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Finding Weakly Simple Closed Quasigeodesics on Polyhedral Spheres

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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 \(\pi \) 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 adaptation 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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Notes

  1. In this proof, we call altitude the distance from E to [AB]. If the triangle AEB has an obtuse angle at its base, then this notion of altitude does not coincide with the usual notion, i.e., the distance from E to (AB).

  2. If there is an infinite number of them, they are parameterized in \(\beta \) by a closed interval. We then consider the representative closest to the interior of the star.

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Acknowledgements

We thank Francis Lazarus for insightful discussions, and Joseph O’Rourke and the anonymous reviewers for very helpful comments. This research was partially supported by the ANR project Min-Max (ANR-19-CE40-0014)), the ANR project SoS (ANR-17-CE40-0033), and the Bézout Labex, funded by ANR, reference ANR-10-LABX-58.

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  1. Univ. Paris-Est Créteil, CNRS, LAMA, 94010, Créteil, France

    Jean Chartier

  2. LIGM, CNRS, Univ. Gustave Eiffel, ESIEE Paris, 77454, Marne-la-Vallée, France

    Arnaud de Mesmay

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  1. Jean Chartier
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Correspondence to Jean Chartier or Arnaud de Mesmay.

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Chartier, J., de Mesmay, A. Finding Weakly Simple Closed Quasigeodesics on Polyhedral Spheres. Discrete Comput Geom 71, 95–120 (2024). https://doi.org/10.1007/s00454-023-00511-2

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