In Euclidean spaces ( \({\mathbb {R}}^{d}\) , \(d \geqslant 2\) ), rotations preserve distances and angles; they are also bijective. These important properties are no longer guaranteed when rotations are considered in discrete spaces. This is especially the case in the Cartesian spaces ( \({\mathbb {Z}}^{d}\) , \(d \geqslant 2\) ), where rotations are, however, crucial for various applications, from image processing to computer graphics. In this article, we deal with the issue of bijectivity of discrete rotations in the 2-dimensional case ( \({\mathbb {Z}}^{2}\) ). We contribute to the state of the art from two points of view. First, we investigate the structure of the (finite and infinite) rotations in \({\mathbb {Z}}^{2}\) , and we shed light on the nature of this combinatorial space, which is analogue to a watershed tree. Second, we focus on the finite rotations, i.e. rotations that act on (finite) Euclidean balls instead of the (infinite) set \({\mathbb {Z}}^{2}\) . Under these hypotheses, we investigate the bijective rotations either as the restrictions of bijective rotations on \({\mathbb {Z}}^{2}\) , or as injective rotations on the Euclidean balls (thus bijective from their domain to their image). We provide two algorithmic schemes for building the combinatorial space of these finite rotations. Codes are freely available at the following url: https://github.com/ngophuc/DiscreteRotationSpace.
