Category:Geometric Reflections
This category contains results about reflections in the context of Euclidean Geometry.
Definitions specific to this category can be found in Definitions/Geometric Reflections.
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) $O$.
Subcategories
This category has the following 2 subcategories, out of 2 total.
E
R
- Reflectional Symmetries (empty)
Pages in category "Geometric Reflections"
The following 10 pages are in this category, out of 10 total.