\begin{definition}[Definition:Reflection (Geometry)/Plane]
A '''reflection''' $\phi_{AB}$ in the plane is an isometry on the Euclidean Space $\Gamma = \R^2$ as follows.
Let $AB$ be a distinguished straight line in $\Gamma$, which has the property that:
:$\forall P \in AB: \map {\phi_{AB} } P = P$
That is, every point on $AB$ maps to itself.
Let $P \in \Gamma$ such that $P \notin AB$.
Let a straight line be constructed from $P$ to $O$ on $AB$ such that $OP$ is perpendicular to $AB$.
Let $PO$ be produced to $P'$ such that $OP = OP'$.
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Then:
:$\map {\phi_{AB} } P = P'$
Thus $\phi_{AB}$ is a '''reflection (in the plane) in (the axis of reflection) $AB$'''.
\end{definition}