I was recently playing with a CAD 3D modelling program. Once you rotate a part using the mouse and let go, it keeps rotating the part using the rotation matrix provided by the mouse input.
The rotation is therefore given by:
$ \begin{align} R_t &= R_x \cdot R_y \cdot R_z \cdot t \\ &= \left[ \begin{matrix} 1 & 0 & 0 \\ 0 & \cos(\phi) & \sin(\phi) \\ 0 & -\sin(\phi) & \cos(\phi) \end{matrix} \right] \cdot \left[ \begin{matrix} \cos(\theta) & 0 & -\sin(\theta) \\ 0 & 1 & 0 \\ \sin(\theta) & 0 & \cos(\theta) \end{matrix} \right] \cdot \left[ \begin{matrix} \cos(\psi) & \sin(\psi) & 0 \\ -\sin(\psi) & \cos(\psi) & 0 \\ 0 & 0 & 1 \end{matrix} \right] \cdot t \\ &= \left[ \begin{matrix} \cos(\psi) \cos(\theta) & \sin(\psi) \cos(\theta) & -\sin(\psi) \\ -\sin(\psi) \cos(\phi) + \cos(\psi)\sin(\theta)\sin(\phi) & \cos(\psi) \cos(\phi) + \sin(\psi)\sin(\theta)\sin(\phi) & \cos(\theta) \sin(\phi) \\ \sin(\psi) \sin(\phi) + \cos(\psi)\sin(\theta)\cos(\phi) & -\cos(\psi) \sin(\phi) + \sin(\psi)\sin(\theta)\cos(\phi) & \cos(\theta) \cos(\phi) \end{matrix} \right] \cdot t \end{align} $ Where $t$ is the current time (ever increasing) and $\phi, \theta, \psi$ are the rotational inputs given by the mouse.
My question is rather simple;
- Does there exist a single combination of $\phi, \theta, \psi$ such that all possible rotations can be achieved by increasing the time $t$?
Note: I'm not from a mathematical background, so my apologies for any incorrections in my question statement.
$R_t$ is not a rotation except if $t=1$. A rotation depends exactly on $3$ parameters, for example $\phi,\theta,\psi$.
EDIT. I think that, when you play, you select (with the mouse) an angle amongst the $3$ above, for instance $\phi$; then $\phi$ is changed with $\phi\pm t$ where $t$ is the time. If you want to reach any rotation, then you must successively make a change of all $3$ angles.