Euler angles

Remarks

Every $R \in SO(3)$ has a triple of Euler angles, and the triple is unique when $0 < \beta < \pi$. The angle $\beta$ is determined by $\cos\beta = R_{33}$, the entry in the third row and third column.

Example: the triple $(\pi/2, 0, 0)$ gives $R = R_z(\pi/2)$, the quarter turn about the $z$-axis. But $(0, 0, \pi/2)$ and $(\pi/4, 0, \pi/4)$ give the same $R$, since $R_x(0)$ is the identity; at $\beta = 0$ only $\alpha + \gamma$ is determined (gimbal lock).

Non-example: the product $R_z(\alpha) R_z(\beta) R_z(\gamma) = R_z(\alpha + \beta + \gamma)$ is not of the form in clause 2. Such products only give rotations about the $z$-axis, so three rotations about one axis cannot serve as Euler angles.

Other conventions use other axis sequences. Proper Euler angles have equal first and third axes, such as $z$-$y$-$z$. Tait–Bryan angles use three distinct axes, such as yaw, pitch and roll about $z$, $y$, $x$. The product $R_z(\alpha) R_x(\beta) R_z(\gamma)$ can be read as rotations about the fixed axes in the order $\gamma$, $\beta$, $\alpha$. It can also be read as rotations about the moving body axes in the order $\alpha$, $\beta$, $\gamma$.