Definition:Reflection (Geometry)/Space

Definition
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 $OP = OP'$.


 * Reflection-in-Space.png

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) $S$.