LIPIcs.SoCG.2017.67.pdf
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On Balls in a Hilbert Polygonal Geometry (Multimedia Contribution)
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33rd International Symposium on Computational Geometry (SoCG 2017)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Symposium on Computational Geometry (SoCG) - License:
Creative Commons Attribution 3.0 Unported license
- Publication Date: 2017-06-20
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Frank Nielsen and Laetitia Shao. On Balls in a Hilbert Polygonal Geometry (Multimedia Contribution). In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 67:1-67:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.SoCG.2017.67
https://doi.org/10.4230/LIPIcs.SoCG.2017.67
Abstract
Hilbert geometry is a metric geometry that extends the hyperbolic Cayley-Klein geometry. In this video, we explain the shape of balls and their properties in a convex polygonal Hilbert geometry. First, we study the combinatorial properties of Hilbert balls, showing that the shapes of Hilbert polygonal balls depend both on the center location and on the complexity of the Hilbert domain but not on their radii. We give an explicit description of the Hilbert ball for any given center and radius. We then study the intersection of two Hilbert balls. In particular, we consider the cases of empty intersection and internal/external tangencies.
Keywords
- Projective geometry
- Hilbert geometry
- balls
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References
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Curtis T. McMullen. Coxeter groups, Salem numbers and the Hilbert metric. Publications mathématiques de l'IHÉS, 95:151-183, 2002.
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Frank Nielsen, Boris Muzellec, and Richard Nock. Classification with mixtures of curved Mahalanobis metrics. In IEEE International Conference on Image Processing (ICIP), pages 241-245. IEEE, 2016.
- Frank Nielsen, Boris Muzellec, and Richard Nock. Large margin nearest neighbor classification using curved Mahalanobis distances. CoRR, abs/1609.07082, 2016. URL: http://arxiv.org/abs/1609.07082.
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Frank Nielsen and Ke Sun. Clustering in Hilbert simplex geometry. ArXiv 1704.00454, April 2017.
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Jürgen Richter-Gebert. Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry. Springer Publishing Company, Incorporated, 1st edition, 2011.