The Escherization problem involves finding a closed figure that tiles the plane that is most similar to a given goal figure. In Koizumi and Sugihara's formulation of the Escherization problem, the tile and goal figures are represented as $n$-point polygons where the similarity between them is measured based on the difference in the positions between the corresponding points. This paper presents alternative similarity measures (distance functions) suitable for this problem. The proposed distance functions focus on the similarity of local structures in several different manners. The designed distance functions are incorporated into a recently developed framework of the exhaustive search of the templates for the Escherization problem. Efficient exhaustive and incomplete search algorithms for the formulated problems are also developed to obtain results within a reasonable computation time. Experimental results showed that the proposed algorithms found satisfactory tile shapes for fairly complicated goal figures in a reasonable computation time.
翻译:分离问题涉及找到一个封闭数字,将平面打成一个与特定目标数字最相似的平面。在小泉和杉原对分离问题的表述中,瓷砖和目标数字以美元点多边形表示,根据相应点之间的位置差异来测量它们之间的相似性。本文件介绍了适合于这一问题的相似性替代措施(远程功能)。拟议的远程功能以不同方式侧重于地方结构的相似性。设计远程功能被纳入最近开发的彻底搜索分离问题模板的框架。还开发了高效的详尽和不完整的预测问题搜索算法,以便在合理的计算时间内获得结果。实验结果表明,拟议的算法发现,在合理的计算时间里,相当复杂的目标数字的瓷形形状是令人满意的。