Rotation on the digital plane

TL;DR

This paper investigates the properties of lattice point transformations involving rotations and rounding functions, establishing conditions under which these transformations are neither surjective nor injective and analyzing their density.

Contribution

It provides new theoretical results on the behavior of rotated lattice points under FLOOR and ROUND functions, including density estimates and conditions for non-surjectivity and non-injectivity.

Findings

Transformations are neither surjective nor injective for most rotation angles.

Density of image sets is positive except for specific rational-related angles.

Provides bounds on the size of image and preimage sets under these transformations.

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 $\sin \varphi \not= \pm \cos \varphi + r, r\in \mathbb{Q}$.