Rotation on the digital plane

Abstract

Let $A_{\varphi}$ denote the matrix of rotation with angle $\varphi$ of the Euclidean plane, FLOOR the function, which rounds a real point to the nearest lattice point down on the left and ROUND the function for rounding off a vector to the nearest node of the lattice. We prove under the natural assumption $\varphi\not= k\frac{\pi}{2}$ that the functions $FLOOR \circ A_{\varphi}$ and $ROUND \circ A_{\varphi}$ are neither surjective nor injective. More precisely we prove lower and upper estimates for the size of the sets of lattice points, which are the image of two lattice points as well as of lattice points, which have no preimages. It turns out that the density of that sets are positive except when $\sin \varphi \not= \pm \cos \varphi + r, r\in \mathbb{Q}$.