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
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O'Rourke, Joseph

Edge-Unfolding Nearly Flat Convex Caps

LIPIcs-SoCG-2018-64.pdf (3 MB)


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

  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 =		{},
  URN =		{urn:nbn:de:0030-drops-87777},
  doi =		{10.4230/LIPIcs.SoCG.2018.64},
  annote =	{Keywords: polyhedra, unfolding}

Keywords: polyhedra, unfolding
Collection: 34th International Symposium on Computational Geometry (SoCG 2018)
Issue Date: 2018
Date of publication: 08.06.2018

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