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Average and Expected Distortion of Voronoi Paths and Scapes

Herbert Edelsbrunner, Anton Nikitenko

Abstract

The approximation of a circle with the edges of a fine square grid distorts the perimeter by a factor about $\tfrac{4}π$. 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.

Average and Expected Distortion of Voronoi Paths and Scapes

Abstract

The approximation of a circle with the edges of a fine square grid distorts the perimeter by a factor about . 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.

Paper Structure

This paper contains 7 sections, 9 theorems, 10 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Random Projections
  3. Tiling the Space of Point-Directions
  4. Mixed Regularity
  5. Mixed Cells
  6. Average and Expected Distortion
  7. Discussion

Key Result

lemma 1

Let $d \geq 0$ and $0 \leq p \leq d$. Then

Figures (1)

  • Figure 1: On the left, the (pink) projection of the tile defined by a Delaunay edge, $\gamma$, and its dual Voronoi edge, $\gamma^*$, has the topology of a disk, while on the right, its projection has the topology of a pinched annulus. In both cases, it is contained in the disk with radius $R_0$ centered at $z_1$, and this disk and therefore also the projection of the tile is contained in the disk with radius $2 R_0$ centered at $z_2$.

Theorems & Definitions (16)

  • lemma 1: Projection Moments
  • proof
  • lemma 2: Uniqueness of Tile
  • proof
  • lemma 3: Volume of Tile
  • proof
  • lemma 4: Projection of Tile
  • definition 1: Mixed Regularity
  • lemma 5: Sufficient Conditions
  • lemma 6: Mixed Regularity in Expectation
  • ...and 6 more