We 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  .
We introduce and study the Hilbert geometry induced by the Siegel disk domain , an open bounded convex set of complex square matrices. This Hilbert geometry yields a generalization of the Klein disk model of hyperbolic geometry, which we term the Siegel-Klein disk model to differentiate it with the classic Siegel upper plane and disk domains. In the Siegel-Klein disk, geodesics are by construction always Euclidean straight and unique , allowing one to design efficient geometric algorithms and data-structures from computational geometry. As an illustrating example, we show how to approximate the Smallest Enclosing Ball (SEB) of a set of complex square matrices in the Siegel disk domains: We compare two implementations of a generalization of the iterative core-set 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 disk origin required at each iteration of the SEB algorithm in the Siegel-Poincar\'e disk model, and (ii) to approximate fast and numerically the Siegel-Klein distance with guaranteed lower and upper bounds derived from nested Hilbert geometries .
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domain
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square
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naturally
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disk
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usual
classic
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Siegel
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straight
Euclidean straight and unique
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build
design
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For
As an illustrating
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square
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disk
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core-set
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origin (Siegel translations)
disk origin
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numerically fast the Siegel
fast and numerically the Siegel-Klein
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derived from nested Hilbert geometries