Today I read a paper titled “Incremental and Transitive Discrete Rotations”
The abstract is:
A discrete rotation algorithm can be apprehended as a parametric application $f\_\alpha$ from $\ZZ[i]$ to $\ZZ[i]$, 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 rotate d copies of an image for angles in-between 0 and a destination angle.
The di scretized rotation consists in the composition of an Euclidean rotation with a discretization; the aim of this article is to describe an algorithm whic h computes incrementally a discretized rotation.
The suggested method uses o nly integer arithmetic and does not compute any sine nor any cosine.
More pr ecisely, its design relies on the analysis of the discretized rotation as a step function: the precise description of the discontinuities turns to be th e key ingredient that will make the resulting procedure optimally fast and e xact.
A complete description of the incremental rotation process is provided, also this result may be useful in the specification of a consistent set of defin itions for discrete geometry.