Characterization of bijective discretized rotations by Gaussian integers

A discretized rotation is the composition of an Euclidean rotation with a rounding operation. It is well known that not all discretized rotations are bijective: e.g. two distinct points may have the same image by a given discretized rotation. Nevertheless, for a certain subset of rotation angles, the discretized rotations are bijective. In the regular square grid, the bijective discretized rotations have been fully characterized by Nouvel and Remila (IWCIA'2005). We provide a simple proof that uses the arithmetical properties of Gaussian integers.