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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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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.

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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

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.

Subject Classification

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

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Related Versions

  • 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.

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

  1. Samuel J. Blank. Extending Immersions of the Circle. PhD thesis, Brandeis University, 1967. Google Scholar
  2. Erin Chambers and David Letscher. On the height of a homotopy. In 21st Canadian Conference on Computational Geometry, pages 103-106, 2009. Google Scholar
  3. Erin Chambers and Yusu Wang. Measuring similarity between curves on 2-manifolds via homotopy area. In Proc. Proc. 29th Symposium on Computational Geometry, pages 425-434, 2013. Google Scholar
  4. David Eppstein and Elena Mumford. Self-overlapping curves revisited. SODA '09 Proceedings of the twentieth annual ACM-SIAM Symposium on Discrete Algorithms, 2009. Google Scholar
  5. Parker Evans. On self-overlapping curves, interior boundaries, and minimum area homotopies. Undergraduate Thesis, Tulane University, 2018. Google Scholar
  6. Parker Evans, Andrea Burns, and Carola Wenk. Homotopy visualizer. http://www.cs.tulane.edu/~carola/research/code.html, 2016. URL: http://www.cs.tulane.edu/~carola/research/code.html.
  7. Parker Evans and Carola Wenk. Combinatorial properties of self-overlapping curves and interior boundaries, 2020. URL: http://arxiv.org/abs/2003.13595.
  8. Brittany Terese Fasy, Selcuk Karakoc, and Carola Wenk. On minimum area homotopies of normal curves in the plane, 2017. URL: http://arxiv.org/abs/1707.02251.
  9. Carl Friedrich Gauß. Zur Geometria Situs. In Nachlass. I. Teubner-Archiv zur Mathematik, ca. 1900. Google Scholar
  10. Jack Graver and Gerald Cargo. When does a curve bound a distorted disk? SIAM Journal on Discrete Mathematics, 25(1):280-305, 2011. Google Scholar
  11. Selcuk Karakoc. On Minimum Homotopy Areas. PhD thesis, Tulane University, 2017. Google Scholar
  12. Morris Marx. The branch point structure of extensions of interior boundaries. Transactions of the American Mathematical Society, 131(1):79-98, 1968. Google Scholar
  13. Morris Marx. Extending immersions of 𝕊^1 to ℝ². Transactions of the American Mathematical Society, 187:309-326, 1974. Google Scholar
  14. Uddipan Mukherjee. Self-overlapping curves: Analaysis and applications. 2013 SIAM Conference on Geometric and Physical Modeling, 2013. Google Scholar
  15. Uddipan Mukherjee, M. Gopi, and Jarek Rossignac. Immersion and embedding of self-crossing loops. Proc. ACM/Eurographics Symposium on Sketch-Based Interfaces and Modeling, pages 31-38, 2011. Google Scholar
  16. Herbert Seifert. Konstruktion dreidimensionaler geschlossener Raume. PhD thesis, Saxon Academy of Sciences Leipzig, 1931. Google Scholar
  17. Peter W. Shor and Christopher Van Wyk. Detecting and decomposing self-overlapping curves. Computational Geometry: Theory and Applications, 2(1):31-50, 1992. Google Scholar
  18. Charles Titus. A theory of normal curves and some applications. Pacific J. Math, 10(3):1083-1096, 1960. Google Scholar
  19. Charles Titus. The combinatorial topology of analytic functions of the boundary of a disk. Acta Mathematica, 106(1):45-64, 1961. Google Scholar
  20. Hassler Whitney. On regular closed curves in the plane. Compositio Math. 4, pages 276-284, 1937. Google Scholar
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