Rotation on the digital plane

Abstract Let $$A_{\varphi }$$ A φ denote the matrix of rotation with angle $$\varphi $$ Euclidean plane, FLOOR function which rounds a real point to nearest lattice down on left and ROUND for rounding off vector node lattice. We prove under natural assumption \not = k\frac{\pi }{2}$$ ≠ k π 2 that functions $${{\,\mathrm{FLOOR}\,}}\circ A_{\varphi FLOOR ∘ $${{\,\mathrm{ROUND}\,}}\circ ROUND are neither surjective nor injective. More precisely we lower upper estimates size sets points, image two points as well have no preimages. It turns out densities those positive.