Average and Expected Distortion of Voronoi Paths and Scapes

TL;DR

This paper proves that the average distortion factor of approximating curves with Voronoi paths is universal across dimensions, extending classical circle approximation results to general rectifiable curves and Voronoi scapes.

Contribution

It establishes a universal average distortion factor for Voronoi path approximations of rectifiable curves in any dimension, generalizing known 2D results to higher dimensions.

Findings

The average distortion factor is approximately 4/π for all rectifiable curves.

The results extend from 2D to all dimensions, applying to Voronoi scapes.

The distortion factor is proven in the ergodic sense for non-exotic Delaunay mosaics.

Abstract

The approximation of a circle with the edges of a fine square grid distorts the perimeter by a factor about $\tfrac{4}{\pi}$. We prove that this factor is the same on average (in the ergodic sense) for approximations of any rectifiable curve by the edges of any non-exotic Delaunay mosaic (known as Voronoi path), and extend the results to all dimensions, generalizing Voronoi paths to Voronoi scapes.