Finding closed quasigeodesics on convex polyhedra
Name
LIPIcs-SoCG-2020-33.pdf
Description
Published version
Size
683.56 KB
Format
Adobe PDF
Checksum (MD5)
db0d46fbd9986e5fcff9031fdbc92bc1
Author(s)
Demaine, Erik D
Hesterberg, Adam Classen
Ku, Jason S
Date Issued
June 2020
Journal
Leibniz International Proceedings in Informatics, LIPIcs
Publisher
Schloss Dagstuhl, Leibniz Center for Informatics
Citation
Demaine, Erik D. et al. “Finding closed quasigeodesics on convex polyhedra.” 36th International Symposium on Computational Geometry, June 2020, virtual, Schloss Dagstuhl and Leibniz Center for Informatics, June 2020. © 2020 The Author(s)
Version
Final published version
Abstract
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 ( nε22 L` b ) time, where ε is the minimum curvature of a vertex, L is the length of the longest edge, ` 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).
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Terms of Use
Creative Commons Attribution 3.0 unported license
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.4230/LIPIcs.SoCG.2020.33
