An Escher-like tiling is a tiling consisting of one or a few artistic shapes of tile. This article proposes a method for generating Escher-like tilings consisting of two distinct shapes (dihedral Escher-like tilings) that are as similar as possible to the two goal shapes specified by the user. This study is an extension of a previous study that successfully generated Escher-like tilings consisting of a single tile shape for a single goal shape. Building upon the previous study, our method attempts to exhaustively search for which parts of the discretized tile contours are adjacent to each other to form a tiling. For each configuration, two tile shapes are optimized to be similar to the given two goal shapes. By evaluating the similarity based on as-rigid-as possible deformation energy, the optimized tile shapes preserve the local structures of the goal shapes, even if substantial deformations are necessary to form a tiling. However, in the dihedral case, this approach is seemingly unrealistic because it suffers from the complexity of the energy function and the combinatorial explosion of the possible configurations. We developed a method to address these issues and show that the proposed algorithms can generate satisfactory dihedral Escher-like tilings in a realistic computation time, even for somewhat complex goal shapes.

埃舍尔式瓷砖是由一个或几个艺术形状的瓷砖组成的瓷砖。本文提出了一种生成埃舍尔式平铺的方法，该方法由两个不同的形状（二面埃舍尔式平铺）组成，这些形状与用户指定的两个目标形状尽可能相似。这项研究是先前研究的延伸，该研究成功生成了由单个目标形状的单个瓷砖形状组成的埃舍尔式瓷砖。基于之前的研究，我们的方法尝试详尽地搜索离散瓦片轮廓的哪些部分彼此相邻以形成瓦片。对于每种配置，两个图块形状被优化为与给定的两个球门形状相似。通过基于尽可能刚性的变形能量评估相似性，优化的瓦片形状保留了目标形状的局部结构，即使需要大量变形来形成瓦片。然而，在二面角的情况下，这种方法似乎不切实际，因为它受到能量函数的复杂性和可能配置的组合爆炸的影响。我们开发了一种方法来解决这些问题，并表明所提出的算法可以在实际计算时间内生成令人满意的二面埃舍尔式平铺，即使对于有些复杂的目标形状也是如此。