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.
We study the Hilbert geometry induced by the Siegel disk domain, an open bounded convex set of complex square matrices of operator norm strictly less than one . This Hilbert geometry yields a generalization of the Klein disk model of hyperbolic geometry, henceforth called the Siegel-Klein disk model to differentiate it with the classical Siegel upper plane and disk domains. In the Siegel-Klein disk, geodesics are by construction always unique and Euclidean straight , allowing one to design efficient geometric algorithms and data-structures from computational geometry. For example, we show how to approximate the smallest enclosing ball of a set of complex square matrices in the Siegel disk domains: We compare two generalizations of the iterative core-set algorithm of Badoiu and Clarkson (BC) 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 BC 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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introduce and
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of operator norm strictly less than one
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which we term
henceforth called
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classic
classical
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Euclidean straight and unique
unique and Euclidean straight
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As an illustrating
For
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Smallest Enclosing Ball (SEB)
smallest enclosing ball
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implementations of a generalization of
generalizations of
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, 2003
(BC)
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.
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SEB
BC