提出可保持连续性的平面对称模式生成方法，实现任意对称群下的图案设计。
Planar Symmetric Pattern Generation
- 基于数学构造将任意2D连续表示转为对称形式
- 在图案、剪纸、拓扑与材料设计中验证对称控制效果
- 适合需要精确对称设计的视觉创作与材料工程场景
在多种现实场景中，生成具有特定对称性的物体至关重要。然而，现有二维连续表示难以有效施加平面群对称性，因为非反射对称元素的变换可能破坏连续性。为此，我们提出一种适用于任意平面群的对称化框架。该方法可将任意二维连续表示转化为对称表示，同时保持连续性。我们提供了该表示的数学形式，证明其对称函数的逼近能力，并详述构建方法。通过三项视觉设计任务（图案设计、剪纸设计、风格化拓扑设计）和一项材料设计任务验证该方法。实验表明，该表示能有效实现对称控制，并展现出广泛适用性。
原文摘要 · Abstract (English)
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.
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