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DOI: 10.4230/LIPIcs.SoCG.2022.27

Finding Weakly Simple Closed Quasigeodesics on Polyhedral Spheres

Authors: Jean Chartier and Arnaud de Mesmay

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

Abstract

A closed quasigeodesic on a convex polyhedron is a closed curve that is locally straight outside of the vertices, where it forms an angle at most π on both sides. While the existence of a simple closed quasigeodesic on a convex polyhedron has been proved by Pogorelov in 1949, finding a polynomial-time algorithm to compute such a simple closed quasigeodesic has been repeatedly posed as an open problem. Our first contribution is to propose an extended definition of quasigeodesics in the intrinsic setting of (not necessarily convex) polyhedral spheres, and to prove the existence of a weakly simple closed quasigeodesic in such a setting. Our proof does not proceed via an approximation by smooth surfaces, but relies on an adapation of the disk flow of Hass and Scott to the context of polyhedral surfaces. Our second result is to leverage this existence theorem to provide a finite algorithm to compute a weakly simple closed quasigeodesic on a polyhedral sphere. On a convex polyhedron, our algorithm computes a simple closed quasigeodesic, solving an open problem of Demaine, Hersterberg and Ku.

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Jean Chartier and Arnaud de Mesmay. Finding Weakly Simple Closed Quasigeodesics on Polyhedral Spheres. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chartier_et_al:LIPIcs.SoCG.2022.27,
  author =	{Chartier, Jean and de Mesmay, Arnaud},
  title =	{{Finding Weakly Simple Closed Quasigeodesics on Polyhedral Spheres}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{27:1--27:16},
  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.2022.27},
  URN =		{urn:nbn:de:0030-drops-160350},
  doi =		{10.4230/LIPIcs.SoCG.2022.27},
  annote =	{Keywords: Quasigeodesic, polyhedron, curve-shortening process, disk flow, weakly simple}
}

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Finding Weakly Simple Closed Quasigeodesics on Polyhedral Spheres

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