2026-07-24T06:45:45Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:16061 2024-03-06T10:56:52Z ddc:004 open_access

A Universal Triangulation for Flat Tori Lazarus, Francis Tallerie, Florent Triangulation flat torus isometric embedding 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Francis Lazarus and Florent Tallerie 2022 Is Part Of LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2022.53 urn:nbn:de:0030-drops-160617 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.53 eng https://creativecommons.org/licenses/by/4.0/legalcode