--- _id: '201' abstract: - lang: eng text: '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.' alternative_title: - ISTA Thesis article_processing_charge: No author: - first_name: Mabel full_name: Iglesias Ham, Mabel id: 41B58C0C-F248-11E8-B48F-1D18A9856A87 last_name: Iglesias Ham citation: ama: Iglesias Ham M. Multiple covers with balls. 2018. doi:10.15479/AT:ISTA:th_1026 apa: Iglesias Ham, M. (2018). Multiple covers with balls. Institute of Science and Technology Austria. https://doi.org/10.15479/AT:ISTA:th_1026 chicago: Iglesias Ham, Mabel. “Multiple Covers with Balls.” Institute of Science and Technology Austria, 2018. https://doi.org/10.15479/AT:ISTA:th_1026. ieee: M. Iglesias Ham, “Multiple covers with balls,” Institute of Science and Technology Austria, 2018. ista: Iglesias Ham M. 2018. Multiple covers with balls. Institute of Science and Technology Austria. mla: Iglesias Ham, Mabel. Multiple Covers with Balls. Institute of Science and Technology Austria, 2018, doi:10.15479/AT:ISTA:th_1026. short: M. Iglesias Ham, Multiple Covers with Balls, Institute of Science and Technology Austria, 2018. date_created: 2018-12-11T11:45:10Z date_published: 2018-06-11T00:00:00Z date_updated: 2023-09-07T12:25:32Z day: '11' ddc: - '514' - '516' degree_awarded: PhD department: - _id: HeEd doi: 10.15479/AT:ISTA:th_1026 file: - access_level: closed checksum: dd699303623e96d1478a6ae07210dd05 content_type: application/zip creator: kschuh date_created: 2019-02-05T07:43:31Z date_updated: 2020-07-14T12:45:24Z file_id: '5918' file_name: IST-2018-1025-v2+5_ist-thesis-iglesias-11June2018(1).zip file_size: 11827713 relation: source_file - access_level: open_access checksum: ba163849a190d2b41d66fef0e4983294 content_type: application/pdf creator: kschuh date_created: 2019-02-05T07:43:45Z date_updated: 2020-07-14T12:45:24Z file_id: '5919' file_name: IST-2018-1025-v2+4_ThesisIglesiasFinal11June2018.pdf file_size: 4783846 relation: main_file file_date_updated: 2020-07-14T12:45:24Z has_accepted_license: '1' language: - iso: eng month: '06' oa: 1 oa_version: Published Version page: '171' publication_identifier: issn: - 2663-337X publication_status: published publisher: Institute of Science and Technology Austria publist_id: '7712' pubrep_id: '1026' status: public supervisor: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 title: Multiple covers with balls type: dissertation user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 year: '2018' ... --- _id: '530' abstract: - lang: eng text: 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. article_processing_charge: No author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Mabel full_name: Iglesias Ham, Mabel id: 41B58C0C-F248-11E8-B48F-1D18A9856A87 last_name: Iglesias Ham citation: ama: 'Edelsbrunner H, Iglesias Ham M. Multiple covers with balls I: Inclusion–exclusion. Computational Geometry: Theory and Applications. 2018;68:119-133. doi:10.1016/j.comgeo.2017.06.014' apa: 'Edelsbrunner, H., & Iglesias Ham, M. (2018). Multiple covers with balls I: Inclusion–exclusion. Computational Geometry: Theory and Applications. Elsevier. https://doi.org/10.1016/j.comgeo.2017.06.014' chicago: 'Edelsbrunner, Herbert, and Mabel Iglesias Ham. “Multiple Covers with Balls I: Inclusion–Exclusion.” Computational Geometry: Theory and Applications. Elsevier, 2018. https://doi.org/10.1016/j.comgeo.2017.06.014.' ieee: 'H. Edelsbrunner and M. Iglesias Ham, “Multiple covers with balls I: Inclusion–exclusion,” Computational Geometry: Theory and Applications, vol. 68. Elsevier, pp. 119–133, 2018.' ista: 'Edelsbrunner H, Iglesias Ham M. 2018. Multiple covers with balls I: Inclusion–exclusion. Computational Geometry: Theory and Applications. 68, 119–133.' mla: 'Edelsbrunner, Herbert, and Mabel Iglesias Ham. “Multiple Covers with Balls I: Inclusion–Exclusion.” Computational Geometry: Theory and Applications, vol. 68, Elsevier, 2018, pp. 119–33, doi:10.1016/j.comgeo.2017.06.014.' short: 'H. Edelsbrunner, M. Iglesias Ham, Computational Geometry: Theory and Applications 68 (2018) 119–133.' date_created: 2018-12-11T11:46:59Z date_published: 2018-03-01T00:00:00Z date_updated: 2023-09-13T08:59:00Z day: '01' ddc: - '000' department: - _id: HeEd doi: 10.1016/j.comgeo.2017.06.014 ec_funded: 1 external_id: isi: - '000415778300010' file: - access_level: open_access checksum: 1c8d58cd489a66cd3e2064c1141c8c5e content_type: application/pdf creator: dernst date_created: 2019-02-12T06:47:52Z date_updated: 2020-07-14T12:46:38Z file_id: '5953' file_name: 2018_Edelsbrunner.pdf file_size: 708357 relation: main_file file_date_updated: 2020-07-14T12:46:38Z has_accepted_license: '1' intvolume: ' 68' isi: 1 language: - iso: eng month: '03' oa: 1 oa_version: Preprint page: 119 - 133 project: - _id: 255D761E-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '318493' name: Topological Complex Systems publication: 'Computational Geometry: Theory and Applications' publication_status: published publisher: Elsevier publist_id: '7289' quality_controlled: '1' scopus_import: '1' status: public title: 'Multiple covers with balls I: Inclusion–exclusion' type: journal_article user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 volume: 68 year: '2018' ... --- _id: '312' abstract: - lang: eng text: 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. acknowledgement: This work was partially supported by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics,” through grant I02979-N35 of the Austrian Science Fund (FWF). article_processing_charge: No article_type: original author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Mabel full_name: Iglesias Ham, Mabel id: 41B58C0C-F248-11E8-B48F-1D18A9856A87 last_name: Iglesias Ham citation: ama: Edelsbrunner H, Iglesias Ham M. On the optimality of the FCC lattice for soft sphere packing. SIAM J Discrete Math. 2018;32(1):750-782. doi:10.1137/16M1097201 apa: Edelsbrunner, H., & Iglesias Ham, M. (2018). On the optimality of the FCC lattice for soft sphere packing. SIAM J Discrete Math. Society for Industrial and Applied Mathematics . https://doi.org/10.1137/16M1097201 chicago: Edelsbrunner, Herbert, and Mabel Iglesias Ham. “On the Optimality of the FCC Lattice for Soft Sphere Packing.” SIAM J Discrete Math. Society for Industrial and Applied Mathematics , 2018. https://doi.org/10.1137/16M1097201. ieee: H. Edelsbrunner and M. Iglesias Ham, “On the optimality of the FCC lattice for soft sphere packing,” SIAM J Discrete Math, vol. 32, no. 1. Society for Industrial and Applied Mathematics , pp. 750–782, 2018. ista: Edelsbrunner H, Iglesias Ham M. 2018. On the optimality of the FCC lattice for soft sphere packing. SIAM J Discrete Math. 32(1), 750–782. mla: Edelsbrunner, Herbert, and Mabel Iglesias Ham. “On the Optimality of the FCC Lattice for Soft Sphere Packing.” SIAM J Discrete Math, vol. 32, no. 1, Society for Industrial and Applied Mathematics , 2018, pp. 750–82, doi:10.1137/16M1097201. short: H. Edelsbrunner, M. Iglesias Ham, SIAM J Discrete Math 32 (2018) 750–782. date_created: 2018-12-11T11:45:46Z date_published: 2018-03-29T00:00:00Z date_updated: 2023-09-13T09:34:38Z day: '29' department: - _id: HeEd doi: 10.1137/16M1097201 external_id: isi: - '000428958900038' intvolume: ' 32' isi: 1 issue: '1' language: - iso: eng main_file_link: - open_access: '1' url: http://pdfs.semanticscholar.org/d2d5/6da00fbc674e6a8b1bb9d857167e54200dc6.pdf month: '03' oa: 1 oa_version: Submitted Version page: 750 - 782 project: - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: SIAM J Discrete Math publication_identifier: issn: - '08954801' publication_status: published publisher: 'Society for Industrial and Applied Mathematics ' publist_id: '7553' quality_controlled: '1' scopus_import: '1' status: public title: On the optimality of the FCC lattice for soft sphere packing type: journal_article user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 volume: 32 year: '2018' ... --- _id: '1295' abstract: - lang: eng text: 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. acknowledgement: This work is partially supported by the Toposys project FP7-ICT-318493-STREP, and by ESF under the ACAT Research Network Programme. author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Mabel full_name: Iglesias Ham, Mabel id: 41B58C0C-F248-11E8-B48F-1D18A9856A87 last_name: Iglesias Ham citation: ama: 'Edelsbrunner H, Iglesias Ham M. Multiple covers with balls II: Weighted averages. Electronic Notes in Discrete Mathematics. 2016;54:169-174. doi:10.1016/j.endm.2016.09.030' apa: 'Edelsbrunner, H., & Iglesias Ham, M. (2016). Multiple covers with balls II: Weighted averages. Electronic Notes in Discrete Mathematics. Elsevier. https://doi.org/10.1016/j.endm.2016.09.030' chicago: 'Edelsbrunner, Herbert, and Mabel Iglesias Ham. “Multiple Covers with Balls II: Weighted Averages.” Electronic Notes in Discrete Mathematics. Elsevier, 2016. https://doi.org/10.1016/j.endm.2016.09.030.' ieee: 'H. Edelsbrunner and M. Iglesias Ham, “Multiple covers with balls II: Weighted averages,” Electronic Notes in Discrete Mathematics, vol. 54. Elsevier, pp. 169–174, 2016.' ista: 'Edelsbrunner H, Iglesias Ham M. 2016. Multiple covers with balls II: Weighted averages. Electronic Notes in Discrete Mathematics. 54, 169–174.' mla: 'Edelsbrunner, Herbert, and Mabel Iglesias Ham. “Multiple Covers with Balls II: Weighted Averages.” Electronic Notes in Discrete Mathematics, vol. 54, Elsevier, 2016, pp. 169–74, doi:10.1016/j.endm.2016.09.030.' short: H. Edelsbrunner, M. Iglesias Ham, Electronic Notes in Discrete Mathematics 54 (2016) 169–174. date_created: 2018-12-11T11:51:12Z date_published: 2016-10-01T00:00:00Z date_updated: 2021-01-12T06:49:41Z day: '01' department: - _id: HeEd doi: 10.1016/j.endm.2016.09.030 ec_funded: 1 intvolume: ' 54' language: - iso: eng month: '10' oa_version: None page: 169 - 174 project: - _id: 255D761E-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '318493' name: Topological Complex Systems publication: Electronic Notes in Discrete Mathematics publication_status: published publisher: Elsevier publist_id: '5976' quality_controlled: '1' scopus_import: 1 status: public title: 'Multiple covers with balls II: Weighted averages' type: journal_article user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87 volume: 54 year: '2016' ... --- _id: '1495' abstract: - lang: eng text: '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. ' author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Mabel full_name: Iglesias Ham, Mabel id: 41B58C0C-F248-11E8-B48F-1D18A9856A87 last_name: Iglesias Ham - first_name: Vitaliy full_name: Kurlin, Vitaliy last_name: Kurlin citation: ama: 'Edelsbrunner H, Iglesias Ham M, Kurlin V. Relaxed disk packing. In: Proceedings of the 27th Canadian Conference on Computational Geometry. Vol 2015-August. Queen’s University; 2015:128-135.' apa: 'Edelsbrunner, H., Iglesias Ham, M., & Kurlin, V. (2015). Relaxed disk packing. In Proceedings of the 27th Canadian Conference on Computational Geometry (Vol. 2015–August, pp. 128–135). Ontario, Canada: Queen’s University.' chicago: Edelsbrunner, Herbert, Mabel Iglesias Ham, and Vitaliy Kurlin. “Relaxed Disk Packing.” In Proceedings of the 27th Canadian Conference on Computational Geometry, 2015–August:128–35. Queen’s University, 2015. ieee: H. Edelsbrunner, M. Iglesias Ham, and V. Kurlin, “Relaxed disk packing,” in Proceedings of the 27th Canadian Conference on Computational Geometry, Ontario, Canada, 2015, vol. 2015–August, pp. 128–135. ista: 'Edelsbrunner H, Iglesias Ham M, Kurlin V. 2015. Relaxed disk packing. Proceedings of the 27th Canadian Conference on Computational Geometry. CCCG: Canadian Conference on Computational Geometry vol. 2015–August, 128–135.' mla: Edelsbrunner, Herbert, et al. “Relaxed Disk Packing.” Proceedings of the 27th Canadian Conference on Computational Geometry, vol. 2015–August, Queen’s University, 2015, pp. 128–35. short: H. Edelsbrunner, M. Iglesias Ham, V. Kurlin, in:, Proceedings of the 27th Canadian Conference on Computational Geometry, Queen’s University, 2015, pp. 128–135. conference: end_date: 2015-08-12 location: Ontario, Canada name: 'CCCG: Canadian Conference on Computational Geometry' start_date: 2015-08-10 date_created: 2018-12-11T11:52:21Z date_published: 2015-08-01T00:00:00Z date_updated: 2021-01-12T06:51:09Z day: '01' department: - _id: HeEd ec_funded: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1505.03402 month: '08' oa: 1 oa_version: Submitted Version page: 128-135 project: - _id: 255D761E-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '318493' name: Topological Complex Systems publication: Proceedings of the 27th Canadian Conference on Computational Geometry publication_status: published publisher: Queen's University publist_id: '5684' quality_controlled: '1' scopus_import: 1 status: public title: Relaxed disk packing type: conference user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87 volume: 2015-August year: '2015' ... --- _id: '2012' abstract: - lang: eng text: 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. acknowledgement: We thank Herbert Edelsbrunner for his valuable discussions and ideas on the topic of this paper. The second author has been supported by the Max Planck Center for Visual Computing and Communication article_number: '1401.0468' article_processing_charge: No author: - first_name: Mabel full_name: Iglesias Ham, Mabel id: 41B58C0C-F248-11E8-B48F-1D18A9856A87 last_name: Iglesias Ham - first_name: Michael full_name: Kerber, Michael last_name: Kerber orcid: 0000-0002-8030-9299 - first_name: Caroline full_name: Uhler, Caroline id: 49ADD78E-F248-11E8-B48F-1D18A9856A87 last_name: Uhler orcid: 0000-0002-7008-0216 citation: ama: Iglesias Ham M, Kerber M, Uhler C. Sphere packing with limited overlap. arXiv. doi:10.48550/arXiv.1401.0468 apa: Iglesias Ham, M., Kerber, M., & Uhler, C. (n.d.). Sphere packing with limited overlap. arXiv. https://doi.org/10.48550/arXiv.1401.0468 chicago: Iglesias Ham, Mabel, Michael Kerber, and Caroline Uhler. “Sphere Packing with Limited Overlap.” ArXiv, n.d. https://doi.org/10.48550/arXiv.1401.0468. ieee: M. Iglesias Ham, M. Kerber, and C. Uhler, “Sphere packing with limited overlap,” arXiv. . ista: Iglesias Ham M, Kerber M, Uhler C. Sphere packing with limited overlap. arXiv, 1401.0468. mla: Iglesias Ham, Mabel, et al. “Sphere Packing with Limited Overlap.” ArXiv, 1401.0468, doi:10.48550/arXiv.1401.0468. short: M. Iglesias Ham, M. Kerber, C. Uhler, ArXiv (n.d.). date_created: 2018-12-11T11:55:12Z date_published: 2014-01-01T00:00:00Z date_updated: 2023-10-18T08:06:45Z day: '01' department: - _id: HeEd - _id: CaUh doi: 10.48550/arXiv.1401.0468 external_id: arxiv: - '1401.0468' language: - iso: eng main_file_link: - open_access: '1' url: http://cccg.ca/proceedings/2014/papers/paper23.pdf month: '01' oa: 1 oa_version: Submitted Version publication: arXiv publication_status: submitted publist_id: '5064' status: public title: Sphere packing with limited overlap type: preprint user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2014' ...