Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Analysis of Dynamic Voronoi Diagrams in the Hilbert Metric

Madeline Bumpus, Xufeng Caesar Dai, Auguste H. Gezalyan, Sam Munoz, Renita Santhoshkumar, Songyu Ye, David M. Mount

TL;DR

This work addresses computing and understanding dynamic Voronoi diagrams under the Hilbert metric on polygonal convex bodies. It develops a sector-based analytic framework that expresses bisectors in each sector as conics $A x^2+B x y+C y^2+D x+E y+F=0$, using projective transformations to canonical forms for efficient analysis. It identifies when bisectors can contain a 2D region, provides a necessary and sufficient condition in terms of vanishing-point rays, and describes a quadrilateral region $Z$ between two sites. It implements a quadratic-time dynamic insertion algorithm and accompanying visualization software, enabling interactive exploration of Hilbert Voronoi diagrams on user-specified convex polygons, with potential applications in convex approximation, quantum information, optimization, and network analysis.

Abstract

The Hilbert metric is a projective metric defined on a convex body which generalizes the Cayley-Klein model of hyperbolic geometry to any convex set. In this paper we analyze Hilbert Voronoi diagrams in the Dynamic setting. In addition we introduce dynamic visualization software for Voronoi diagrams in the Hilbert metric on user specified convex polygons.

Analysis of Dynamic Voronoi Diagrams in the Hilbert Metric

TL;DR

This work addresses computing and understanding dynamic Voronoi diagrams under the Hilbert metric on polygonal convex bodies. It develops a sector-based analytic framework that expresses bisectors in each sector as conics , using projective transformations to canonical forms for efficient analysis. It identifies when bisectors can contain a 2D region, provides a necessary and sufficient condition in terms of vanishing-point rays, and describes a quadrilateral region between two sites. It implements a quadratic-time dynamic insertion algorithm and accompanying visualization software, enabling interactive exploration of Hilbert Voronoi diagrams on user-specified convex polygons, with potential applications in convex approximation, quantum information, optimization, and network analysis.

Abstract

The Hilbert metric is a projective metric defined on a convex body which generalizes the Cayley-Klein model of hyperbolic geometry to any convex set. In this paper we analyze Hilbert Voronoi diagrams in the Dynamic setting. In addition we introduce dynamic visualization software for Voronoi diagrams in the Hilbert metric on user specified convex polygons.
Paper Structure (10 sections, 7 theorems, 3 equations, 6 figures, 1 table)

This paper contains 10 sections, 7 theorems, 3 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Previous Work
  4. Defining the Hilbert Metric
  5. Voronoi Diagrams in the Hilbert Metric
  6. Hilbert Bisector analysis
  7. Finding Bisectors in a Sector
  8. Analysis of Hilbert Voronoi Diagrams
  9. Software
  10. Concluding Remarks

Key Result

Theorem 1

The equation of the bisector between two sites in any particular sector is of the form $Ax^2+Bxy+Cy^2+Dx+Ey+F = 0$, with coefficients depending only on the line equations of the four edges and the sites.

Figures (6)

  • Figure 1: Illustration of different sectors.
  • Figure 3: Illustration for the proof of Lemma \ref{['lem:3edgeAllConics']} (a bisector in the three-edge case with shared edge in front of both sites).
  • Figure 4: The bisector conics for the cases given in Table \ref{['tab:3edgeAllConics']} visualized.
  • Figure 5: The point $p$ is equidistant from both sites $s$ and $t$ by the invariance of the cross ratio.
  • Figure 6: A discontinuity in the Hilbert metric Voronoi.
  • ...and 1 more figures

Theorems & Definitions (9)

  • Definition 1: Cross Ratio
  • Definition 2: Hilbert metric
  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7