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Homotopic Curve Shortening and the Affine Curve-Shortening Flow

Authors Sergey Avvakumov, Gabriel Nivasch

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Author Details

Sergey Avvakumov
  • Institute of Science and Technology Austria (IST Austria), Am Campus 1, 3400 Klosterneuburg, Austria
Gabriel Nivasch
  • Ariel University, Ariel, Israel

Funding

  • Avvakumov, Sergey: Supported by the Austrian Science Fund (FWF), Project P31312-N35.

Acknowledgements

Thanks to Arseniy Akopyan, Imre Bárány, Jeff Erickson, Radoslav Fulek, Jeremy Schiff, Arkadiy Skopenkov, and Peter Synak for useful discussions. Thanks also to the referees for their useful comments.

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Sergey Avvakumov and Gabriel Nivasch. Homotopic Curve Shortening and the Affine Curve-Shortening Flow. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.SoCG.2020.12

Abstract

We define and study a discrete process that generalizes the convex-layer decomposition of a planar point set. Our process, which we call homotopic curve shortening (HCS), starts with a closed curve (which might self-intersect) in the presence of a set P⊂ ℝ² of point obstacles, and evolves in discrete steps, where each step consists of (1) taking shortcuts around the obstacles, and (2) reducing the curve to its shortest homotopic equivalent.
We find experimentally that, if the initial curve is held fixed and P is chosen to be either a very fine regular grid or a uniformly random point set, then HCS behaves at the limit like the affine curve-shortening flow (ACSF). This connection between HCS and ACSF generalizes the link between "grid peeling" and the ACSF observed by Eppstein et al. (2017), which applied only to convex curves, and which was studied only for regular grids.
We prove that HCS satisfies some properties analogous to those of ACSF: HCS is invariant under affine transformations, preserves convexity, and does not increase the total absolute curvature. Furthermore, the number of self-intersections of a curve, or intersections between two curves (appropriately defined), does not increase. Finally, if the initial curve is simple, then the number of inflection points (appropriately defined) does not increase.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Geometric topology
Keywords
  • affine curve-shortening flow
  • shortest homotopic path
  • integer grid
  • convex-layer decomposition

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