Polygonal network disorder and the turning distance

TL;DR

The paper introduces turning disorders for polygonal networks by averaging turning distances $d_2$ between network faces and ordered shapes such as circles and regular polygons. It develops computationally efficient formulas, including an $O((m+n)\log(m+n))$ time bound for 2-turning distances between regular polygons and closed forms for divisibility cases, along with circle–polygon distances. It provides exact expressions for Archimedean lattices and analyzes two stochastic network processes (T1 spring networks and polygonal rupturing), showing how the choice of ordered shape and area weighting affects measured disorder. The framework yields a flexible, geometry-driven metric for characterizing microstructure disorder with potential applications in materials science, foams, and related visualization or ML tasks.

Abstract

The turning distance is a well-studied metric for measuring the similarity between two polygons. This metric is constructed by taking an $L^p$ distance between step functions which track each shape's tangent angle of a path tracing its boundary. In this study, we introduce \textit{turning disorders} for polygonal planar networks, defined by averaging turning distances between network faces and "ordered" shapes (regular polygons or circles). We derive closed-form expressions of turning distances for special classes of regular polygons, related to the divisibility of $m$ and $n$, and also between regular polygons and circles. These formulas are used to show that the time for computing the 2-turning distances reduces to $O((m+n) \log(m+n))$ when both shapes are regular polygons, an improvement from $O(mn\log(mn))$ operations needed to compute distances between general polygons of $n$ and $m$ sides. We also apply these formulas to several examples of network microstructure with varying disorder. For Archimedean lattices, a class of regular tilings, we can express turning disorders with exact expressions. We also consider turning disorders applied to two examples of stochastic processes on networks: spring networks evolving under T1 moves and polygonal rupture processes. We find that the two aspects of defining different turning disorders, the choice of ordered shape and whether to apply area-weighting, can capture different notions of network disorder.

Figures (10)

• Figure 1: Turning distances of ordered shapes compared against rectangles $P_a$ with aspect ratios $1 \le a \le 20$
• Figure 2: A spiral shape with two rotations wound around a circle (shown dashed) of radius $1/(4\pi n)$. The turning function begins at the origin (black dot). The interior of the spiral shape is shaded in gray.
• Figure 3: Three examples of Archimedean lattices, with a fundamental region for each lattice shaded in gray.
• Figure 4: A T1 move.
• Figure 5: Snapshots of the T1 process with initial conditions of a Voronoi diagram with 1000 cells after the Tutte algorithm. From top left to bottom right: the process after 0, 1000, 2000, and 3000 T1 moves.

Theorems & Definitions (22)

• Proposition 1
• proof
• Corollary 1
• Proposition 2
• Proposition 3
• Corollary 2
• Proposition 4
• proof
• Remark 1
• Proposition 5