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28 March 2024
 
  » arxiv » 2004.8160

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Hilbert geometry of the Siegel disk: The Siegel-Klein disk model
Frank Nielsen ;
Date 17 Apr 2020
AbstractWe introduce and study the Hilbert geometry induced by the Siegel disk, an open bounded convex set of complex matrices. This Hilbert geometry naturally yields a generalization of the Klein disk model of hyperbolic geometry, which we term the Siegel-Klein model to differentiate it with the usual Siegel upper plane and Siegel disk domains. In the Siegel-Klein disk, geodesics are by construction always straight, allowing one to build efficient geometric algorithms and data-structures from computational geometry. For example, we show how to approximate the Smallest Enclosing Ball (SEB) of a set of complex matrices in the Siegel domains: We compare two implementations of a generalization of the iterative algorithm of [Badoiu and Clarkson, 2003] in the Siegel-Poincar’e disk and in the Siegel-Klein disk. We demonstrate that geometric computing in the Siegel-Klein disk allows one (i) to bypass the time-costly recentering operations to the origin (Siegel translations) required at each iteration of the SEB algorithm in the Siegel-Poincar’e disk model, and (ii) to approximate numerically fast the Siegel distance with guaranteed lower and upper bounds.
Source arXiv, 2004.8160
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