# 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.

## Key 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.