Grid Peeling and the Affine Curve-Shortening Flow

David Eppstein, Sariel Har-Peled, Gabriel Nivasch

Research output: Contribution to journalArticlepeer-review

Abstract

In this article, we study an experimentally-observed connection between two seemingly unrelated processes, one from computational geometry and the other from differential geometry. The first one (which we call grid peeling) is the convex-layer decomposition of subsets (Formula presented.) of the integer grid, previously studied for the particular case G = {1, …, m}2 by Har-Peled and Lidický. The second one is the affine curve-shortening flow (ACSF), first studied by Alvarez et al. and Sapiro and Tannenbaum. We present empirical evidence that, in a certain well-defined sense, grid peeling behaves at the limit like ACSF on convex curves. We offer some theoretical arguments in favor of this conjecture. We also pay closer attention to the simple case where (Formula presented.) is a quarter-infinite grid. This case corresponds to ACSF starting with an infinite L-shaped curve, which when transformed using the ACSF becomes a hyperbola for all times t > 0. We prove that, in the grid peeling of (Formula presented.), (1) the number of grid points removed up to iteration n is Θ(n3/2log n); and (2) the boundary at iteration n is sandwiched between two hyperbolas that are separated from each other by a constant factor.

Original languageEnglish (US)
Pages (from-to)306-316
Number of pages11
JournalExperimental Mathematics
Volume29
Issue number3
DOIs
StatePublished - Sep 1 2020

Keywords

  • 11H06
  • 11P21
  • 52A10
  • 53C44
  • 68U05
  • affine curve-shortening flow
  • convex-layer decomposition
  • curve-shortening flow
  • integer grid
  • onion decomposition

ASJC Scopus subject areas

  • General Mathematics

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