Mapping/Examples/Rotation through 30 Degrees

Example of Mapping
Let $\Gamma$ be the Cartesian plane.

The rotation $R_{30 \degrees}$ of $\Gamma$ anticlockwise through an angle of $30 \degrees$ about the origin $O$ is a mapping from $\Gamma$ to $\Gamma$.

Proof
For every point $P$ in $\Gamma$ which is not the origin $O$ it is possible to:
 * construct a straight line $OP$
 * construct a straight line $OP'$ at an angle of $30 \degrees$ to $OP$ measured in an anticlockwise direction
 * construct the point $P'$ such that $\len \paren {OP} = \len \paren {OP'}$.


 * Rotation-through-30-degrees.png

It can be seen that for every $P$ there is a unique $P'$ to which $R_{30 \degrees}$ maps $P$, apart from $O$ which is mapped to $O$.