I How can Euler angles be visualized using a polar plot?

AI Thread Summary
The discussion focuses on visualizing Euler angles using a polar plot, specifically when projecting an ellipsoid onto the xy-plane. The user describes a process of rotating the ellipsoid around the z-axis and then the y-axis, resulting in a function for the Euler angle gamma based on angles alpha and beta. They express a preference for a stereographic net for visualization but encounter challenges due to the mapping of points at beta=0. Ultimately, the user proposes a solution by reformulating the rotation matrices, allowing them to plot the combined angle of alpha and gamma against the new rotation axis orientation and angle beta on a polar plot. This approach aims to effectively visualize the relationship between the Euler angles.
DrDu
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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?
 
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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.
 
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