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Arithmetic Discrete Hyperspheres and Separatingness

Christophe Fiorio 1 Jean-Luc Toutant 1
1 ARITH - Arithmétique informatique
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
Abstract : In the framework of the arithmetic discrete geometry, a discrete object is provided with its own analytical definition corresponding to a discretization scheme. It can thus be considered as the equivalent, in a discrete space, of a Euclidean object. Linear objects, namely lines and hyperplanes, have been widely studied under this assumption and are now deeply understood. This is not the case for discrete circles and hyperspheres for which no satisfactory definition exists. In the present paper, we try to fill this gap. Our main results are a general definition of discrete hyperspheres and the characterization of the k-minimal ones thanks to an arithmetic definition based on a non-constant thickness function. To reach such topological properties, we link adjacency and separatingness with norms.
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https://hal-lirmm.ccsd.cnrs.fr/lirmm-00128277
Contributor : Christophe Fiorio <>
Submitted on : Wednesday, January 31, 2007 - 3:15:55 PM
Last modification on : Wednesday, September 18, 2019 - 1:22:15 AM
Long-term archiving on: : Tuesday, April 6, 2010 - 10:56:09 PM

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Christophe Fiorio, Jean-Luc Toutant. Arithmetic Discrete Hyperspheres and Separatingness. DGCI: Discrete Geometry for Computer Imagery, Oct 2006, Szeged, Hungary. pp.425-436, ⟨10.1007/11907350_36⟩. ⟨lirmm-00128277⟩

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