Rotation

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

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.

The clock hands are rotating, the center of the clock being the fixed point. $$$$ The (perpendicular) distances of any of the points inside or on the surface of the earth from the axis do not change. $$$$ Multiplying a complex number by a complex number of unit magnitude rotates it about 0.

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,

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

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.

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?

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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. $_\square$