Bijectivity Certification of 3D Digitized Rotations

  • Kacper PlutaEmail author
  • Pascal Romon
  • Yukiko Kenmochi
  • Nicolas Passat
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9667)


Euclidean rotations in \(\mathbb {R}^n\) are bijective and isometric maps. Nevertheless, they lose these properties when digitized in \(\mathbb {Z}^n\). For \(n=2\), the subset of bijective digitized rotations has been described explicitly by Nouvel and Rémila and more recently by Roussillon and Cœurjolly. In the case of 3D digitized rotations, the same characterization has remained an open problem. In this article, we propose an algorithm for certifying the bijectivity of 3D digitized rational rotations using the arithmetic properties of the Lipschitz quaternions.


Time Complexity Rational Rotation Rigid Motion Integer Lattice Arithmetic Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The authors express their thanks to Éric Andres of Université de Poitiers for his very helpful feedback and comments which allowed us to improve the article.

The research leading to these results has received funding from the Programme d’Investissements d’Avenir (LabEx Bézout, ANR-10-LABX-58).


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Kacper Pluta
    • 1
    • 2
    Email author
  • Pascal Romon
    • 2
  • Yukiko Kenmochi
    • 1
  • Nicolas Passat
    • 3
  1. 1.Université Paris-Est, LIGM, CNRS, ESIEEParisFrance
  2. 2.Université Paris-Est, LAMA, UPEMParisFrance
  3. 3.Université de Reims Champagne-Ardenne, CReSTICReimsFrance

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