Designing an algorithm that transforms each pixel back to motif in a fundamental region.

Abstract

Current algorithms to create repeating hyperbolic patterns transform the motif about the hyperbolic plane to points in the Poincaré circle model. This is inefficient near the bounding circle since the entire motif is drawn, even though it covers only few pixels. To avoid this shortcoming, we designed another algorithm that transforms each pixel in a motif in a fundamental region and then colors the original pixel using a color permutation of the color of the final pixel. This solves the inefficiency problems of the previous algorithms.