TY - GEN

T1 - Homotopic curve shortening and the affine curve-shortening flow

AU - Avvakumov, Sergey

AU - Nivasch, Gabriel

N1 - Publisher Copyright:
© Sergey Avvakumov and Gabriel Nivasch; licensed under Creative Commons License CC-BY 36th International Symposium on Computational Geometry (SoCG 2020).

PY - 2020/6/1

Y1 - 2020/6/1

N2 - 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.

AB - 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.

KW - Affine curve-shortening flow

KW - Convex-layer decomposition

KW - Integer grid

KW - Shortest homotopic path

UR - http://www.scopus.com/inward/record.url?scp=85086501434&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.SoCG.2020.12

DO - 10.4230/LIPIcs.SoCG.2020.12

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AN - SCOPUS:85086501434

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 36th International Symposium on Computational Geometry, SoCG 2020

A2 - Cabello, Sergio

A2 - Chen, Danny Z.

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 36th International Symposium on Computational Geometry, SoCG 2020

Y2 - 23 June 2020 through 26 June 2020

ER -