# Definition:Reflection (Geometry)

## Definition

A **reflection** in the context of Euclidean geometry is an isometry from a Euclidean Space $\R^n$ as follows.

A **reflection** is defined usually for either:

- $n = 2$, representing the plane

or:

- $n = 3$, representing ordinary space.

### Reflection in the Plane

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) $AB$**.

### Reflection in 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 $OP = OP'$.

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) $S$**.

### Point Reflection in Space

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) $O$**.

## Also see

- Results about
**geometric reflections**can be found**here**.

## Linguistic Note

In older texts, you sometimes see the word **reflexion**, which is merely an archaic spelling of **reflection**.

Contemporary authors in various more-or-less literary genres occasionally affect this outdated spelling, but in mathematics texts written since the middle of the $20$th century, the **reflexion** spelling is vanishingly rare.

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 34$. Examples of groups: $(5)$