Average and Expected Distortion of Voronoi Paths and Scapes

TL;DR

This paper shows that approximating curves with grid edges distorts their length by a consistent factor, regardless of the curve or grid used.

Contribution

The paper proves that the distortion factor is universally constant for Voronoi paths and extends the concept to Voronoi scapes in all dimensions.

Findings

The distortion factor for Voronoi paths is consistently $\tfrac{4}{\pi}$ on average.

This result generalizes to all rectifiable curves and non-exotic Delaunay mosaics.

The concept is extended to Voronoi scapes in any dimension.

Abstract

The approximation of a circle with the edges of a fine square grid distorts the perimeter by a factor about \documentclass[12pt]{minimal}
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\begin{document}$$\tfrac{4}{\pi }$$\end{document}4π. 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.

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