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Finding Weakly Simple Closed Quasigeodesics on Polyhedral Spheres

Authors Jean Chartier, Arnaud de Mesmay



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Author Details

Jean Chartier
  • Univ. Paris Est Creteil, CNRS, LAMA, F-94010 Creteil, France
Arnaud de Mesmay
  • LIGM, CNRS, Univ. Gustave Eiffel, ESIEE Paris, F-77454 Marne-la-Vallée, France

Acknowledgements

We thank Francis Lazarus for insightful discussions, and Joseph O'Rourke and the anonymous reviewers for helpful comments.

Cite AsGet BibTex

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)
https://doi.org/10.4230/LIPIcs.SoCG.2022.27

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Quasigeodesic
  • polyhedron
  • curve-shortening process
  • disk flow
  • weakly simple

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