How can Euler angles be visualized using a polar plot?

[DrDu]

Dear Forum,

say I am projecting an ellipsoid along the z-axis to the xy-Plane. The resulting ellipsis is rotated around the z-axis by the angle gamma until the principal axes coincide with the x- and y axis.
Now before projecting, I rotate the ellipsoid first around the z- and then around the y-axis by angles alpha and beta, respectively.
In effect, I get the Euler angle gamma as a function of alpha and beta and I would like to visualise this. Of course, I could plot gamma over alpha and beta, but intuitively, I would prefer to plot over a stereographic net with angular coordinates alpha and beta. However, In a stereographic projection, all points with different angle alpha at beta=0 are mapped to one point, but gamma becomes proportional to alpha, so this does not work.
I suppose this kind of problem of visualising Euler angles is not new. Do you have any ideas?

 

[DrDu]

I think I solved my problem: Writing $R_z(\alpha)R_y(\beta)R_z(\gamma)$ as $R_z(\alpha)R_y(\beta)R_z(-\alpha)R_z(\gamma+\alpha)=R_{y'(\alpha)}(\beta) R_z (\gamma+\alpha)$, I can plot $\alpha+\gamma$ as a function of the orientation of the new rotation axis $y'(\alpha)$ and the rotation angle $\beta$ on a polar plot.