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.
@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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