Planar Symmetric Pattern Generation
It provides a general method for enforcing planar symmetries in continuous representations, addressing a bottleneck in generative design for symmetric patterns.
The paper proposes a symmetrization framework that transforms any 2D continuous representation into one with planar group symmetry while preserving continuity, validated across four design tasks.
Generating objects with specific symmetries is essential in various real-world scenarios. However, adapting existing 2D continuous representations to enforce planar group symmetry remains a challenge, as the transformation of non-reflective group elements may disrupt continuity. To overcome this limitation, we propose a symmetrization framework for arbitrary planar groups. Our method transforms any 2D continuous representation into a symmetric one while preserving continuity. We provide the mathematical formulation of this representation, demonstrate its approximation capability for symmetric functions, and detail the construction methodology. We validate our approach through three visual design tasks (pattern design, paper-cutting design and stylized topology design) and one material design task. Experiments confirm that our representation enables effective symmetry control and demonstrate its broader applicability.