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Characterization of Bijective Discretized Rotations

  • Bertrand Nouvel
  • Eric Rémila
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3322)

Abstract

A discretized rotation is the composition of an Euclidean rotation with the rounding operation. For 0 < α < π/4, we prove that the discretized rotation [ r α ] is bijective if and only if there exists a positive integer k such as

\(\{cos\alpha, sin\alpha\}=\{\frac{2k+1}{2k^{2}+2k+1},\frac{2k^{2}+2k}{2k^{2}+2k+1}\}\)

The proof uses a particular subgroup of the torus \((\mathbb{R/Z})^{2}\).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Bertrand Nouvel
    • 1
  • Eric Rémila
    • 1
    • 2
  1. 1.Laboratoire de l’Informatique du Parallélisme, UMR 5668 (CNRS – ENS Lyon – UCB Lyon – INRIA)Ecole Normale Supérieure de LyonLyon cedex 07France
  2. 2.IUT Roanne (Université de Saint-Etienne)Roanne CedexFrance

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