What is the uniqueness of Euler angles?

In summary: Agreed that the first two steps are simple... even simpler if implemented in something like mathematica..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...Just have a look at the resulting matrix in wikipedia...In summary, 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). However, the theorem mentioned in the article requires proof and lectures can provide this proof.
  • #1
krishna mohan
117
0
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.
 
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  • #2
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
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
I found a book..

Biedenharn, L. C.; Louck, J. D. (1981), Angular Momentum in Quantum Physics, ReadingOne 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
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! :biggrin: Worth doing this calculation once in a life.
 
  • #6
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
krishna mohan said:
Just have a look at the resulting matrix in wikipedia...

If you copy and paste that matrix here we'll have it a look...:wink:
 
  • #8
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]
 
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  • #9
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.
 

1. What are Euler angles and why are they important in science?

Euler angles are a set of three angles that describe the orientation of a rigid body in three-dimensional space. They are important because they provide a convenient way to represent the rotation of an object, which is a common phenomenon in science and engineering.

2. How are Euler angles different from other ways of representing rotations?

Euler angles are unique in that they use a sequence of three rotations around different axes to describe a rotation, while other representations, such as quaternions, use a single rotation and a vector to describe a rotation. This makes Euler angles easier to visualize and interpret.

3. What is the uniqueness of Euler angles?

The uniqueness of Euler angles refers to the fact that there are multiple ways to represent the same rotation using different sequences of three angles. This means that there is not a single set of Euler angles that can fully describe all possible rotations, making it important to carefully define the sequence of rotations used.

4. How are Euler angles used in real-world applications?

Euler angles are commonly used in fields such as aerospace engineering, robotics, and computer graphics to describe the orientation of objects. They are also used in motion capture systems and in navigation and control systems for vehicles and aircraft.

5. What are some limitations of using Euler angles?

One limitation of Euler angles is that they can suffer from gimbal lock, where one of the angles becomes undefined and the representation breaks down. They also have issues with numerical instability and can be difficult to use for certain types of rotations, such as those involving multiple axes at once.

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