# Rotations

A rotation on the plane with center $O$ and angle $\alpha$ is a direct movement that makesa point $P$ map to another point $P'$ so that:

$$\overline{PO}=\overline{P'O'} \quad \text{ and } \quad \widehat{POP'}=\alpha$$

We represent the function rotation as $g(O, \alpha)$. The angle $\alpha$ is known by the name of argument. As the rotation is a direct and isometric transformation, we can associate a two-dimensional equations system:

$$\begin{pmatrix} x'_1 \\ x'_2 \end{pmatrix} = \begin{pmatrix} \cos(\alpha) & -\sin(\alpha) \\ \sin(\alpha) & \cos(\alpha) \end{pmatrix} \cdot \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$$

where the matrix:

$A=\begin{pmatrix} \cos(\alpha) & -\sin(\alpha) \\ \sin(\alpha) & \cos(\alpha) \end{pmatrix}$

is the rotation matrix and it says how the elements of the plane move by means of the transformation. Notice, also, that thanks to the fact that the rotation matrix of the system is composed by sines and cosines, the rotation transformation is periodic of reason $360^\circ$.

Since we have already said previously, the rotation is a direct and isometric transformation, since its determinant is $1$, which indicates that it satisfies the following equality:

$$d (P, Q) = d (g (P), g (Q)) = d (P ', Q')$$

being $g$ the function turned with an $\alpha$ arbitrary angle.

Finally, we observe that the inverse of the rotation is the same rotation but with the angle of rotation sign, that is to say, that its argument is $-\alpha$.

To finish with this section, we will tell how to proceed when calculating the transformed of the three most elementary objects that exist on the plane, as we have already done in the case of the translations.

• Rotation of segments: To calculate the rotation of a segment, it is enough to calculate the transformed of the endpoints and to join them to obtain the transformed segment.
• Straight lines rotation: It is enough to calculate the transformed of two points of the straight line and join them to obtain the transformation of the straight line.
• Angle rotation: As an angle is given by the intersection of two sides, it is enough to apply the rotation to each of its sides to obtain the transformed angle.

We want to calculate the centre rotation $O$ and angle $\alpha= 60^\circ$ of the vector $x = (2,2)$.

By means of the formulation with matrices, we realize that first we have to calculate the value of the rotation matrix:

$$A=\begin{pmatrix} \cos(\frac{\pi}{3}) & -\sin(\frac{\pi}{3}) \\ \sin(\frac{\pi}{3}) & \cos(\frac{\pi}{3}) \end{pmatrix}= \begin{pmatrix} \frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix}$$

Note that in radians, $60^\circ$ are $\dfrac{\pi}{3}$.

Therefore, the transformed is:

$$\begin{pmatrix} y_1 \\ y_2 \end{pmatrix} = \begin{pmatrix} \frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix} \cdot \begin{pmatrix} 2 \\ 2 \end{pmatrix} = \begin{pmatrix} 1-\sqrt{3} \\ 1+\sqrt{3} \end{pmatrix}$$