\begin{definition}[Definition:Rotation (Geometry)/Plane]
A '''rotation''' $r_\alpha$ in the plane is an isometry on the Euclidean Space $\Gamma = \R^2$ as follows.
Let $O$ be a distinguished point in $\Gamma$, which has the property that:
:$\map {r_\alpha} O = O$
That is, $O$ maps to itself.
Let $P \in \Gamma$ such that $P \ne O$.
Let $OP$ be joined by a straight line.
Let a straight line $OP'$ be constructed such that:
:$(1): \quad OP' = OP$
:$(2): \angle POP' = \alpha$ such that $OP \to OP'$ is in the anticlockwise direction:
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Then:
:$\map {r_\alpha} P = P'$
Thus $r_\alpha$ is a '''rotation (in the plane) of (angle) $\alpha$ about (the axis) $O$'''.
\end{definition}