Planar Symmetric Pattern Generation
2026-06-01 • Machine Learning
Machine Learning
AI summaryⓘ
The authors address the challenge of creating 2D shapes that follow specific patterns of symmetry without breaking their smoothness. They developed a method that can take any 2D continuous shape and make it symmetric, while keeping it smooth and continuous. They tested their approach on various design tasks, like patterns and materials, showing it works well to control symmetry. Their method applies to many types of 2D symmetry groups.
planar group symmetry2D continuous representationsymmetrizationnon-reflective symmetrypattern designtopology designmaterial designcontinuityapproximation theory
Authors
Ning Lin, Luxi Chen, Huaguan Chen, Jiacheng Cen, Chongxuan Li, Wenbing Huang, Hao Sun
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.