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Zone Theorem for Arrangements in three dimensions

Sanjeev Saxena

TL;DR

In this note, a simple description of zone theorem in three dimensions is given.

Abstract

In this note, a simple description of zone theorem in three dimensions is given.Arrangements in three dimensions are useful for constructing higher-order Voronoi diagrams in plane. An elementary and very intuitive treatment of this result is also given.

Zone Theorem for Arrangements in three dimensions

TL;DR

In this note, a simple description of zone theorem in three dimensions is given.

Abstract

In this note, a simple description of zone theorem in three dimensions is given.Arrangements in three dimensions are useful for constructing higher-order Voronoi diagrams in plane. An elementary and very intuitive treatment of this result is also given.

Paper Structure

This paper contains 6 sections, 2 theorems, 6 equations.

Table of Contents

  1. Abstract
  2. 1. Introduction
  3. 2. Definitions and basic properties
  4. 3. Zone Theorem in $3$-dimensions
  5. 4.$k$-Nearest Neighbours and Arrangements

Key Result

Theorem 1

Assume ${\cal{A}}$ is an arrangement of $n$ planes, $Q$ is a plane in ${\cal{A}}$, and $S$ is a plane not in ${\cal{A}}$. Let $C$ be a cell in zone$(S)$. Let $f$ be a face of cell $C$ not lying in plane $Q$. Then total number of such pairs $(f,C)$ (of face $f$ and cell $C$) is at most the sum of

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2