How to Compute Euler Angles for Rotating Reference Frames

[matteo86bo]

I know this is rather trivial question but it's not homework!
I need this as part of a bigger project.

What I have to do is rotate my reference frame to another one. I want my new z-axis to be a vector Z=(Z1,Z2,Z3)

Following the notation of this wikipage:
http://en.wikipedia.org/wiki/Euler_angles

all I need to do is compute the Euler angles, however, I don't manage to get Y3 and therefore the last angle, gamma.

Can you please help?

 

[phyzguy]

Merely giving your new Z-axis only determines 2 of the 3 Euler angles, since it doesn't tell where your new X and Y axes are. You need to specify at least one of the new X or Y axes to determine the third one - that's why you need Y3. Think of it this way - after you've specified your new Z-axis, you can freely rotate about the new Z-axis. This will leave the new Z-axis unchanged, but will change X and Y.

 

[matteo86bo]

Merely giving your new Z-axis only determines 2 of the 3 Euler angles, since it doesn't tell where your new X and Y axes are. You need to specify at least one of the new X or Y axes to determine the third one - that's why you need Y3. Think of it this way - after you've specified your new Z-axis, you can freely rotate about the new Z-axis. This will leave the new Z-axis unchanged, but will change X and Y.

You're right! But sorry if I insist, I just want to get this straight.

If I only know the new z-axis and x and y could be in directions, although forming an orthonormal system, does this means that I could choose my angle gamma to be any value between 0 and 2pi?

 

[phyzguy]

You're right! But sorry if I insist, I just want to get this straight.

If I only know the new z-axis and x and y could be in directions, although forming an orthonormal system, does this means that I could choose my angle gamma to be any value between 0 and 2pi?

Yes.