---
_id: '10773'
abstract:
- lang: eng
text: The Voronoi tessellation in Rd is defined by locally minimizing the power
distance to given weighted points. Symmetrically, the Delaunay mosaic can be defined
by locally maximizing the negative power distance to other such points. We prove
that the average of the two piecewise quadratic functions is piecewise linear,
and that all three functions have the same critical points and values. Discretizing
the two piecewise quadratic functions, we get the alpha shapes as sublevel sets
of the discrete function on the Delaunay mosaic, and analogous shapes as superlevel
sets of the discrete function on the Voronoi tessellation. For the same non-critical
value, the corresponding shapes are disjoint, separated by a narrow channel that
contains no critical points but the entire level set of the piecewise linear function.
acknowledgement: Open access funding provided by the Institute of Science and Technology
(IST Austria).
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Ranita
full_name: Biswas, Ranita
id: 3C2B033E-F248-11E8-B48F-1D18A9856A87
last_name: Biswas
orcid: 0000-0002-5372-7890
- first_name: Sebastiano
full_name: Cultrera Di Montesano, Sebastiano
id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
last_name: Cultrera Di Montesano
orcid: 0000-0001-6249-0832
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Morteza
full_name: Saghafian, Morteza
last_name: Saghafian
citation:
ama: Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Continuous
and discrete radius functions on Voronoi tessellations and Delaunay mosaics. Discrete
and Computational Geometry. 2022;67:811-842. doi:10.1007/s00454-022-00371-2
apa: Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian, M.
(2022). Continuous and discrete radius functions on Voronoi tessellations and
Delaunay mosaics. Discrete and Computational Geometry. Springer Nature.
https://doi.org/10.1007/s00454-022-00371-2
chicago: Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner,
and Morteza Saghafian. “Continuous and Discrete Radius Functions on Voronoi Tessellations
and Delaunay Mosaics.” Discrete and Computational Geometry. Springer Nature,
2022. https://doi.org/10.1007/s00454-022-00371-2.
ieee: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Continuous
and discrete radius functions on Voronoi tessellations and Delaunay mosaics,”
Discrete and Computational Geometry, vol. 67. Springer Nature, pp. 811–842,
2022.
ista: Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2022. Continuous
and discrete radius functions on Voronoi tessellations and Delaunay mosaics. Discrete
and Computational Geometry. 67, 811–842.
mla: Biswas, Ranita, et al. “Continuous and Discrete Radius Functions on Voronoi
Tessellations and Delaunay Mosaics.” Discrete and Computational Geometry,
vol. 67, Springer Nature, 2022, pp. 811–42, doi:10.1007/s00454-022-00371-2.
short: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Discrete
and Computational Geometry 67 (2022) 811–842.
date_created: 2022-02-20T23:01:34Z
date_published: 2022-04-01T00:00:00Z
date_updated: 2023-08-02T14:31:25Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s00454-022-00371-2
external_id:
isi:
- '000752175300002'
file:
- access_level: open_access
checksum: 9383d3b70561bacee905e335dc922680
content_type: application/pdf
creator: dernst
date_created: 2022-08-02T06:07:55Z
date_updated: 2022-08-02T06:07:55Z
file_id: '11718'
file_name: 2022_DiscreteCompGeometry_Biswas.pdf
file_size: 2518111
relation: main_file
success: 1
file_date_updated: 2022-08-02T06:07:55Z
has_accepted_license: '1'
intvolume: ' 67'
isi: 1
language:
- iso: eng
month: '04'
oa: 1
oa_version: Published Version
page: 811-842
publication: Discrete and Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Continuous and discrete radius functions on Voronoi tessellations and Delaunay
mosaics
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 67
year: '2022'
...