Combinatorial Properties of Self-Overlapping Curves and Interior Boundaries
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.
Self-overlapping curves
interior boundaries
minimum homotopy area
immersion
Theory of computation~Computational geometry
41:1-41:17
Regular Paper
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.
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.
Parker
Evans
Parker Evans
Department of Mathematics, Rice University, Houston, TX, USA
Supported by NSF grant CCF-1618469 and by a Goldwater Scholarship from the Goldwater Foundation.
Brittany Terese
Fasy
Brittany Terese Fasy
School of Computing and Department of Mathematical Sciences, Montana State University, Bozeman, MT, USA
https://orcid.org/0000-0003-1908-0154
Supported by NSF-CCF 1618605.
Carola
Wenk
Carola Wenk
Department of Computer Science, Tulane University, New Orleans, LA, USA
https://orcid.org/0000-0001-9275-5336
Supported by NSF grant CCF-1618469.
10.4230/LIPIcs.SoCG.2020.41
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http://www.cs.tulane.edu/~carola/research/code.html
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http://arxiv.org/abs/2003.13595
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Parker Evans, Brittany Terese Fasy, and Carola Wenk
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