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 Nou- vel and Rémila (IWCIA'2005). We provide a simple proof that uses the arithmetical properties of Gaussian integers.
@techreport{dcoeurjo_RRGaussian,
author = {Roussillon, Tristan and Coeurjolly, David},
hal_id = {hal-01259826},
institution = {LIRIS UMR CNRS 5205},
keywords = {Gaussian integers ; Bijective discretized rotations},
month = {January},
pdf = {https://hal.archives-ouvertes.fr/hal-01259826/file/RR.pdf},
title = {Characterization of bijective discretized rotations by Gaussian integers},
type = {Research Report},
url = {https://hal.archives-ouvertes.fr/hal-01259826},
year = {2016}
}