Turning two rotation groups into one

I need to figure out how to turn two rotation groups, each rotating around Z, X then Y into a single rotation group, so that in one set of rotations I might obtain the same positions for a set of rotated points as after two consecutive sets of ZXY rotations.

So for a rotation group given by

$$R_{zxy}(\psi, \varphi, \theta) = \begin{bmatrix} \cos \psi & -\sin \psi & 0 \\ \sin \psi & \cos \psi & 0\\ 0 & 0 & 1\\ \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \varphi & -\sin \varphi \\ 0 & \sin \varphi & \cos \varphi \\ \end{bmatrix} \begin{bmatrix} \cos \theta & 0 & \sin \theta \\ 0 & 1 & 0 \\ -\sin \theta & 0 & \cos \theta \\ \end{bmatrix}$$

I need to find $\psi_3, \varphi_3, \theta_3$ from $\psi_1, \varphi_1, \theta_1$ and $\psi_2, \varphi_2, \theta_2$ so that for a point P

$$P' = P R_{zxy}(\psi_3, \varphi_3, \theta_3) = P R_{zxy}(\psi_1, \varphi_1, \theta_1) R_{zxy}(\psi_2, \varphi_2, \theta_2)$$

• What you mean is that you want to combine two rotations, not rotation groups, into one. A rotation group is an entire set of rotations with a particular structure. Commented Jul 12, 2015 at 18:33
• Oh I looked on Wikipedia and it seemed like that was what a SO(3) rotation group was. But yes I am talking about combining two 3D rotations. Commented Jul 13, 2015 at 9:03