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.
- The paper is 13 pages.
- The paper is available in PDF (657k).
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