\begin{definition}[Definition:Reflection (Geometry)/Point]
A '''point reflection''' $\psi_O$ in space is an isometry on the Euclidean Space $\Gamma = \R^3$ as follows.
Let $O$ be a distinguished point in $\Gamma$, called the '''inversion point''', 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 $PO$ be produced to $P'$ such that $OP = OP'$.
Then:
:$\map {\psi_O} P = P'$
Thus $\phi_S$ is a '''point reflection (in space) in (the inversion point) $O$'''.
\end{definition}