Quaternion coordinates

Coordinates can be introduced into four dimensional space accroding to

$$\begin{eqnarray*} q_0 & = & \cos \theta \\ q_1 & = & \sin \theta \cos \alpha ... ... \sin \theta \cos \beta \\ q_0 & = & \sin \theta \cos \gamma, \end{eqnarray*}$$

as well as the alternative,

$$\begin{eqnarray*} q_0 + iq_3 & = & \cos \rho e^{i\phi} \\ q_1 + i q_2 & = & \sin \rho e^{i \psi}. \end{eqnarray*}$$

The first corresponds to the axis-angle representation, the second to Euler angles [3].