How can Euler angles be visualized using a polar plot?

In summary, the conversation discusses the challenge of visualizing Euler angles and finding a solution for plotting them. The individual is projecting an ellipsoid and rotating it around different axes before projecting it onto a stereographic net. However, this method does not accurately represent the relationship between the Euler angles. The individual proposes a solution by writing the Euler angles in a different form and plotting them on a polar plot.
  • #1
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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  • #2
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.
 

1. What are Euler angles and why are they used in visualisation?

Euler angles are a set of three angles that represent the orientation of a rigid body in three-dimensional space. They are used in visualisation because they provide a simple and intuitive way to describe the rotation of an object.

2. How are Euler angles calculated and what do they represent?

Euler angles are calculated using a sequence of three rotations around the three axes of a coordinate system. The first angle represents the rotation around the z-axis, the second around the new y-axis, and the third around the new x-axis. They represent the orientation of an object in terms of its pitch, yaw, and roll.

3. What are the advantages and disadvantages of using Euler angles in visualisation?

The main advantage of using Euler angles is their simplicity and intuitive interpretation. However, they can also lead to issues such as gimbal lock, where certain orientations cannot be represented. Additionally, they can be prone to numerical errors and can be difficult to interpolate between.

4. How do Euler angles compare to other methods of visualising orientation?

Euler angles are one of several methods for representing orientation, including quaternions and rotation matrices. Each method has its own advantages and disadvantages, but Euler angles are often preferred for their simplicity and ease of use.

5. Can Euler angles be used to represent any orientation in three-dimensional space?

No, Euler angles have limitations and cannot represent all possible orientations in three-dimensional space. For example, gimbal lock and numerical errors can limit their range of representation. Other methods, such as quaternions, may be necessary for more complex orientations.

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