The visualization of hyperbolic patterns from invariant mapping method

A fast algorithm is developed to transform an arbitrary point into fundamental regions, which avoids the intrinsic difficulty of the infiniteness of hyperbolic groups. A flexible construction of invariant mappings, independent of the mapping form, is achieved by imposing the continuity of the mapping on its parameters. The fast algorithm combined with the resulting invariant mapping contributes to a simple and efficient method to generate various aesthetic hyperbolic patterns.

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► A fast algorithm that transforms any point into the fundamental region is proposed.
► A flexible construction of mappings, independent of its form, is developed.
► A simple and efficient method is proposed to generate hyperbolic patterns.