Definition:Reflection (Geometry)/Point
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Definition
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) $O$.
Inversion Point
Let $\psi_O$ be a point reflection in the Euclidean Space $\Gamma = \R^3$.
The point $O$ in $\Gamma$ such that:
- $\map {\psi_O} O = O$
is called the inversion point of $\psi_O$.
Also see
- Results about geometric reflections can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): reflection: 1. (in a point)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): reflection: 1. (in a point)