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

Minimizing The Maximum Distance Traveled To Form Patterns With Systems of Mobile Robots