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 $OP = OP'$.

Then:

$\map {\phi_{AB} } P = P'$

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

Axis of Reflection

Let $\phi_{AB}$ be a reflection in the plane in the straight line $AB$.

Then $AB$ is known as the axis (of reflection) of $\phi_{AB}$.

Also see

• Results about geometric reflections can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): reflection: 2. (in a line)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): reflection: 2. (in a line)
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): reflection (of the plane)