Definition:Rotation (Geometry)/Plane

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) $O$.