Planar Symmetric Pattern Generation

2026-06-01

AI Summary

Existing 2D continuous representations struggle to simultaneously preserve continuity and satisfy arbitrary plane group symmetries, particularly because non-reflection operations often disrupt continuity. This work proposes the first general-purpose symmetrization framework that rigorously enforces full plane group symmetry—including non-reflection operations—while maintaining continuity in 2D continuous representations. By integrating group-theoretic modeling with approximation theory for continuous functions, the method transforms any 2D continuous representation into one that strictly adheres to prescribed symmetries without compromising smoothness. The approach is validated across four diverse applications: pattern design, kirigami art, stylized topology, and material design, demonstrating high-fidelity, controllable generation of symmetric patterns. This study thus achieves, for the first time, full compatibility between general plane group symmetries and continuous 2D representations.

Abstract

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.

Problem

• planar symmetry
• continuous representation
• symmetry enforcement
• 2D pattern generation
• group transformation

Innovation

• planar symmetry
• continuous representation
• symmetrization framework
• group-equivariant
• function approximation