Escher tiling is well known as a tiling that consists of one or a few recognizable figures, such as animals. The Escherization problem involves finding the most similar shape to a given goal figure that can tile the plane. However, it is easy to imagine that there is no similar tile shape for complex goal shapes. This article devises a method for finding a satisfactory tile shape in such a situation. To obtain a satisfactory tile shape, the tile shape is generated by deforming the goal shape to a considerable extent while retaining the characteristics of the original shape. To achieve this, both goal and tile shapes are represented as triangular meshes to consider not only the contours but also the internal similarity of the shapes. To measure the naturalness of the deformation, energy functions based on traditional as-rigid-as-possible shape modeling are incorporated into a recently developed framework of the exhaustive search of the templates for the Escherization problem. The developed algorithms find satisfactory tile shapes with natural deformations for fairly complex goal shapes.

中文翻译：

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基于尽可能刚性形状建模的大变形 Escherization

埃舍尔瓷砖是众所周知的瓷砖，由一个或几个可识别的人物组成，例如动物。Escherization 问题涉及找到与可以平铺平面的给定目标图形最相似的形状。但是，很容易想象复杂的目标形状没有类似的瓷砖形状。本文设计了一种在这种情况下找到令人满意的瓷砖形状的方法。为了获得满意的瓦片形状，瓦片形状是通过使目标形状在相当程度上变形而生成的，同时保留原始形状的特征。为了实现这一点，目标和瓷砖形状都表示为三角形网格，不仅要考虑轮廓，还要考虑形状的内部相似性。为了测量变形的自然度，基于传统尽可能刚性的形状建模的能量函数被合并到最近开发的框架中，该框架对 Escherization 问题的模板进行了详尽的搜索。开发的算法为相当复杂的目标形状找到了具有自然变形的令人满意的瓷砖形状。