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DOI: 10.4230/LIPIcs.SoCG.2022.53

A Universal Triangulation for Flat Tori

Authors: Francis Lazarus and Florent Tallerie

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

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 𝔼³. 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.

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Francis Lazarus and Florent Tallerie. A Universal Triangulation for Flat Tori. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 53:1-53:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@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/entities/document/10.4230/LIPIcs.SoCG.2022.53},
  URN =		{urn:nbn:de:0030-drops-160617},
  doi =		{10.4230/LIPIcs.SoCG.2022.53},
  annote =	{Keywords: Triangulation, flat torus, isometric embedding}
}

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A Universal Triangulation for Flat Tori

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