# Rotation

Rotation is a transformation in which a figure is turned about a given point.

## Definition

A rotation of an object around a point or an axis is a continuous transformation that does not change the distance of any of the points on the object from the point or the axis.

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## Rotation about a Point

## Formula

Set the origin $O=(0,0)$ as the center of rotation. When a point $P=(x,y)$ is rotated counterclockwise by angle $\theta$, the resulting point $P'(x',y')$ can be calculated with the formula

$\begin{pmatrix}x' \\ y'\end{pmatrix} = \begin{pmatrix} \cos \theta & -\sin\theta \\ \sin \theta & \cos \theta\end{pmatrix} \begin{pmatrix}x \\ y \end{pmatrix}.$

In other words,

$\begin{aligned} x' &= x\cos\theta - y\sin\theta \\ y' &= x\sin\theta + y\cos\theta.\end{aligned}$

## Rotational Symmetry

*See also: Symmetry*

**Rotational symmetry** occurs if an object can be rotated less than $360^\circ$ and be unchanged. The center of the rotation is called the *point of symmetry*.

Take for example some of the letters of the alphabet, which have rotational symmetry as they can be rotated $180^\circ$ and remain the same.

## Problem Solving

Consider a rotation $R$ about about point $(25,64)$ by $30^{\circ}$ clockwise. How many rotations will it take for $(81,62)$ to be rotated back to itself?

We have to rotate the point by $360^{\circ}$ to get it back. So the rotations required are $\frac {360^{\circ}}{30^{\circ}}$= $12$ rotations. $_\square$