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

2026-03-27

AI Summary

Existing tools struggle to visualize higher-order Voronoi diagrams and Delaunay tessellations under polygonal metrics, particularly Hilbert geometry. This work proposes the first efficient, dynamically interactive visualization system that unifies the generation and display of arbitrary-order Voronoi diagrams, Delaunay tessellations, and their associated clustering, overlapping, and exterior structures under Hilbert, Funk, and Thompson polygonal metrics, leveraging computational geometry algorithms. The core contributions include an integrated framework for generating and interactively exploring higher-order Voronoi diagrams, the discovery that k-th order Voronoi cells need not be star-shaped, and the establishment of theoretical complexity bounds for the underlying algorithms.

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.

Problem

• higher-order Voronoi diagrams
• Hilbert geometry
• visualization
• Delaunay mosaics
• overlap regions

Innovation

• higher-order Voronoi diagrams
• Hilbert geometry
• interactive visualization
• Delaunay mosaics
• polygonal metrics