Abstract: To address the challenges of weak controllability and poor seam handling in existing hyperbolic pattern generation methods, this paper introduces a quasi-regular pattern model and proposes a novel approach for generating hyperbolic patterns. By combining the negative curvature characteristics of hyperbolic geometry with the quasi-regular pattern model, we construct an efficient pattern generation framework that includes fundamental region algorithms, invariant mapping construction, quasi-regular pattern coloring, and GPU acceleration. First, three types of hyperbolic symmetry groups and their fundamental domains are defined based on the Poincaré disk model, enabling the mapping of arbitrary points into the fundamental domain through symmetry transformations. Second, a two-step invariant mapping technique is proposed to resolve phase mismatch issues at boundary seams in traditional methods while preserving the internal texture features of the fundamental domain. Finally, the quasi-regular pattern model and GPU parallel computing are utilized to optimize algorithm efficiency, and conformal mapping techniques are introduced to expand artistic styles. Experimental results demonstrate that the patterns generated by this method significantly outperform traditional dynamical system models in terms of texture continuity, natural boundary transitions, and parameter control flexibility, achieving a generation efficiency of 0.000223 seconds per frame at 4K×4K resolution. This study provides a new technical pathway for generating complex patterns and highlights its potential in digital art and intelligent design applications.