Characterization of Bijective Discretized Rotations

Abstract : A discretized rotation is the composition of an Euclidean rotation with the rounding operation. For $0 < \alpha < \pi/4$, we prove that the discretized rotation $\round{r_\alpha}$ is bijective if and only if there exists a positive integer $k$ such as $$\{\cos{\alpha},\sin{\alpha} \} = \{\frac{2k+1}{2k2+2k+1},\frac{2k2+2k}{2k2+2k+1} \}$$ The proof uses a particular subgroup of the torus $(\RR/\ZZ)2$.
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Bertrand Nouvel, Eric Rémila. Characterization of Bijective Discretized Rotations. [Research Report] LIP RR-2004-40, Laboratoire de l'informatique du parallélisme. 2004, 2+9p. ⟨hal-02101990⟩

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