Configurations induced by discrete rotations: periodicity and quasi-periodicity properties

  • Authors:

  • Bertrand Nouvel;Éric Rémila

  • Affiliations:

  • Laboratoire de l'Informatique du Parallélisme, UMR CNRS, ENS Lyon, UCB Lyon, INRIA 5668, École Normale Supérieure de Lyon, 46, Allée d'Italie 69364 Lyon Cedex 07, France;Laboratoire de l'Informatique du Parallélisme, UMR CNRS, ENS Lyon, UCB Lyon, INRIA 5668, École Normale Supérieure de Lyon, 46, Allée d'Italie 69364 Lyon Cedex 07, France

  • Venue:
  • Discrete Applied Mathematics - Special issue: Advances in discrete geometry and topology (DGCI 2003)
  • Year:
  • 2005

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Abstract

A discretized rotation acts on a pixel grid: the edges of the neighborhood relation are affected in particular way. Two types of configurations (i.e. applications from Z2 to a finite set of states) are introduced to code locally the transformations of the neighborhood. All the characteristics of discretized rotations are encoded within the configurations. We prove that their structure is linked to a subgroup of the bidimensional torus. Using this link, we obtain a characterization of periodical configurations and we prove their quasi-periodicity for any angle.