In my engineering problem, I'm considering rotations of a solid body about two fixed axes (say, $\alpha$ about $x$ and $\beta$ about $y$, while $z$ is excluded). To my understanding, those are not the first two Euler angles from any convention, since in Euler's convention the consecutive rotations are performed about rotated axes, not the ones given in the original coordinate system.
I am trying to come up with a nice graphical representation of all the final positions of the solid body. Clearly, the $\alpha-\beta$ plane is not useful, since these two rotations do not commute.
I think, the best illustration would be a sphere corresponding to all the possible final positions of the body. Why a sphere? Well, have we used all the three angles, that would be the $SO(3)$ torsor, $\mathbb{RP}^3$. However, excluding a rotation about a fixed axis is equivalent to factoring over $S^1$, so we end up with $S^2$.
Were I initially using Euler $(\theta, \phi, \psi=0)$ angles, the sphere I'm talking about would correspond to treating $\theta$ and $\phi$ as angles in spherical coordinates. However, I cannot use this convention because omitting the last Euler angle is equivalent to excluding the rotation about the $z''$ axis, while in my problem the rotations about the original $z$ are excluded.
So how do I get such a sphere of all possible final positions in my case?
My guess is that I should simply track some fixed vector $\vec{v}$ attached to the solid body, and draw the tip of $R(\alpha, \beta)\vec{v}$. If so, how do I get the $R(\alpha, \beta)$ matrix?