License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2018.64
URN: urn:nbn:de:0030-drops-87777
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8777/
O'Rourke, Joseph
Edge-Unfolding Nearly Flat Convex Caps
Abstract
The main result of this paper is a proof that a nearly flat, acutely triangulated convex cap C in R^3 has an edge-unfolding to a non-overlapping polygon in the plane. A convex cap is the intersection of the surface of a convex polyhedron and a halfspace. "Nearly flat" means that every outer face normal forms a sufficiently small angle f < F with the z^-axis orthogonal to the halfspace bounding plane. The size of F depends on the acuteness gap a: if every triangle angle is at most pi/2 {-} a, then F ~~ 0.36 sqrt{a} suffices; e.g., for a=3°, F ~~ 5°. The proof employs the recent concepts of angle-monotone and radially monotone curves. The proof is constructive, leading to a polynomial-time algorithm for finding the edge-cuts, at worst O(n^2); a version has been implemented.
BibTeX - Entry
@InProceedings{orourke:LIPIcs:2018:8777,
author = {Joseph O'Rourke},
title = {{Edge-Unfolding Nearly Flat Convex Caps}},
booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)},
pages = {64:1--64:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-066-8},
ISSN = {1868-8969},
year = {2018},
volume = {99},
editor = {Bettina Speckmann and Csaba D. T{\'o}th},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/8777},
URN = {urn:nbn:de:0030-drops-87777},
doi = {10.4230/LIPIcs.SoCG.2018.64},
annote = {Keywords: polyhedra, unfolding}
}
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Keywords: |
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polyhedra, unfolding |
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Collection: |
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34th International Symposium on Computational Geometry (SoCG 2018) |
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Issue Date: |
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2018 |
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Date of publication: |
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08.06.2018 |
