Are Tait and Euler angles a complete parametrization of 3D space?

[Trying2Learn]

TL;DR Summary

How do you know these angles cover all orientations?

Hi

With displacements, I KNOW that three orthogonal axes cover all of 3D Space.

What about rotations?

How do I KNOW that the Tait or Euler angles cover all orientations?

For Tait, I would almost "expect" it.
The object rotates about the local body axes in order of: one axis, then a second then a unique third.

For Euler, one rotates about one angle, then a second, then one repeats the first.

I would never have expected that, but I can see it with the gyroscope or a top.

Aside from OBSERVING that we can assert all orientations (ignoring the issue of gimbal lock) by the Euler angle sequence, how would one KNOW that that a repitition of the first axis rotation (but about the new local frame) covers all orientations?

For example, if one multiplies the rotation matrices of the three cases, can one make a statement about the final composite rotation matrix, to assert that it covers all orientations?

 

[kuruman]

Summary:: How do you know these angles cover all orientations?

Aside from OBSERVING that we can assert all orientations (ignoring the issue of gimbal lock) by the Euler angle sequence, how would one KNOW that that a repitition of the first axis rotation (but about the new local frame) covers all orientations?

We know by observing and applying logic. If three Euler angles can be found to relate an arbitrary orientation of a frame relative to another frame, then this can be done for any relative frame orientation.

 

[Trying2Learn]

We know by observing and applying logic. If three Euler angles can be found to relate an arbitrary orientation of a frame relative to another frame, then this can be done for any relative frame orientation.

I am sorry. I am a bit dense. Could you clarify this?When you say "any arbitrary" to another, how do you know it is truly arbitrary and not some special case?

How do you choose these two test cases?

 

[kuruman]

I am sorry. I am a bit dense. Could you clarify this?When you say "any arbitrary" to another, how do you know it is truly arbitrary and not some special case?

How do you choose these two test cases?

If the transformation angles are expressed symbolically as $\alpha,~ \beta,~\gamma$ and you can substitute any value you please for each one of them, then you can safely say that their values are not special because they can be anything. Anything is no special thing.

 

[Trying2Learn]

If the transformation angles are expressed symbolically as $\alpha,~ \beta,~\gamma$ and you can substitute any value you please for each one of them, then you can safely say that their values are not special because they can be anything. Anything is no special thing.

OK, your response helped a lot.

And I follow it.

But it enabled me to focus on exactly the issue that frustrates me.

Since it is now a new issue, I reposted it as a new thread. Could I ask you to turn there?

 

[kuruman]

OK, your response helped a lot.

And I follow it.

But it enabled me to focus on exactly the issue that frustrates me.

Since it is now a new issue, I reposted it as a new thread. Could I ask you to turn there?

I saw the new thread and I see that your actual question was how you set up an arbitrary rotation matrix $R(\alpha,\beta,\gamma)$ not how one is sure that it is arbitrary. Anyway, I am glad you got that settled.

 

[Trying2Learn]

I saw the new thread and I see that your actual question was how you set up an arbitrary rotation matrix $R(\alpha,\beta,\gamma)$ not how one is sure that it is arbitrary. Anyway, I am glad you got that settled.

Not a problem. It was my fault -- I just did not know what I was asking.

But your comment helped.

So thank you!

 

[vanhees71]

Here's an elementary proof for Euler angles as a complete parametrization of the SO(3). I hope that helps.

https://itp.uni-frankfurt.de/~hees/pf-faq/euler-angles.pdf