标题： 通过闭式解的埃舍尔瓷砖变形 显示英文标题

标题： Escher Tile Deformation via Closed-Form Solution

摘要： 我们提出了一种实时变形方法用于埃舍尔瓷砖——这种互锁的有机形态遵循对称规则，能够无缝地铺满平面。 我们将问题表述为确定一个周期性位移场。 目标是在不引入间隙或重叠的情况下变形埃舍尔瓷砖。 通过解析解，位移场以闭合形式获得。 我们的方法处理17种墙纸群的瓷砖，涵盖图像和网格等多种表示形式。 我们不仅仅将瓷砖视为边界，而是将其视为有纹理的形状，确保边界和内部同时变形。 为了实现精细的艺术输入，我们的交互式工具具有用户可控的自适应衰减参数，允许精确调整局部性，并支持具有有意义语义控制的变形。 我们通过各种示例展示了方法的有效性，包括照片编辑和形状雕刻，展示了其在制造和动画等应用中的使用。 显示英文摘要

摘要： We present a real-time deformation method for Escher tiles -- interlocking organic forms that seamlessly tessellate the plane following symmetry rules. We formulate the problem as determining a periodic displacement field. The goal is to deform Escher tiles without introducing gaps or overlaps. The resulting displacement field is obtained in closed form by an analytical solution. Our method processes tiles of 17 wallpaper groups across various representations such as images and meshes. Rather than treating tiles as mere boundaries, we consider them as textured shapes, ensuring that both the boundary and interior deform simultaneously. To enable fine-grained artistic input, our interactive tool features a user-controllable adaptive fall-off parameter, allowing precise adjustment of locality and supporting deformations with meaningful semantic control. We demonstrate the effectiveness of our method through various examples, including photo editing and shape sculpting, showing its use in applications such as fabrication and animation.