My question may be simple compared to some of the questions I see here but it has me stumped.
I have a 3"disc with an X, Y, Z origin at the center. X and Y form the plane of the disc and Z extends vertically. I rotate 10° about the X-axis, 20° about the Y-axis and 5° in Z-axis.
The disc disc position after these moves needs to be recreated using only Y and Z axis. How can the new angles for Y and Z be determined that will put the disc in the same position?
A rotation around the x-axis is a conjugate of a rotation around the y axis as follows.
$$ R(\frac{\pi}{2},z) R(\theta , x) R( \frac{-\pi}{2}, z) = R(\theta , y)\\ R (\theta , x) = R(\frac{-\pi}{2},z) R(\theta , y) R(\frac{\pi}{2},z)\\ R(5^\circ , z) R(20^\circ , y ) R(10^\circ , x) = R(5^\circ , z) R(20^\circ , y )\\ * R(\frac{-\pi}{2},z) R(10^\circ, y) R(\frac{\pi}{2},z)\\ $$
so you can just substitute that conjugation in for the corresponding x rotation to leave only y and z rotations.