License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2022.53
URN: urn:nbn:de:0030-drops-160617
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16061/
Lazarus, Francis ;
Tallerie, Florent
A Universal Triangulation for Flat Tori
Abstract
A result due to Burago and Zalgaller states that every orientable polyhedral surface, one that is obtained by gluing Euclidean polygons, has an isometric piecewise linear (PL) embedding into Euclidean space ?³. A flat torus, resulting from the identification of the opposite sides of a Euclidean parallelogram, is a simple example of polyhedral surface. In a first part, we adapt the proof of Burago and Zalgaller, which is partially constructive, to produce PL isometric embeddings of flat tori. In practice, the resulting embeddings have a huge number of vertices, moreover distinct for every flat torus. In a second part, based on another construction of Zalgaller and on recent works by Arnoux et al., we exhibit a universal triangulation with 5974 triangles which can be embedded linearly on each triangle in order to realize the metric of any flat torus.
BibTeX - Entry
@InProceedings{lazarus_et_al:LIPIcs.SoCG.2022.53,
author = {Lazarus, Francis and Tallerie, Florent},
title = {{A Universal Triangulation for Flat Tori}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {53:1--53:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16061},
URN = {urn:nbn:de:0030-drops-160617},
doi = {10.4230/LIPIcs.SoCG.2022.53},
annote = {Keywords: Triangulation, flat torus, isometric embedding}
}
Keywords: |
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Triangulation, flat torus, isometric embedding |
Collection: |
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38th International Symposium on Computational Geometry (SoCG 2022) |
Issue Date: |
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2022 |
Date of publication: |
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01.06.2022 |