Euler, Tait-Bryan, Tait, proper, Improper

[Trying2Learn]

TL;DR Summary

Euler, Tait-Bryan, Tait, proper, Improper: total confusion

Can I try again?

I have seen (on the web), all these names, DISTINCTLY: Euler, Tait-Bryan, Tait, proper, Improper

I am still trying to make sense of this and am facing some strange naming conventions. I now can see this (the actual math does not concern me--it is only the names that cause me confusion):

There are 12 possible rotations

121 131 212 232 313 323
123 132 213 231 312 321

All are called Euler angles (which is odd to me)

If the third axis is NOT repeated (red ones, above): they are called Tait-Bryan angles
If third one is repeated, they are called Euler angles (YES; USING THE SAME NAME--should this have been called Euler ROTATION PROCESS to distinguish from Euler Angles)

Then there are proper/intrinsic vs improper/extrinsic:

When the rotations are improper/ EXtrinsic: it means they happen about fixed spatial axes (the axes are EXternal to the body)

When the rotations are proper/INtrinsic: it means they happen about fixed spatial axes (the axes are INternal to the body--attached to it)

This suggests FOUR cases:

1. Improper Euler
2. Proper Euler
3. Improper Tait-Bryan
4. Proper Tait-Bryan

Sometimes, I read about Tait angles (without any mention of Bryan) and some seem to call them Euler, but I assume they are referencing the very nature of angles (12 sets) and not the distinction on whether an axis is repeated. I think.

Can someone comment on this?
I cannot quote a source but so many websites do this.
It seems to me that Wikipedia gets it right: https://en.wikipedia.org/wiki/Euler_angles
But they fail to address other possibilities
And they fail to explain how and why each is used (limits, advantages)

Do I have it correctly?

Finally, with planes, we talk about Tait-Bryan, but with a condition: 1-faces forward, 2-to the right and 3-down In other words, these angles are mapped to how planes fly, and the names, in order of the sentence just above this one, is: pitch, yaw, roll

What is the convention with ships?

My head his spinning. ChatGPT, I think, makes this even worse and gets it all wrong.

I now also think that one cannot discuss steady precession when an axis is repeated: it only happens in the BLUE set, above.
In other words, there is no corresponding notion of steady precession with Tait-Bryan because all axes are different.

 

[vanhees71]

Naming conventions are just, well, conventions and may depend on the author using them. One just has to make sure to check the definition of the angles used to describe rotations.