Bijectivity Analysis of Finite Rotations on  \({\mathbb Z}^2\) : A Hierarchical Approach

In this article, we investigate the rotations on \(\mathbb Z^2\) (a.k.a discrete rotations). In particular, we focus on the finite rotations that act on finite subsets of \({\mathbb Z}^2\) , especially Euclidean balls. We shed light on the hierarchical structure of these rotations, induced by their discontinuity (characterized by hinge angles) and the size of the considered ball. We propose efficient algorithmic schemes leading to the construction of combinatorial models (trees) of the bijective finite rotations. These algorithms and structures open the way to a better understanding of the notion of bijectivity with respect to finite vs. infinite discrete rotations.