# Uniqueness of Euler angles

1. Aug 5, 2010

### krishna mohan

Hi..

The wikipedia article on euler angles claims that the Euler angles in zxz convention are unique if we constrain the range they are allowed to take (except in the case of the gimbal lock).

This seems reasonable. But can someone give me a reference... a book or a paper where this is stated?

On searching the web, I was able to find some lecture notes which proved the above assertion. But it did not have references.

2. Aug 5, 2010

### Petr Mugver

Sorry, I don't get a thing... If you found lectures that proved the assertion, then in the same lectures the assertion must also have been stated, in something like this:

Theorem: (assertion)

Proof: (proof)

...or not?

3. Aug 5, 2010

### krishna mohan

Yes... the proof is there...

And I do believe the statement.. the theorem seems alright..

But just want to be sure....and would like to have something to give as a reference when I use this fact... I cannot give lecture notes as reference...

That is why I need a book or a paper....

4. Aug 6, 2010

### krishna mohan

I found a book..

Biedenharn, L. C.; Louck, J. D. (1981), Angular Momentum in Quantum Physics, Reading

One of the references in the wikipedia article....

The way I understand the theorem is like this...

The angle between the initial z axis and the final Z axis is beta...

If we know the position of the initial z axis and the final Z axes, then the two axes together form a plane... beta rotation must have been performed about an axis perpendicular to this plane...

This leaves two choices for the axis of rotation...either along z x Z direction or along Z x z direction.... choosing either of the two directions perependicular to the plane as the positive axis...

For one choice, if beta=theta....then for the other choice, beta= - theta.....

Also, the two choices are related by N(the line of nodes) going to - N....

Which can be accomplished by alpha going to alpha+ 180 deg....

Thus, assuming beta is not zero or 180.... the relative position of z and Z axes can be achieved by two choices of alpha and beta..

1) alpha and beta...

2) alpha+180 and -beta (or 360- beta)...

for beta zero or 180...it is easy to see that alpha angle is inconsequential...

We can fix one choice by requiring beta to be between 0 and 180....then alpha is also fixed..
[wikipedia article seems to suggest that beta between -90 and 90 will also work..but this goes against my argument...using mathematica, I got the result that the sets (alpha,beta,gamma) as (135,60,270) and (315,-60,90) gave the same rotation matrix...so this particular point is most probably wrong...]

Once alpha and beta are fixed, it is easy to see that gamma is unique... of course all the while assuming that we are considering only the range 0 to 360...

5. Aug 6, 2010

### Petr Mugver

An idea:

(1) write the 3 matrices corresponding the 3 zxz rotations. This is easy.
(2) multiply them in the correct order. Also easy, but you get a lot of sin and cos.
(3) verify that the matrix obtained is different for different values of the 3 angles, in their range. Very easy.

...so...it should be easy! Worth doing this calculation once in a life.

6. Aug 7, 2010

### krishna mohan

Agreed that the first two steps are simple... even simpler if implemented in something like mathematica..
lt
The resulting matrix is already given in the wikipedia article...

But I do not see how the third step can be easy...

Just have a look at the resulting matrix in wikipedia....

7. Aug 7, 2010

### Petr Mugver

If you copy and paste that matrix here we'll have it a look...

8. Aug 7, 2010

### krishna mohan

I think I do understand.. Your idea must be something like the one detailed in the link below...

http://www.gregslabaugh.name/publications/euler.pdf [Broken]

Last edited by a moderator: May 4, 2017
9. Aug 8, 2010

### Petr Mugver

Yes. The calculation in that link is a bit "longer" of what I meant because, not only it shows the "essential" uniqueness of Euler angles, but it actually calculates them for an arbitrary rotation.