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DOI: 10.4230/LIPIcs.SOCG.2015.390
URN: urn:nbn:de:0030-drops-51380
URL: http://drops.dagstuhl.de/opus/volltexte/2015/5138/
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Kiazyk, Stephen ; Lubiw, Anna

Star Unfolding from a Geodesic Curve

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Abstract

There are two known ways to unfold a convex polyhedron without overlap: the star unfolding and the source unfolding, both of which use shortest paths from vertices to a source point on the surface of the polyhedron. Non-overlap of the source unfolding is straightforward; non-overlap of the star unfolding was proved by Aronov and O'Rourke in 1992. Our first contribution is a much simpler proof of non-overlap of the star unfolding. Both the source and star unfolding can be generalized to use a simple geodesic curve instead of a source point. The star unfolding from a geodesic curve cuts the geodesic curve and a shortest path from each vertex to the geodesic curve. Demaine and Lubiw conjectured that the star unfolding from a geodesic curve does not overlap. We prove a special case of the conjecture. Our special case includes the previously known case of unfolding from a geodesic loop. For the general case we prove that the star unfolding from a geodesic curve can be separated into at most two non-overlapping pieces.

BibTeX - Entry

@InProceedings{kiazyk_et_al:LIPIcs:2015:5138,
  author =	{Stephen Kiazyk and Anna Lubiw},
  title =	{{Star Unfolding from a Geodesic Curve}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{390--404},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Lars Arge and J{\'a}nos Pach},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5138},
  URN =		{urn:nbn:de:0030-drops-51380},
  doi =		{10.4230/LIPIcs.SOCG.2015.390},
  annote =	{Keywords: unfolding, convex polyhedra, geodesic curve}
}

Keywords: unfolding, convex polyhedra, geodesic curve
Seminar: 31st International Symposium on Computational Geometry (SoCG 2015)
Issue Date: 2015
Date of publication: 11.06.2015


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