I need a little help to follow a paper I am reading for a project.
I have a parallelogram made by sides $A$ and $B$ and an acute angle $\gamma$ described on the plane $XZ$. I aim to rotate this parallelogram by $\alpha$ on the plane $YX$ pivoting on point $q$.
This results in the parallelogram to "penetrate" the plane $XZ$ (orange section)...
...which can be fixed by rotating the parallelogram along the red axis (seen below) by $\beta$ until point $p$ snaps back to plane $XZ$.
I would like to know how to establish a relationship between these parameters so I can find $\beta$ by inputting the other variables. After this step, I might be able to find the dihedral angle between the parallelogram and plane $XZ$.
Since you’re pivoting around the point $q$, you might as well place the origin there. Also, w.l.o.g we can take the length of the side $pq$ to be one. The problem is then reduced to working with the unit vector $(-\sin\gamma,0,-\cos\gamma)$.
The first rotation is through an angle of $\alpha$ about the $z$-axis. Using Rodrigues’ rotation formula or a basic rotation matrix, the resulting vector is $$(-\cos\alpha\sin\gamma,-\sin\alpha\sin\gamma,-\cos\gamma).$$ We now rotate this about the axis $(\cos\alpha,\sin\alpha,0)$ through the unknown angle $\beta$ to obtain $$(-\sin\alpha\sin\beta\cos\gamma-\cos\alpha\sin\gamma,\cos\alpha\sin\beta\cos\gamma-\sin\alpha\sin\gamma,-\cos\beta\cos\gamma).$$ We want the $y$-coordinate of this vector to vanish. If neither $\cos\alpha$ nor $\cos\gamma$ vanishes, then $$\sin\beta = \tan\alpha\tan\gamma.$$ Note that for some values of $\alpha$ and $\gamma$, the value of the right-hand side is outside the range of the $\sin$ function, so it’s not always possible to “fix” the paralellogram. On the other hand, if either $\cos\alpha$ or $\cos\gamma$ vanishes, then the $y$-coordinate reduces to $-\sin\alpha\sin\gamma$ and either the edge is already on the $x$-$z$ plane or there’s no rotation about the adjacent edge that will place it there.