Paper by Erik D. Demaine

Erik D. Demaine, Adam C. Hesterberg, and Jason S. Ku, “Finding Closed Quasigeodesics on Convex Polyhedra”, in Proceedings of the 36th International Symposium on Computational Geometry (SoCG 2020), June 23–26, 2020, 33:1–33:13.

A closed quasigeodesic is a closed loop on the surface of a polyhedron with at most 180° of surface on both sides at all points; such loops can be locally unfolded straight. In 1949, Pogorelov proved that every convex polyhedron has at least three (non-self-intersecting) closed quasigeodesics, but the proof relies on a nonconstructive topological argument. We present the first finite algorithm to find a closed quasigeodesic on a given convex polyhedron, which is the first positive progress on a 1990 open problem by O'Rourke and Wyman. The algorithm's running time is pseudopolynomial, namely O((n2 / ε2) (L / ℓ) b) time, where ε is the minimum curvature of a vertex, L is the length of the longest edge, ℓ is the smallest distance within a face between a vertex and a nonincident edge (minimum feature size of any face), and b is the maximum number of bits of an integer in a constant-size radical expression of a real number representing the polyhedron. We take special care in the model of computation and needed precision, showing that we can achieve the stated running time on a pointer machine supporting constant-time w-bit arithmetic operations where w = Ω(lg b).

This paper is also available from LIPIcs. Our presentation is available on YouTube.

The paper is 13 pages.

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Last updated October 19, 2020 by Erik Demaine.