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Visualizing Higher Order Structures, Overlap Regions, and Clustering in the Hilbert Geometry (Media Exposition)

Authors 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. Mount

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LIPIcs.SoCG.2026.100.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Hilbert metric
  • Funk metric
  • Voronoi diagrams

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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.

Cite As Get BibTex

Hridhaan Banerjee, Soren Brown, June Cagan, Auguste H. Gezalyan, Megan Hunleth, Veena Kailad, Chaewoon Kyoung, Rowan Shigeno, Yasmine Tajeddin, Andrew Wagger, Kelin Zhu, and David M. Mount. Visualizing Higher Order Structures, Overlap Regions, and Clustering in the Hilbert Geometry (Media Exposition). In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 100:1-100:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.100

Author Details

Hridhaan Banerjee
  • Thomas Jefferson High School for Science and Technology, Alexandria, VA, USA
Soren Brown
  • Department of Computer Science, University of Maryland, College Park, MD, USA
June Cagan
  • Department of Computer Science, University of Maryland, College Park, MD, USA
Auguste H. Gezalyan
  • Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France
Megan Hunleth
  • Montgomery Blair High School, Silver Spring, MD, USA
Veena Kailad
  • Montgomery Blair High School, Silver Spring, MD, USA
Chaewoon Kyoung
  • Montgomery Blair High School, Silver Spring, MD, USA
Rowan Shigeno
  • Department of Mathematics, Haverford College, PA, USA
Yasmine Tajeddin
  • Department of Computer Science, University of Maryland, College Park, MD, USA
Andrew Wagger
  • Department of Computer Science, University of Maryland, College Park, MD, USA
Kelin Zhu
  • Department of Mathematics, University of Maryland, College Park, MD, USA
David M. Mount
  • Department of Computer Science, University of Maryland, College Park, MD, USA

Funding

REU-CAAR 2025

Supplementary Materials

  • Software (Web Application) https://andrew.wagger.net/hilbert/ 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. Mount. Visualizing Higher Order Structures, Overlap Regions, and Clustering in the Hilbert Geometry (Software, Web Application). Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/artifacts.25998
  • Audiovisual (Video) https://youtu.be/yhJn--3Qgks 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. Mount. Visualizing Higher Order Structures, Overlap Regions, and Clustering in the Hilbert Geometry (Audiovisual, Video). Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/artifacts.25999

References

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  3. Hridhaan Banerjee, Soren Brown, June Cagan, Auguste H. Gezalyan, Megan Hunleth, Veena Kailad, Chaewoon Kyoung, Rowan Shigeno, Yasmine Tajeddin, Andrew Wagger, Kelin Zhu, and David M. Mount. Visualizing Higher Order Structures, Overlap Regions, and Clustering in the Hilbert Geometry. Software (visited on 2026-05-12). URL: https://andrew.wagger.net/hilbert/
    full metadata available at: https://doi.org/10.4230/artifacts.25998
  4. Hridhaan Banerjee, Soren Brown, June Cagan, Auguste H. Gezalyan, Megan Hunleth, Veena Kailad, Chaewoon Kyoung, Rowan Shigeno, Yasmine Tajeddin, Andrew Wagger, Kelin Zhu, and David M. Mount. Visualizing Higher Order Structures, Overlap Regions, and Clustering in the Hilbert Geometry. Audiovisual (visited on 2026-05-12). URL: https://youtu.be/yhJn--3Qgks
    full metadata available at: https://doi.org/10.4230/artifacts.25999
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