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Hilbert geometry of the Siegel disk: The Siegel-Klein disk model | Frank Nielsen
; | Date: |
17 Apr 2020 | Abstract: | 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. | Source: | arXiv, 2004.8160 | Services: | Forum | Review | PDF | Favorites |
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