Definition:Reflection (Geometry)/Point

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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 $OP = OP'$.


$\map {\psi_O} P = P'$

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.