  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}.
