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

TL;DR

This paper introduces an interactive visualization tool for higher-order Voronoi diagrams and Delaunay mosaics in polygonal metrics, addressing a gap in existing visualization capabilities.

Contribution

The authors present a new software tool that visualizes complex geometric structures in Hilbert geometry, including higher-order diagrams and clustering, with proven complexity bounds.

Findings

Voronoi cells are not always star-shaped.

Algorithm generates all order Voronoi diagrams simultaneously.

Software extends visualization to Hilbert, Funk, and Thompson geometries.

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.