I need some help with working out the tilt angle V and rotation angle U of this 5 axis design. I want the head pointing at a desired tilt angle (say 45 degrees) to a plane perpendicular to the X-Y plane but at an angle alpha to the X-Z plane. For example I want to make a cut on a material at 45 degrees to the X-Y plane (in the direction X).

My guess is that this is an issue that has been solved before but I don't know how to do the translation and rotation calculations of these joints/planes.

I cannot work out the angles. I am coding in C# and I need to work out the angles U and V as a function of the angle of the cutter in relation to the X-Z plane.