Grid Peeling of Parabolas

TL;DR

This work proves a conjectured link between discrete grid peeling and the affine curve-shortening flow (ACSF) for a natural class of curves—parabolas with vertical axis—by introducing grid parabolas P_t and their horizontal period H_t, and showing that grid peeling reproduces the reference parabola Pi_t with bounded deviation. The authors derive precise asymptotics H_t ∼ (2 ζ(3)/π^2) t^3, compute the constant c_g = ⎷⎸π^2/(2 ζ(3))⎸^{1/3}, and establish an error bound for the vertical distance between grid-peeled curves and the ACSF trajectory, scaling as O((T a^{2/3}+a^{-2/3})/n^{1/3} log(n/a)). They also show that parabolic grid-peeling speeds align with ACSF speeds: for a in appropriate ranges, the average vertical speed equals 1/t (or lies within a narrow interval depending on parity), and the results extend to parabolas with axis rational slopes via unimodular transformations. The paper connects discrete convex-layer processes with a continuous affine-flow, provides rigorous analysis, and opens avenues to general convex-curvature-based evolution problems, with potential implications for discrete geometry and related algorithms.

Abstract

Grid peeling is the process of repeatedly removing the convex hull vertices of the grid-points that lie inside a given convex curve. It has been conjectured that, for a more and more refined grid, grid peeling converges to a continuous process, the affine curve-shortening flow, which deforms the curve based on the curvature. We prove this conjecture for one class of curves, parabolas with a vertical axis, and we determine the value of the constant factor in the formula that relates the two processes.

Figures (20)

• Figure 1: Left: The Affine Curve- Shortening Flow (ACSF). The velocity is indicated by arrows, whose length is proportional to $\kappa^{1/3}$. Right: A convex curve and the first three steps of grid peeling.
• Figure 2: ACSF (left) and grid peeling (right) of a semicircle of diameter 1. The left figure shows 10 snapshots of ACSF with regular time increments; the right figure shows every 2714th convex layer for a grid of spacing $1/5000$. The increment 2714 corresponds to the conjectured formula \ref{['m_conj']} with a value $c_{\mathrm{g}}\approx 1.587$. (From ehn-gpacs-20, by permission from the authors.)
• Figure 3: The set $V_{11}$ of vectors $(x,y)$ from which $P_{11}$ is formed, shown as green dots. The vector with slope $s=2/5$ is highlighted. The points of $V_{11}$ extend indefinitely to the top and to the bottom. The picture shows the range $-1\le y\le9$.
• Figure 4: The grid parabola $P_t$ for $t=5$. The inset in the upper left corner shows some vectors of the set $V_5$, at a slightly enlarged scale. $\Pi_t$ is the reference parabola defined by $y= \frac{x^2}{2H_t}$. Here it is shifted down by some offset $\gamma$.
• Figure 5: Consecutive peelings of $P_5$. Since consecutive peelings share many vertices, it is not easy to distinguish the curves. In the lower part, we have therefore vertically separated the consecutive peelings. This has the effect that some grid points appear in several copies with small vertical offsets, and horizontal grid lines get a curved appearance.

Theorems & Definitions (26)

• Theorem 1
• Theorem 2
• Proposition 1
• proof
• Lemma 1
• Lemma 2
• proof
• Lemma 3
• Lemma 4
• Proposition 2