In the pattern formation problem, robots in a system must self-coordinate to form a given pattern, regardless of translation, rotation, uniform-scaling, and/or reflection. In other words, a valid final configuration of the system is a formation that is \textit{similar} to the desired pattern. While there has been no shortage of research in the pattern formation problem under a variety of assumptions, models, and contexts, we consider the additional constraint that the maximum distance traveled among all robots in the system is minimum. Existing work in pattern formation and closely related problems are typically application-specific or not concerned with optimality (but rather feasibility). We show the necessary conditions any optimal solution must satisfy and present a solution for systems of three robots. Our work also led to an interesting result that has applications beyond pattern formation. Namely, a metric for comparing two triangles where a distance of $0$ indicates the triangles are similar, and $1$ indicates they are \emph{fully dissimilar}.

中文翻译：

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使用移动机器人系统最小化形成图案的最大行驶距离

在图案形成问题中，系统中的机器人必须自协调以形成给定的图案，而不管平移、旋转、均匀缩放和/或反射。换句话说，系统的有效最终配置是与所需模式 \textit{similar} 的构造。虽然在各种假设、模型和背景下对模式形成问题的研究并不缺乏，但我们考虑了系统中所有机器人之间的最大行进距离最小的附加约束。模式形成方面的现有工作和密切相关的问题通常是特定于应用程序的或不关心最优性（而是可行性）。我们展示了任何最佳解决方案必须满足的必要条件，并为三个机器人的系统提供了一个解决方案。我们的工作还导致了一个有趣的结果，该结果具有超出模式形成的应用范围。即，用于比较两个三角形的度量，其中距离 $0$ 表示三角形相似，$1$ 表示它们 \emph{完全不同}。