Definition:Reflection (Geometry)/Plane

Definition
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 $OP = OP'$.


 * Reflection-in-Plane.png

Then:
 * $\map {\phi_{AB} } P = P'$

Thus $\phi_{AB}$ is a reflection (in the plane) in (the axis of reflection) $AB$.