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DOI: 10.4230/LIPIcs.SOCG.2015.390

Star Unfolding from a Geodesic Curve

Authors: Stephen Kiazyk and Anna Lubiw

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

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.

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Stephen Kiazyk and Anna Lubiw. Star Unfolding from a Geodesic Curve. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 390-404, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kiazyk_et_al:LIPIcs.SOCG.2015.390,
  author =	{Kiazyk, Stephen and Lubiw, Anna},
  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 =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.390},
  URN =		{urn:nbn:de:0030-drops-51380},
  doi =		{10.4230/LIPIcs.SOCG.2015.390},
  annote =	{Keywords: unfolding, convex polyhedra, geodesic curve}
}

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Star Unfolding from a Geodesic Curve

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