Change in angle due to rotation

[Laiva-59]

Hi,

I have the following problem:

A rigid body, to which an axis system xyz is attached, has an orientation defined by Euler angles phi, theta and psi with respect to an intertial frame XYZ.

The Euler angles are defined as explained on wikipedia and can be thought of as a rotation matrix as shown in following link:
http://en.wikipedia.org/wiki/Rotati...)#Conversion_formulae_between_representations


Now, on this fixed body, I would like to find (on a given point), the angle the outer surface normal makes with the XY-plane. I found the outer surface normal in the body-fixed coordinates N = [nx, ny, nz].

Q's:

1. How now to find the angle with plane XY (Z=0)? (call it beta)
2. If we rotate the body along phi,theta and psi, how does this beta change?

So basically: If we have defined this angle, how does the angle change when we rotate the axes?

 

[pat_connell]

Thanks in advance for any help!Answer:1. The angle beta can be calculated using the dot product of the normal vector and the unit vector in the xy-plane (i.e. [0, 0, 1]) as follows: beta = acos(N . [0, 0, 1] / (|N| * |[0, 0, 1]|))2. If the body is rotated along phi, theta, and psi, then the normal vector in body-fixed coordinates will rotate accordingly and its orientation with respect to the xy-plane will change. To calculate the corresponding angle, you need to use the rotation matrix defined by the Euler angles to obtain the new orientation of the normal vector in the inertial frame XYZ, and then calculate the angle between this vector and the unit vector in the xy-plane as described above.