Incremental and Transitive Discrete Rotations

Abstract

A discrete rotation algorithm can be apprehended as a parametric map f _α from $\mathbb Z[i]$ to $\mathbb Z[i]$, whose resulting permutation “looks like” the map induced by an Euclidean rotation. For this kind of algorithm, to be incremental means to compute successively all the intermediate rotated copies of an image for angles in-between 0 and a destination angle. The discretized rotation consists in the composition of an Euclidean rotation with a discretization; the aim of this article is to describe an algorithm which computes incrementally a discretized rotation. The suggested method uses only integer arithmetic and does not compute any sine nor any cosine. More precisely, its design relies on the analysis of the discretized rotation as a step function: the precise description of the discontinuities turns to be the key ingredient that makes the resulting procedure optimally fast and exact. A complete description of the incremental rotation process is provided, also this result may be useful in the specification of a consistent set of definitions for discrete geometry.