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DOI: 10.4230/LIPIcs.SoCG.2017.67

On Balls in a Hilbert Polygonal Geometry (Multimedia Contribution)

Authors: Frank Nielsen and Laetitia Shao

Published in: LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

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.

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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)

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@InProceedings{nielsen_et_al:LIPIcs.SoCG.2017.67,
  author =	{Nielsen, Frank and Shao, Laetitia},
  title =	{{On Balls in a Hilbert Polygonal Geometry}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{67:1--67:4},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Aronov, Boris and Katz, Matthew J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.67},
  URN =		{urn:nbn:de:0030-drops-72443},
  doi =		{10.4230/LIPIcs.SoCG.2017.67},
  annote =	{Keywords: Projective geometry, Hilbert geometry, balls}
}

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On Balls in a Hilbert Polygonal Geometry (Multimedia Contribution)

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