Finding Weakly Simple Closed Quasigeodesics on Polyhedral Spheres

TL;DR

This paper extends the concept of quasigeodesics to polyhedral spheres, proves their existence in this broader setting, and provides a finite algorithm to compute such curves, solving a longstanding open problem.

Contribution

It introduces an extended definition of quasigeodesics on polyhedral spheres, proves their existence, and develops a finite algorithm for computing them.

Findings

Existence of weakly simple closed quasigeodesics on polyhedral spheres.

A finite algorithm to compute such quasigeodesics.

Solution to an open problem for convex polyhedra.

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 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.