# Definition:Rotation (Geometry)/Plane

< Definition:Rotation (Geometry)(Redirected from Definition:Plane Rotation)

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## Definition

A **rotation** $r_\alpha$ in the plane is an isometry on the Euclidean Space $\Gamma = \R^2$ as follows.

Let $O$ be a distinguished point in $\Gamma$, 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 a straight line $OP'$ be constructed such that:

- $(1): \quad OP' = OP$
- $(2): \angle POP' = \alpha$ such that $OP \to OP'$ is in the anticlockwise direction:

Then:

- $\map {r_\alpha} P = P'$

Thus $r_\alpha$ is a **rotation (in the plane) of (angle) $\alpha$ about (the axis) $O$**.