Euler Angles between two versors

1. Sep 9, 2008

larry77

Dear All,
Suppose you have two versors x and y. You would like to calculate the 3 Euler angles between them. How would you do that?
You know that
x=Ry where R is the rotation matrix (see e..g Goldstein).
Now the point is that I know both x and y and want to get phi, theta and psi (no quaternion solutions, please).
Many thanks

Larry77

2. Sep 9, 2008

Ben Niehoff

Quaternion solutions would be the easiest. But since you don't want to use them, do it this way:

Write

$$R = R_{\theta}R_{\phi}R_{\psi}$$

(put the factors in whatever order is consistent with your convention for the Euler angles).

Then multiply these out. You will get a rotation matrix with three unknown parameters. Writing

x = Ry

gives you three equations. Then you need to solve them.

What is a versor, precisely? I'm assuming it is a vector with an orientation? If you know the axis of the object (which is an ordinary vector), then you merely need to rotate the axes into each other, and then rotate around the axis. This should allow you to find one of the Euler angles independently of the other two. Might save you some algebra.