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Contemporary authors in various more-or-less literary genres occasionally affect this outdated spelling, but in mathematics texts written since the middle of the $20$th century, the reflexion spelling is vanishingly rare.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: $(5)$