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| Main Authors: | , , , , , , , , , , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.27009 |
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| _version_ | 1866910081698758656 |
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| author | Banerjee, Hridhaan Brown, Soren Cagan, June Gezalyan, Auguste H. Hunleth, Megan Kailad, Veena Kyoung, Chaewoon Shigeno, Rowan Tajeddin, Yasmine Wagger, Andrew Zhu, Kelin Moun, David M. |
| author_facet | Banerjee, Hridhaan Brown, Soren Cagan, June Gezalyan, Auguste H. Hunleth, Megan Kailad, Veena Kyoung, Chaewoon Shigeno, Rowan Tajeddin, Yasmine Wagger, Andrew Zhu, Kelin Moun, David M. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27009 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Visualizing Higher Order Structures, Overlap Regions, and Clustering in the Hilbert Geometry Banerjee, Hridhaan Brown, Soren Cagan, June Gezalyan, Auguste H. Hunleth, Megan Kailad, Veena Kyoung, Chaewoon Shigeno, Rowan Tajeddin, Yasmine Wagger, Andrew Zhu, Kelin Moun, David M. Computational Geometry 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. |
| title | Visualizing Higher Order Structures, Overlap Regions, and Clustering in the Hilbert Geometry |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2603.27009 |