How do you KNOW the Euler (Tait) angles cover orienations

[observer1]

Well, that question just about states my issue.

We have a body and we rotate about, say, the 3-axis of its body frame.
Then, we must do the next rotation about the 1 or 2 axis.

Let me say we choose the 1-axis
Then we have a choice: continue on to the 2 axis or repeat the 3 axis.

One set is called the Euler angles: precession, nutation, spin
The other set is called the Tait-Bryan angles: pitch, yaw, roll

So now my question is HOW do you KNOW that you have covered all rotations?

I mean, I can see by a geometric argument.
But how do you KNOW it other than by geometry?
Because it seems rather odd, from a distance, that that final choice of either going to the last axis (2) or repeating the first axis (3) SHOULD cover all orientations.

(I understand the proof of how the space of orientations is a 3 dimensional sub manifold of R9. I get that... I follow the proof. I just don't get how one can be so sure, without testing it or without geometry, that the two sequences Euler or Tait can define the orientation (and actually be the three variables)
'

 

[Dale]

So now my question is HOW do you KNOW that you have covered all rotations?

I mean, I can see by a geometric argument.
But how do you KNOW it other than by geometry?
...

(I understand the proof of how the space of orientations is a 3 dimensional sub manifold of R9. I get that... I follow the proof.

Hmm, I am not sure what you expect. You understand the geometry and you understand the more abstract proof. What more could we possibly add to that? I am also not sure what you mean by without geometry since it is inherently a geometric question.

I mean, can you describe what form of an answer you are looking for? I just think that it is unlikely we can give a helpful answer as it is.

 

[observer1]

Hmm, I am not sure what you expect. You understand the geometry and you understand the more abstract proof. What more could we possibly add to that? I am also not sure what you mean by without geometry since it is inherently a geometric question.

YES, I do understand the abstract proof, BUT that proof only proves that a rotation matrix is a 3D submanifold

It does NOT prove WHICH three PHYSICAL angles you need.

Well, perhaps you answered my question with your sincere and transparent objection.

But just in case (if you are so inclined to make one more pass), I suppose I am "startled" that you can rotate about 3, then, 1 and back to 3 and define the orientation.

I suppose, in my "brute" ignorance, I would be content with thinking: 3, then 1 then 2.

But even for both, i suppose I am surprised that either work. I mean, these are intrinsic rotations and, ASIDE from geometry and proving that you can define the orientation, how do you KNOW in advance, that they do?

Or perhaps you are right. Perhaps my incredulity is, in itself, naive.

I don't know.

 

[Dale]

But even for both, i suppose I am surprised that either work. I mean, these are intrinsic rotations and, aside from geometry and proving that you can define the orientation, how do you KNOW in advance, that they do?

Well, chances are that your sense of surprise (an emotional response) cannot be resolved by a mathematical proof. But you could at least convince yourself logically by mapping these intrinsic rotations to the complete space of extrinsic rotations.

 

[observer1]

Well, chances are that your sense of surprise (an emotional response) cannot be resolved by a mathematical proof. But you could at least convince yourself logically by mapping these intrinsic rotations to the complete space of extrinsic rotations.

So is that really it?
I suppose I can accept it.
I can accept that, geometrically, I can prove it.
I suppose I was expecting something more algebraic.

Like maybe: if I multiply out the three different rotation matrices, I get a new rotation matrix where all nine terms are different, so we do cover the space.
Or this
We can use euler parameters and map the result to either euler angles or tait angles.

I only can inuit such a proof right now. I suppose I can work it out later. But I am just surprised that there is no simple algebraic statement that.. well.. in choosing Tait or Euler... "we chose wisely."

 

[Dale]

I suppose I was expecting something more algebraic.

Like maybe: if I multiply out the three different rotation matrices, I get a new rotation matrix where all nine terms are different

I suspect that does exist. I may be able to work that out later.

 

[observer1]

I suspect that does exist. I may be able to work that out later.

I suspect that does exist. I may be able to work that out later.

And not to put too much pressure on you... And I will be waiting :-) (with gracious patience and hope)
Because this issue is freaking me out.