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Software for the Thompson and Funk Polygonal Geometry

Hridhaan Banerjee, Carmen Isabel Day, Auguste H. Gezalyan, Olga Golovatskaia, Megan Hunleth, Sarah Hwang, Nithin Parepally, Lucy Wang, David M. Mount

TL;DR

The paper addresses understanding and visualizing polygonal Thompson, Funk, reverse Funk, and Hilbert geometries in convex polygons. It develops dynamic Javascript tools to manipulate B_F(p,r) and B_{rF}(p,r), with B_T(p,r) computed as the intersection of forward and reverse Funk balls, and it provides a Hilbert-geometry traversal visualization that preserves H(Omega) via a projective-affine movement map phi_v(p)=p/(1+p·v) together with John ellipsoid normalization. Key contributions include explicit metric definitions, linear-time computation of Thompson-ball intersections, nesting B_H(p,r/2) ⊆ B_T(p,r) ⊆ B_H(p,r), and open-source software at https://github.com/nithin1527/funk-geo-visualizer with a live app at https://funk-geo-visualizer.vercel.app/. The work provides educational and practical infrastructure for exploring polygonal metric geometries and their interrelations.

Abstract

Metric spaces defined within convex polygons, such as the Thompson, Funk, reverse Funk, and Hilbert metrics, are subjects of recent exploration and study in computational geometry. This paper contributes an educational piece of software for understanding these unique geometries while also providing a tool to support their research. We provide dynamic software for manipulating the Funk, reverse Funk, and Thompson balls in convex polygonal domains. Additionally, we provide a visualization program for traversing the Hilbert polygonal geometry.

Software for the Thompson and Funk Polygonal Geometry

TL;DR

The paper addresses understanding and visualizing polygonal Thompson, Funk, reverse Funk, and Hilbert geometries in convex polygons. It develops dynamic Javascript tools to manipulate B_F(p,r) and B_{rF}(p,r), with B_T(p,r) computed as the intersection of forward and reverse Funk balls, and it provides a Hilbert-geometry traversal visualization that preserves H(Omega) via a projective-affine movement map phi_v(p)=p/(1+p·v) together with John ellipsoid normalization. Key contributions include explicit metric definitions, linear-time computation of Thompson-ball intersections, nesting B_H(p,r/2) ⊆ B_T(p,r) ⊆ B_H(p,r), and open-source software at https://github.com/nithin1527/funk-geo-visualizer with a live app at https://funk-geo-visualizer.vercel.app/. The work provides educational and practical infrastructure for exploring polygonal metric geometries and their interrelations.

Abstract

Metric spaces defined within convex polygons, such as the Thompson, Funk, reverse Funk, and Hilbert metrics, are subjects of recent exploration and study in computational geometry. This paper contributes an educational piece of software for understanding these unique geometries while also providing a tool to support their research. We provide dynamic software for manipulating the Funk, reverse Funk, and Thompson balls in convex polygonal domains. Additionally, we provide a visualization program for traversing the Hilbert polygonal geometry.

Paper Structure

This paper contains 3 sections, 5 theorems, 4 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Traversing the Hilbert Geometry

Key Result

Lemma 2

The open forward Funk ball of radius $r$ around a point $p$ is the image of $\Omega$ under Euclidean homothety around the point $p$ with dilation factor of $(1-e^{-r})$papadopoulos2014handbook.

Figures (3)

  • Figure 1: The Hilbert (a), Funk (b), Reverse Funk (c), and Thompson (d) balls around a point
  • Figure 2: (a) Thompson balls are not pseudo-disks, (b) a Thompson ball of radius $1$ between two Hilbert balls of radii $1/2$ and $1$.
  • Figure 3: (a) Travelling in a hendecagon and (b) after travelling some distance in the square.

Theorems & Definitions (8)

  • Definition 1: Funk weak metric
  • Lemma 2
  • Lemma 3
  • Definition 4: Hilbert metric
  • Definition 5: Thompson metric
  • Lemma 6
  • Lemma 7
  • Lemma 8