Calculating Euler Angles from Two Frames of Reference

[muonnoum]

Fairly straight forward question. If you have a set of three vectors specifying a frame of reference and a second set of 3 vectors stating another frame of reference. How do you get the Euler angles associated with that rotation?

More generally I am considering the relative orientation of one water molecule with another and require the Euler angles associated.

Thanks in advance.

 

[Bill_K]

1) Rotate about z axis until z' axis is in the x-z plane. This is the first angle φ.
2) Rotate about y' axis until z' coincides with z. This is the second angle θ.
3) Rotate about z axis until x' and y' coincide with x and y. This is the third angle ψ.

There's more than one set of conventions in use, so these may or may not be exactly the ones you want.

 

[muonnoum]

I'm not quite sure what you mean but I think I have worked out what I need to do.

I think I need to calculate the transform matrix from the matrix of frame 1 and the inverse matrix of frame 2. This matrix corresponds to the Euler rotation matrix and so I simply have to equate the matrix elements and solve them simultaneously with the constraints imposed by the Euler angle conventions. I confirmed that this works in 2 dimensions but if you could confirm that there are no complications in going to 3 dimensions I would appreciate it.

Many thanks