---
_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
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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'
...