Visualizing Higher Order Structures, Overlap Regions, and Clustering in the Hilbert Geometry
This tool fills a gap in computational geometry by enabling visualization of higher-order structures in polygonal metrics, benefiting researchers in geometry and visualization.
The authors introduce a dynamic interactive software tool for visualizing higher-order Voronoi diagrams and Delaunay mosaics in the Hilbert polygonal metric, proving that k-th order Voronoi cells are not always star-shaped and establishing complexity bounds for their algorithm.
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.