\begin{definition}[Definition:Reflection (Geometry)/Space]
A '''reflection''' $\phi_S$ in space is an isometry on the Euclidean Space $\Gamma = \R^3$ as follows.
Let $S$ be a distinguished plane in $\Gamma$, which has the property that:
:$\forall P \in S: \map {\phi_S} P = P$
That is, every point on $S$ maps to itself.
Let $P \in \Gamma$ such that $P \notin S$.
Let a straight line be constructed from $P$ to $O$ on $S$ such that $OP$ is perpendicular to $S$.
Let $PO$ be produced to $P'$ such that $OP = OP'$.
:600px
In the above diagram, $ABCD$ is in the plane of $S$.
Then:
:$\map {\phi_S} P = P'$
Thus $\phi_S$ is a '''reflection (in space) in (the plane of reflection) $S$'''.
\end{definition}