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Combinatorial Properties of Self-Overlapping Curves and Interior Boundaries

Authors Parker Evans, Brittany Terese Fasy , Carola Wenk

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  • This paper is based on the honors thesis of the first author [Parker Evans, 2018]. A full version of this paper [Parker Evans and Carola Wenk, 2020] is available at https://arxiv.org/abs/2003.13595.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Self-overlapping curves
  • interior boundaries
  • minimum homotopy area
  • immersion

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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.

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Parker Evans, Brittany Terese Fasy, and Carola Wenk. Combinatorial Properties of Self-Overlapping Curves and Interior Boundaries. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 41:1-41:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.SoCG.2020.41

Author Details

Parker Evans
  • Department of Mathematics, Rice University, Houston, TX, USA
Brittany Terese Fasy
  • School of Computing and Department of Mathematical Sciences, Montana State University, Bozeman, MT, USA
Carola Wenk
  • Department of Computer Science, Tulane University, New Orleans, LA, USA

Funding

  • Evans, Parker: Supported by NSF grant CCF-1618469 and by a Goldwater Scholarship from the Goldwater Foundation.
  • Fasy, Brittany Terese: Supported by NSF-CCF 1618605.
  • Wenk, Carola: Supported by NSF grant CCF-1618469.

Supplementary Materials

  • 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.

References

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  7. Parker Evans and Carola Wenk. Combinatorial properties of self-overlapping curves and interior boundaries, 2020. URL: http://arxiv.org/abs/2003.13595.
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