Title: Creating Symmetric Patterns on a Plane

Abstract: The ability to generate objects with specific symmetries is crucial for numerous practical applications. Yet, adapting existing 2D continuous representations to enforce planar group symmetry presents a significant hurdle, particularly because transforming non-reflective group elements can compromise continuity. To address this issue, we introduce a symmetrization framework designed for arbitrary planar groups. Our technique converts any 2D continuous representation into a symmetric version without sacrificing continuity. We present the mathematical foundation of this representation, prove its capacity to approximate symmetric functions, and outline the construction process. We evaluate our method across three visual design tasks—pattern design, paper-cutting design, and stylized topology design—as well as one material design task. Our results confirm that this representation allows for precise symmetry control and highlights its wide-ranging potential.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC