Delaunay triangulations on orientable surfaces of low genus
Identifieur interne : 000010 ( Main/Exploration ); précédent : 000009; suivant : 000011Delaunay triangulations on orientable surfaces of low genus
Auteurs : Mikhail Bogdanov [France] ; Monique Teillaud [France] ; Gert Vegter [Pays-Bas]Source :
English descriptors
Abstract
Earlier work on Delaunay triangulation of point sets on the 2D flat torus, which is locally isometric to the Euclidean plane, was based on lifting the point set to a locally isometric 9-sheeted covering space of the torus. Under mild conditions the Delaunay triangulation of the lifted point set, consisting of 9 copies of the input set, projects to the Delaunay triangulation of the input set. We improve and generalize this work. First we present a new construction based on an 8-sheeted covering space, which shows that eight copies suffice for the standard flat torus. Then we generalize this construction to the context of compact orientable surfaces of higher genus, which are locally isometric to the hyperbolic plane. We investigate more thoroughly the Bolza surface, homeomorphic to a sphere with two handles, both because it is the hyperbolic surface with lowest genus, and because triangulations on the Bolza surface have applications in various fields such as neuromathematics and cosmological models. While the general properties (existence results of appropriate covering spaces) show similarities with the results for the flat case, explicit constructions and their proofs are much more complex, even in the case of the apparently simple Bolza surface. One of the main reasons is the fact that two hyperbolic translations do not commute in general. To the best of our knowledge, the results in this paper are the first ones of this kind. The interest of our contribution lies not only in the results, but most of all in the construction of covering spaces itself and the study of their properties.
Url:
DOI: 10.4230/LIPIcs.SoCG.2016.20
Affiliations:
Links toward previous steps (curation, corpus...)
- to stream Hal, to step Corpus: 001B42
- to stream Hal, to step Curation: 001B42
- to stream Hal, to step Checkpoint: 000010
- to stream Main, to step Merge: 000010
- to stream Main, to step Curation: 000010
Le document en format XML
<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Delaunay triangulations on orientable surfaces of low genus</title><author><name sortKey="Bogdanov, Mikhail" sort="Bogdanov, Mikhail" uniqKey="Bogdanov M" first="Mikhail" last="Bogdanov">Mikhail Bogdanov</name><affiliation wicri:level="1"><hal:affiliation type="researchteam" xml:id="struct-56048" status="VALID"><idno type="RNSR">200318408A</idno><orgName>Geometric computing</orgName><orgName type="acronym">GEOMETRICA</orgName><desc><address><addrLine>2004 route des Lucioles BP 93 F-06902 Sophia Antipolis (France) / Bât Alan Turing, Campus de l'Ecole Polytechnique, 1 rue Honoré d'Estienne d'orves, 91120 Palaiseau (France)</addrLine><country key="FR"></country></address><ref type="url">http://www-sop.inria.fr/geometrica/</ref></desc><listRelation><relation active="#struct-34586" type="direct"></relation><relation active="#struct-300009" type="indirect"></relation><relation active="#struct-118511" type="direct"></relation></listRelation><tutelles><tutelle active="#struct-34586" type="direct"><org type="laboratory" xml:id="struct-34586" status="VALID"><idno type="RNSR">198318250R</idno><orgName>Inria Sophia Antipolis - Méditerranée </orgName><orgName type="acronym">CRISAM</orgName><desc><address><addrLine>2004 route des Lucioles BP 93 06902 Sophia Antipolis</addrLine><country key="FR"></country></address><ref type="url">http://www.inria.fr/centre/sophia/</ref></desc><listRelation><relation active="#struct-300009" type="direct"></relation></listRelation></org></tutelle><tutelle active="#struct-300009" type="indirect"><org type="institution" xml:id="struct-300009" status="VALID"><orgName>Institut National de Recherche en Informatique et en Automatique</orgName><orgName type="acronym">Inria</orgName><desc><address><addrLine>Domaine de VoluceauRocquencourt - BP 10578153 Le Chesnay Cedex</addrLine><country key="FR"></country></address><ref type="url">http://www.inria.fr/en/</ref></desc></org></tutelle><tutelle active="#struct-118511" type="direct"><org type="laboratory" xml:id="struct-118511" status="VALID"><idno type="RNSR">200818248E</idno><orgName>INRIA Saclay - Ile de France</orgName><desc><address><addrLine>1 rue Honoré d'Estienne d'OrvesBâtiment Alan TuringCampus de l'École Polytechnique91120 Palaiseau</addrLine><country key="FR"></country></address><ref type="url">http://www.inria.fr/centre/saclay</ref></desc><listRelation><relation active="#struct-300009" type="direct"></relation></listRelation></org></tutelle></tutelles></hal:affiliation><country>France</country></affiliation></author><author><name sortKey="Teillaud, Monique" sort="Teillaud, Monique" uniqKey="Teillaud M" first="Monique" last="Teillaud">Monique Teillaud</name><affiliation wicri:level="1"><hal:affiliation type="researchteam" xml:id="struct-205129" status="VALID"><idno type="RNSR">200518305E</idno><orgName>Effective Geometric Algorithms for Surfaces and Visibility</orgName><orgName type="acronym">VEGAS</orgName><desc><address><country key="FR"></country></address><ref type="url">http://www.inria.fr/equipes/vegas</ref></desc><listRelation><relation active="#struct-129671" type="direct"></relation><relation active="#struct-300009" type="indirect"></relation><relation active="#struct-423083" type="direct"></relation><relation active="#struct-206040" type="indirect"></relation><relation active="#struct-413289" type="indirect"></relation><relation name="UMR7503" active="#struct-441569" type="indirect"></relation></listRelation><tutelles><tutelle active="#struct-129671" type="direct"><org type="laboratory" xml:id="struct-129671" status="VALID"><idno type="RNSR">198618246Y</idno><orgName>INRIA Nancy - Grand Est</orgName><desc><address><addrLine>615 rue du Jardin Botanique 54600 Villers-lès-Nancy</addrLine><country key="FR"></country></address><ref type="url">http://www.inria.fr/nancy</ref></desc><listRelation><relation active="#struct-300009" type="direct"></relation></listRelation></org></tutelle><tutelle active="#struct-300009" type="indirect"><org type="institution" xml:id="struct-300009" status="VALID"><orgName>Institut National de Recherche en Informatique et en Automatique</orgName><orgName type="acronym">Inria</orgName><desc><address><addrLine>Domaine de VoluceauRocquencourt - BP 10578153 Le Chesnay Cedex</addrLine><country key="FR"></country></address><ref type="url">http://www.inria.fr/en/</ref></desc></org></tutelle><tutelle active="#struct-423083" type="direct"><org type="department" xml:id="struct-423083" status="VALID"><orgName>Department of Algorithms, Computation, Image and Geometry</orgName><orgName type="acronym">LORIA - ALGO</orgName><desc><address><country key="FR"></country></address><ref type="url">http://www.loria.fr/la-recherche-en/departements/algorithmics</ref></desc><listRelation><relation active="#struct-206040" type="direct"></relation><relation active="#struct-300009" type="indirect"></relation><relation active="#struct-413289" type="indirect"></relation><relation name="UMR7503" active="#struct-441569" type="indirect"></relation></listRelation></org></tutelle><tutelle active="#struct-206040" type="indirect"><org type="laboratory" xml:id="struct-206040" status="VALID"><idno type="IdRef">067077927</idno><idno type="RNSR">198912571S</idno><idno type="IdUnivLorraine">[UL]RSI--</idno><orgName>Laboratoire Lorrain de Recherche en Informatique et ses Applications</orgName><orgName type="acronym">LORIA</orgName><date type="start">2012-01-01</date><desc><address><addrLine>Campus Scientifique BP 239 54506 Vandoeuvre-lès-Nancy Cedex</addrLine><country key="FR"></country></address><ref type="url">http://www.loria.fr</ref></desc><listRelation><relation active="#struct-300009" type="direct"></relation><relation active="#struct-413289" type="direct"></relation><relation name="UMR7503" active="#struct-441569" type="direct"></relation></listRelation></org></tutelle><tutelle active="#struct-413289" type="indirect"><org type="institution" xml:id="struct-413289" status="VALID"><idno type="IdRef">157040569</idno><idno type="IdUnivLorraine">[UL]100--</idno><orgName>Université de Lorraine</orgName><orgName type="acronym">UL</orgName><date type="start">2012-01-01</date><desc><address><addrLine>34 cours Léopold - CS 25233 - 54052 Nancy cedex</addrLine><country key="FR"></country></address><ref type="url">http://www.univ-lorraine.fr/</ref></desc></org></tutelle><tutelle name="UMR7503" active="#struct-441569" type="indirect"><org type="institution" xml:id="struct-441569" status="VALID"><idno type="ISNI">0000000122597504</idno><idno type="IdRef">02636817X</idno><orgName>Centre National de la Recherche Scientifique</orgName><orgName type="acronym">CNRS</orgName><date type="start">1939-10-19</date><desc><address><country key="FR"></country></address><ref type="url">http://www.cnrs.fr/</ref></desc></org></tutelle></tutelles></hal:affiliation><country>France</country><placeName><settlement type="city">Nancy</settlement><settlement type="city">Metz</settlement><region type="region" nuts="2">Grand Est</region><region type="old region" nuts="2">Lorraine (région)</region></placeName><orgName type="university">Université de Lorraine</orgName></affiliation></author><author><name sortKey="Vegter, Gert" sort="Vegter, Gert" uniqKey="Vegter G" first="Gert" last="Vegter">Gert Vegter</name><affiliation wicri:level="1"><hal:affiliation type="laboratory" xml:id="struct-48400" status="VALID"><orgName>Johann Bernoulli Institute for Mathematics and Computer Science</orgName><desc><address><addrLine>Nijenborgh 9 9747 AG Groningen The Netherlands</addrLine><country key="NL"></country></address><ref type="url">http://www.rug.nl/wiskunde/onderzoek/overonderzoek/index</ref></desc><listRelation><relation active="#struct-131861" type="direct"></relation></listRelation><tutelles><tutelle active="#struct-131861" type="direct"><org type="institution" xml:id="struct-131861" status="VALID"><orgName>University of Groningen [Groningen]</orgName><desc><address><addrLine>PO Box 72 9700 AB Groningen</addrLine><country key="NL"></country></address><ref type="url">http://www.rug.nl/?lang=en</ref></desc></org></tutelle></tutelles></hal:affiliation><country>Pays-Bas</country></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">HAL</idno><idno type="RBID">Hal:hal-01276386</idno><idno type="halId">hal-01276386</idno><idno type="halUri">https://hal.inria.fr/hal-01276386</idno><idno type="url">https://hal.inria.fr/hal-01276386</idno><idno type="doi">10.4230/LIPIcs.SoCG.2016.20</idno><date when="2016-06-14">2016-06-14</date><idno type="wicri:Area/Hal/Corpus">001B42</idno><idno type="wicri:Area/Hal/Curation">001B42</idno><idno type="wicri:Area/Hal/Checkpoint">000010</idno><idno type="wicri:explorRef" wicri:stream="Hal" wicri:step="Checkpoint">000010</idno><idno type="wicri:Area/Main/Merge">000010</idno><idno type="wicri:Area/Main/Curation">000010</idno><idno type="wicri:Area/Main/Exploration">000010</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en">Delaunay triangulations on orientable surfaces of low genus</title><author><name sortKey="Bogdanov, Mikhail" sort="Bogdanov, Mikhail" uniqKey="Bogdanov M" first="Mikhail" last="Bogdanov">Mikhail Bogdanov</name><affiliation wicri:level="1"><hal:affiliation type="researchteam" xml:id="struct-56048" status="VALID"><idno type="RNSR">200318408A</idno><orgName>Geometric computing</orgName><orgName type="acronym">GEOMETRICA</orgName><desc><address><addrLine>2004 route des Lucioles BP 93 F-06902 Sophia Antipolis (France) / Bât Alan Turing, Campus de l'Ecole Polytechnique, 1 rue Honoré d'Estienne d'orves, 91120 Palaiseau (France)</addrLine><country key="FR"></country></address><ref type="url">http://www-sop.inria.fr/geometrica/</ref></desc><listRelation><relation active="#struct-34586" type="direct"></relation><relation active="#struct-300009" type="indirect"></relation><relation active="#struct-118511" type="direct"></relation></listRelation><tutelles><tutelle active="#struct-34586" type="direct"><org type="laboratory" xml:id="struct-34586" status="VALID"><idno type="RNSR">198318250R</idno><orgName>Inria Sophia Antipolis - Méditerranée </orgName><orgName type="acronym">CRISAM</orgName><desc><address><addrLine>2004 route des Lucioles BP 93 06902 Sophia Antipolis</addrLine><country key="FR"></country></address><ref type="url">http://www.inria.fr/centre/sophia/</ref></desc><listRelation><relation active="#struct-300009" type="direct"></relation></listRelation></org></tutelle><tutelle active="#struct-300009" type="indirect"><org type="institution" xml:id="struct-300009" status="VALID"><orgName>Institut National de Recherche en Informatique et en Automatique</orgName><orgName type="acronym">Inria</orgName><desc><address><addrLine>Domaine de VoluceauRocquencourt - BP 10578153 Le Chesnay Cedex</addrLine><country key="FR"></country></address><ref type="url">http://www.inria.fr/en/</ref></desc></org></tutelle><tutelle active="#struct-118511" type="direct"><org type="laboratory" xml:id="struct-118511" status="VALID"><idno type="RNSR">200818248E</idno><orgName>INRIA Saclay - Ile de France</orgName><desc><address><addrLine>1 rue Honoré d'Estienne d'OrvesBâtiment Alan TuringCampus de l'École Polytechnique91120 Palaiseau</addrLine><country key="FR"></country></address><ref type="url">http://www.inria.fr/centre/saclay</ref></desc><listRelation><relation active="#struct-300009" type="direct"></relation></listRelation></org></tutelle></tutelles></hal:affiliation><country>France</country></affiliation></author><author><name sortKey="Teillaud, Monique" sort="Teillaud, Monique" uniqKey="Teillaud M" first="Monique" last="Teillaud">Monique Teillaud</name><affiliation wicri:level="1"><hal:affiliation type="researchteam" xml:id="struct-205129" status="VALID"><idno type="RNSR">200518305E</idno><orgName>Effective Geometric Algorithms for Surfaces and Visibility</orgName><orgName type="acronym">VEGAS</orgName><desc><address><country key="FR"></country></address><ref type="url">http://www.inria.fr/equipes/vegas</ref></desc><listRelation><relation active="#struct-129671" type="direct"></relation><relation active="#struct-300009" type="indirect"></relation><relation active="#struct-423083" type="direct"></relation><relation active="#struct-206040" type="indirect"></relation><relation active="#struct-413289" type="indirect"></relation><relation name="UMR7503" active="#struct-441569" type="indirect"></relation></listRelation><tutelles><tutelle active="#struct-129671" type="direct"><org type="laboratory" xml:id="struct-129671" status="VALID"><idno type="RNSR">198618246Y</idno><orgName>INRIA Nancy - Grand Est</orgName><desc><address><addrLine>615 rue du Jardin Botanique 54600 Villers-lès-Nancy</addrLine><country key="FR"></country></address><ref type="url">http://www.inria.fr/nancy</ref></desc><listRelation><relation active="#struct-300009" type="direct"></relation></listRelation></org></tutelle><tutelle active="#struct-300009" type="indirect"><org type="institution" xml:id="struct-300009" status="VALID"><orgName>Institut National de Recherche en Informatique et en Automatique</orgName><orgName type="acronym">Inria</orgName><desc><address><addrLine>Domaine de VoluceauRocquencourt - BP 10578153 Le Chesnay Cedex</addrLine><country key="FR"></country></address><ref type="url">http://www.inria.fr/en/</ref></desc></org></tutelle><tutelle active="#struct-423083" type="direct"><org type="department" xml:id="struct-423083" status="VALID"><orgName>Department of Algorithms, Computation, Image and Geometry</orgName><orgName type="acronym">LORIA - ALGO</orgName><desc><address><country key="FR"></country></address><ref type="url">http://www.loria.fr/la-recherche-en/departements/algorithmics</ref></desc><listRelation><relation active="#struct-206040" type="direct"></relation><relation active="#struct-300009" type="indirect"></relation><relation active="#struct-413289" type="indirect"></relation><relation name="UMR7503" active="#struct-441569" type="indirect"></relation></listRelation></org></tutelle><tutelle active="#struct-206040" type="indirect"><org type="laboratory" xml:id="struct-206040" status="VALID"><idno type="IdRef">067077927</idno><idno type="RNSR">198912571S</idno><idno type="IdUnivLorraine">[UL]RSI--</idno><orgName>Laboratoire Lorrain de Recherche en Informatique et ses Applications</orgName><orgName type="acronym">LORIA</orgName><date type="start">2012-01-01</date><desc><address><addrLine>Campus Scientifique BP 239 54506 Vandoeuvre-lès-Nancy Cedex</addrLine><country key="FR"></country></address><ref type="url">http://www.loria.fr</ref></desc><listRelation><relation active="#struct-300009" type="direct"></relation><relation active="#struct-413289" type="direct"></relation><relation name="UMR7503" active="#struct-441569" type="direct"></relation></listRelation></org></tutelle><tutelle active="#struct-413289" type="indirect"><org type="institution" xml:id="struct-413289" status="VALID"><idno type="IdRef">157040569</idno><idno type="IdUnivLorraine">[UL]100--</idno><orgName>Université de Lorraine</orgName><orgName type="acronym">UL</orgName><date type="start">2012-01-01</date><desc><address><addrLine>34 cours Léopold - CS 25233 - 54052 Nancy cedex</addrLine><country key="FR"></country></address><ref type="url">http://www.univ-lorraine.fr/</ref></desc></org></tutelle><tutelle name="UMR7503" active="#struct-441569" type="indirect"><org type="institution" xml:id="struct-441569" status="VALID"><idno type="ISNI">0000000122597504</idno><idno type="IdRef">02636817X</idno><orgName>Centre National de la Recherche Scientifique</orgName><orgName type="acronym">CNRS</orgName><date type="start">1939-10-19</date><desc><address><country key="FR"></country></address><ref type="url">http://www.cnrs.fr/</ref></desc></org></tutelle></tutelles></hal:affiliation><country>France</country><placeName><settlement type="city">Nancy</settlement><settlement type="city">Metz</settlement><region type="region" nuts="2">Grand Est</region><region type="old region" nuts="2">Lorraine (région)</region></placeName><orgName type="university">Université de Lorraine</orgName></affiliation></author><author><name sortKey="Vegter, Gert" sort="Vegter, Gert" uniqKey="Vegter G" first="Gert" last="Vegter">Gert Vegter</name><affiliation wicri:level="1"><hal:affiliation type="laboratory" xml:id="struct-48400" status="VALID"><orgName>Johann Bernoulli Institute for Mathematics and Computer Science</orgName><desc><address><addrLine>Nijenborgh 9 9747 AG Groningen The Netherlands</addrLine><country key="NL"></country></address><ref type="url">http://www.rug.nl/wiskunde/onderzoek/overonderzoek/index</ref></desc><listRelation><relation active="#struct-131861" type="direct"></relation></listRelation><tutelles><tutelle active="#struct-131861" type="direct"><org type="institution" xml:id="struct-131861" status="VALID"><orgName>University of Groningen [Groningen]</orgName><desc><address><addrLine>PO Box 72 9700 AB Groningen</addrLine><country key="NL"></country></address><ref type="url">http://www.rug.nl/?lang=en</ref></desc></org></tutelle></tutelles></hal:affiliation><country>Pays-Bas</country></affiliation></author></analytic><idno type="DOI">10.4230/LIPIcs.SoCG.2016.20</idno></biblStruct></sourceDesc></fileDesc><profileDesc><textClass><keywords scheme="mix" xml:lang="en"><term> Fuchsian groups</term><term> finitely presented groups</term><term> hyperbolic surfaces</term><term> systole</term><term>covering spaces</term></keywords></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"> Earlier work on Delaunay triangulation of point sets on the 2D flat torus, which is locally isometric to the Euclidean plane, was based on lifting the point set to a locally isometric 9-sheeted covering space of the torus. Under mild conditions the Delaunay triangulation of the lifted point set, consisting of 9 copies of the input set, projects to the Delaunay triangulation of the input set. We improve and generalize this work. First we present a new construction based on an 8-sheeted covering space, which shows that eight copies suffice for the standard flat torus. Then we generalize this construction to the context of compact orientable surfaces of higher genus, which are locally isometric to the hyperbolic plane. We investigate more thoroughly the Bolza surface, homeomorphic to a sphere with two handles, both because it is the hyperbolic surface with lowest genus, and because triangulations on the Bolza surface have applications in various fields such as neuromathematics and cosmological models. While the general properties (existence results of appropriate covering spaces) show similarities with the results for the flat case, explicit constructions and their proofs are much more complex, even in the case of the apparently simple Bolza surface. One of the main reasons is the fact that two hyperbolic translations do not commute in general. To the best of our knowledge, the results in this paper are the first ones of this kind. The interest of our contribution lies not only in the results, but most of all in the construction of covering spaces itself and the study of their properties.</div></front></TEI><affiliations><list><country><li>France</li><li>Pays-Bas</li></country><region><li>Grand Est</li><li>Lorraine (région)</li></region><settlement><li>Metz</li><li>Nancy</li></settlement><orgName><li>Université de Lorraine</li></orgName></list><tree><country name="France"><noRegion><name sortKey="Bogdanov, Mikhail" sort="Bogdanov, Mikhail" uniqKey="Bogdanov M" first="Mikhail" last="Bogdanov">Mikhail Bogdanov</name></noRegion><name sortKey="Teillaud, Monique" sort="Teillaud, Monique" uniqKey="Teillaud M" first="Monique" last="Teillaud">Monique Teillaud</name></country><country name="Pays-Bas"><noRegion><name sortKey="Vegter, Gert" sort="Vegter, Gert" uniqKey="Vegter G" first="Gert" last="Vegter">Gert Vegter</name></noRegion></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
EXPLOR_STEP=$WICRI_ROOT/Wicri/Lorraine/explor/InforLorV4/Data/Main/Exploration
HfdSelect -h $EXPLOR_STEP/biblio.hfd -nk 000010 | SxmlIndent | more
Ou
HfdSelect -h $EXPLOR_AREA/Data/Main/Exploration/biblio.hfd -nk 000010 | SxmlIndent | more
Pour mettre un lien sur cette page dans le réseau Wicri
{{Explor lien
|wiki= Wicri/Lorraine
|area= InforLorV4
|flux= Main
|étape= Exploration
|type= RBID
|clé= Hal:hal-01276386
|texte= Delaunay triangulations on orientable surfaces of low genus
}}
| This area was generated with Dilib version V0.6.33. |  |