Algebraic proof that Euler angles define a proper rotation matrix

[Trying2Learn]

TL;DR Summary

I am looking for an algebraic proof that the Euler angles can define a proper rotation matrix

I have asked this question twice and each time, while the answers are OK, I am left dissatisfied.

However, now I can state my question properly (due to the last few responses).

Go to this page and scroll down to the matrix for sixth row of the proper Euler angles:
https://en.wikipedia.org/wiki/Euler_angles

Given ANY proper rotation matrix, I can compute the single angle that informs the third row, third column.
Good: one down.

Then, I proceed UP that third column and I find two equations for the first angle.
s1s2 = #
-c1s2 = different number

So assuming a superficial ignorance, I would have objected with: "there is a relationship between the sine and cosine, so those two terms cannot be any value."

BUT then I realize that there are restrictions on a proper rotation matrix: it must be orthogonal.

So if I wrote out the equations of orthogonality for a 3 by 3 matrix, I get nine equations: three that are equal to 1, and six that are equal to 0.

However, the six equal to 0, are not unique due to the transpose. So I really have only six equations for 9 numbers.

This means that 3 numbers can uniquely define a proper rotation matrix. So that is encouraging.

However, now I want to know why the Proper Euler Angles (or for that matter, the Tait-Bryan) can describe any proper rotation matrix.

But here is the issue: I do NOT want to "know" this by "observing" the rotations, geometrically (as everyone has shown me so far, and for which I am grateful). I want to know it with my eyes closed.

I am looking for a ALGEBRAIC proof that any matrix defined by six equations of orthogonality can be obtained by the trigonometric equations that result from the Proper Euler or Tait angles.

(And I am aware of the issue of gimbal lock; so do not focus on that.)

 

[A.T.]

Go to this page and scroll down to the matrix for sixth row of the proper Euler angles:
https://en.wikipedia.org/wiki/Euler_angles

The matrices in that table are the product of 3 basic rotation matrices (also given in the table):
https://en.wikipedia.org/wiki/Rotation_matrix#Basic_rotations

So for your proof you have to show that:
- Those 3 basic rotation matrices are orthogonal
- The product of two orthogonal matrices is also a orthogonal matrix

PS: This should be posted in the Linear Algebra forum.

 

[Trying2Learn]

The matrices in that table are the product of 3 basic rotation matrices (also given in the table):
https://en.wikipedia.org/wiki/Rotation_matrix#Basic_rotations

So for your proof you have to show that:
- Those 3 basic rotation matrices are orthogonal
- The product of two orthogonal matrices is also a orthogonal matrix

Yes, I know -- that I can do. (I can prove it and also realize that they form a Group). But that was not my question.

My question was: "how can you algebraically (without viewing it--without seeing it) prove that the three euler angles can "describe" all such orthogonal matrices."

 

[A.T.]

prove that the three euler angles can define all such orthogonal matrices."

For this you need to find a function that converts any given orthogonal matrix to Euler angles.

 

[Trying2Learn]

For this you need to find a function that converts any given orthogonal matrix to Euler angles.

Exactly! That is the very thing I am looking for. That is what I am hoping someone can show me.

I can see it VISUALLY.

But I am hoping someone can help me show it algebraically.

At the very least, THANK YOU (and all others) for helping me ASK the question.

Now I am just hoping for the answer

 

[A.T.]

Exactly! That is the very thing I am looking for. That is what I am hoping someone can show me.

Have you googled "rotation matrix to euler angles"?

 

[Trying2Learn]

Have you googled "rotation matrix to euler angles"?

Well imagine that! Now that I know what I am asking, I found it!

Thanks for your patience.