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Visualizing Higher Order Structures, Overlap Regions, and Clustering in the Hilbert Geometry

Hridhaan Banerjee, Soren Brown, June Cagan, Auguste H. Gezalyan, Megan Hunleth, Veena Kailad, Chaewoon Kyoung, Rowan Shigeno, Yasmine Tajeddin, Andrew Wagger, Kelin Zhu, David M. Moun

Abstract

Higher-order Voronoi diagrams and Delaunay mosaics in polygonal metrics have only recently been studied, yet no tools exist for visualizing them. We introduce a tool that fills this gap, providing dynamic interactive software for visualizing higher-order Voronoi diagrams and Delaunay mosaics along with clustering and tools for exploring overlap and outer regions in the Hilbert polygonal metric. We prove that $k^{th}$ order Voronoi cells are not always star-shaped and establish complexity bounds for our algorithm, which generates all order Voronoi diagrams at once. Our software unifies and extends previous tools for visualizing the Hilbert, Funk, and Thompson geometries.

Visualizing Higher Order Structures, Overlap Regions, and Clustering in the Hilbert Geometry

Abstract

Higher-order Voronoi diagrams and Delaunay mosaics in polygonal metrics have only recently been studied, yet no tools exist for visualizing them. We introduce a tool that fills this gap, providing dynamic interactive software for visualizing higher-order Voronoi diagrams and Delaunay mosaics along with clustering and tools for exploring overlap and outer regions in the Hilbert polygonal metric. We prove that order Voronoi cells are not always star-shaped and establish complexity bounds for our algorithm, which generates all order Voronoi diagrams at once. Our software unifies and extends previous tools for visualizing the Hilbert, Funk, and Thompson geometries.

Paper Structure

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

Table of Contents

  1. Introduction
  2. Higher order structures
  3. Overlap and Outer Region
  4. Clustering
  5. Conclusion

Key Result

Lemma 1

The circumcenter of any three sites serves as a Voronoi vertex in both the $k^{th}$ and $(k+1)^{th}$ order Hilbert Voronoi diagrams for some $0<k<n-1$.

Figures (4)

  • Figure 1: (a) First order Hilbert Voronoi (b) second order (c) final ($(n-1)^{th}$) order
  • Figure 2: The red region, the Voronoi cell closest to 1 and 2, is not a star-shaped region.
  • Figure 3: Two infinite balls along bisector through two sites (in red) generated by our software.
  • Figure 4: 120 points in 38 clusters using single link clustering.

Theorems & Definitions (6)

  • Lemma 1
  • Lemma 2
  • Corollary 3
  • Definition 5: Infinite Ball
  • Definition 6: Overlap Region
  • Definition 7: Outer Region