Planar Symmetric Pattern Generation

Generating objects with specific symmetries is essential in various real-world scenarios, with the creation of patterns exhibiting planar group symmetries being a representative task. However, adapting existing 2D continuous representations to enforce symmetry remains a significant challenge, as the transformation of non-reflective group elements may disrupt continuity. To overcome this limitation, we propose a novel symmetric continuous representation framework for arbitrary planar groups. Our method transforms any underlying 2D continuous representation into a symmetric one while strictly preserving continuity. We provide the mathematical formulation of this representation, demonstrate its universal approximation capability for symmetric functions, and detail the construction methodology. We validate our approach through three distinct generation tasks: general symmetric pattern design, connectivity-preserving paper-cutting design, and mechanically-constrained geometric stylized patterns. Experimental results confirm that our representation enables precise and effective symmetry control in pattern generation.