Characterization of Bijective Discretized Rotations

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 $(\mathbb{R/Z})^{2}$.