Rotation

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Rotation is a transformation where the coordinates are rotated around the origin of the coordinate system. In this article rotation is first described for two dimensions.

The figure on the right shows an example, where the vector $\vec{\mathbf{q}}_0$ is rotated by $\varphi$ around the origin, what results in vector $\vec{\mathbf{q}}_1$. The length of the vector $\vec{\mathbf{q}}_0$ is assumed as $l$ and so the length of $\vec{\mathbf{q}}_0$ is $l$ as well. The initial angle of $\vec{\mathbf{q}}_0$ relative to the x-axis is $\alpha$. Hence the resulting coordinates $x_1$ and $y_1$ can be computed as follows:

$x_1=l\cdot\cos(\alpha+\varphi)$

$y_1=l\cdot\sin(\alpha+\varphi)$

Using the addition theorems of sine and cosine leads to:

$x_1=l\cdot(\cos\alpha\cos\varphi-\sin\alpha\sin\varphi)$

$y_1=l\cdot(\sin\alpha\cos\varphi-\cos\alpha\sin\varphi)$