TY - THES AB - 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. AU - Iglesias Ham, Mabel ID - 201 SN - 2663-337X TI - Multiple covers with balls ER - TY - JOUR AB - Inclusion–exclusion is an effective method for computing the volume of a union of measurable sets. We extend it to multiple coverings, proving short inclusion–exclusion formulas for the subset of Rn covered by at least k balls in a finite set. We implement two of the formulas in dimension n=3 and report on results obtained with our software. AU - Edelsbrunner, Herbert AU - Iglesias Ham, Mabel ID - 530 JF - Computational Geometry: Theory and Applications TI - Multiple covers with balls I: Inclusion–exclusion VL - 68 ER - TY - JOUR AB - Motivated by biological questions, we study configurations of equal spheres that neither pack nor cover. Placing their centers on a lattice, we define the soft density of the configuration by penalizing multiple overlaps. Considering the 1-parameter family of diagonally distorted 3-dimensional integer lattices, we show that the soft density is maximized at the FCC lattice. AU - Edelsbrunner, Herbert AU - Iglesias Ham, Mabel ID - 312 IS - 1 JF - SIAM J Discrete Math SN - 08954801 TI - On the optimality of the FCC lattice for soft sphere packing VL - 32 ER - TY - JOUR AB - Voronoi diagrams and Delaunay triangulations have been extensively used to represent and compute geometric features of point configurations. We introduce a generalization to poset diagrams and poset complexes, which contain order-k and degree-k Voronoi diagrams and their duals as special cases. Extending a result of Aurenhammer from 1990, we show how to construct poset diagrams as weighted Voronoi diagrams of average balls. AU - Edelsbrunner, Herbert AU - Iglesias Ham, Mabel ID - 1295 JF - Electronic Notes in Discrete Mathematics TI - Multiple covers with balls II: Weighted averages VL - 54 ER - TY - CONF AB - Motivated by biological questions, we study configurations of equal-sized disks in the Euclidean plane that neither pack nor cover. Measuring the quality by the probability that a random point lies in exactly one disk, we show that the regular hexagonal grid gives the maximum among lattice configurations. AU - Edelsbrunner, Herbert AU - Iglesias Ham, Mabel AU - Kurlin, Vitaliy ID - 1495 T2 - Proceedings of the 27th Canadian Conference on Computational Geometry TI - Relaxed disk packing VL - 2015-August ER - TY - GEN AB - The classical sphere packing problem asks for the best (infinite) arrangement of non-overlapping unit balls which cover as much space as possible. We define a generalized version of the problem, where we allow each ball a limited amount of overlap with other balls. We study two natural choices of overlap measures and obtain the optimal lattice packings in a parameterized family of lattices which contains the FCC, BCC, and integer lattice. AU - Iglesias Ham, Mabel AU - Kerber, Michael AU - Uhler, Caroline ID - 2012 T2 - arXiv TI - Sphere packing with limited overlap ER -