TY - JOUR

T1 - Grid Peeling and the Affine Curve-Shortening Flow

AU - Eppstein, David

AU - Har-Peled, Sariel

AU - Nivasch, Gabriel

N1 - Funding Information:
Supported in part by the National Science Foundation under Grants CCF-1228639, CCF-1618301, and CCF-1616248. Work on this paper was partially supported by a NSF AF awards CCF-1421231 and CCF-1217462. The third author would like to thank Franck Assous and Elad Horev for useful conversations.
Publisher Copyright:
© 2018 Taylor & Francis.

PY - 2020/9/1

Y1 - 2020/9/1

N2 - 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.

AB - 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.

KW - 11H06

KW - 11P21

KW - 52A10

KW - 53C44

KW - 68U05

KW - affine curve-shortening flow

KW - convex-layer decomposition

KW - curve-shortening flow

KW - integer grid

KW - onion decomposition

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U2 - 10.1080/10586458.2018.1466379

DO - 10.1080/10586458.2018.1466379

M3 - Article

AN - SCOPUS:85047263679

SN - 1058-6458

VL - 29

SP - 306

EP - 316

JO - Experimental Mathematics

JF - Experimental Mathematics

IS - 3

ER -