3.3.2.1 Euler's Angles

The angles which specify the relative orientation of the two coordinate systems are called Euler's angles. They are defined by the following rules. First a clockwise rotation around the $z$-axis is performed. This angle is usually denoted as $\alpha$. Then a clockwise rotation around the new $y$-axis follows. This second angle is denoted as $\beta$. Finally, a clockwise rotation around the new $z$-axis finishes the transformation. The last angle is denoted as $\gamma$. The range of these angles are determined as follows:

$$0\leqslant\alpha\leqslant 2\pi,$$

$$0\leqslant\beta\leqslant\pi,$$ (3.39)

$$0\leqslant\gamma\leqslant 2\pi.$$

S. Smirnov: