving partner Michael for his support, fruitful discussions and encouragement.
Open access: yes (repository)
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