Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Evans, Parker; Fasy, Brittany Terese; Wenk, Carola https://www.dagstuhl.de/lipics License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
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URN: urn:nbn:de:0030-drops-121993
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Combinatorial Properties of Self-Overlapping Curves and Interior Boundaries

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Abstract

We study the interplay between the recently-defined concept of minimum homotopy area and the classical topic of self-overlapping curves. The latter are plane curves that are the image of the boundary of an immersed disk. Our first contribution is to prove new sufficient combinatorial conditions for a curve to be self-overlapping. We show that a curve γ with Whitney index 1 and without any self-overlapping subcurves is self-overlapping. As a corollary, we obtain sufficient conditions for self-overlapping ness solely in terms of the Whitney index of the curve and its subcurves. These results follow from our second contribution, which shows that any plane curve γ, modulo a basepoint condition, is transformed into an interior boundary by wrapping around γ with Jordan curves. In fact, we show that n+1 wraps suffice, where γ has n vertices. Our third contribution is to prove the equivalence of various definitions of self-overlapping curves and interior boundaries, often implicit in the literature. We also introduce and characterize zero-obstinance curves, a further generalization of interior boundaries defined by optimality in minimum homotopy area.

BibTeX - Entry

@InProceedings{evans_et_al:LIPIcs:2020:12199,
  author =	{Parker Evans and Brittany Terese Fasy and Carola Wenk},
  title =	{{Combinatorial Properties of Self-Overlapping Curves and Interior Boundaries}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{41:1--41:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Sergio Cabello and Danny Z. Chen},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12199},
  URN =		{urn:nbn:de:0030-drops-121993},
  doi =		{10.4230/LIPIcs.SoCG.2020.41},
  annote =	{Keywords: Self-overlapping curves, interior boundaries, minimum homotopy area, immersion}
}

Keywords: Self-overlapping curves, interior boundaries, minimum homotopy area, immersion
Seminar: 36th International Symposium on Computational Geometry (SoCG 2020)
Issue date: 2020
Date of publication: 08.06.2020
Supplementary Material: An accompanying computer program that can determine whether a plane curve is self-overlapping, compute its minimum homotopy area, and display the self-overlapping decomposition associated with a minimum homotopy is available for download [Parker Evans et al., 2016], http://www.cs.tulane.edu/~carola/research/code.html. Figure 10 was created with this program.


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