Homotopic curve shortening and the affine curve-shortening flow

TL;DR

This paper introduces homotopic curve shortening (HCS), a discrete process that generalizes convex-layer decomposition and exhibits behavior similar to the affine curve-shortening flow (ACSF), with proven properties like affine invariance and convexity preservation.

Contribution

The paper defines HCS, connects it to ACSF through experimental evidence, and proves key properties such as affine invariance and curvature preservation, extending prior convex curve results to more general curves.

Findings

HCS behaves like ACSF in experiments with grid and random point sets.

HCS is affine-invariant and preserves convexity.

Number of self-intersections and inflection points does not increase under HCS.

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 $P\subset \mathbb R^2$ 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 ACSF and HCS generalizes the link between ACSF and convex-layer decomposition (Eppstein et al., 2017; Calder and Smart, 2020), which is restricted to convex curves. 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.