Trigonometry to move an arm around axis in 3D space

Question

Hello I have a question about rotating a Rod hinged to a fixed point (so it swings all around), and targets. short form, I would like to know how to find a rotation that would point my rod (while still attached to the fixed point) towards the position of the target.

I would also like to find the rotation as a quaternion, and/or axis-angle to prevent gimbal lock.

I am using unity

The image I made shows my problem with clarity.

Answer 1

Assuming your cylinder is oriented so that its local y+ axis (green arrow on the local transform gizmo) points along the length of the cylinder (like it does for the default Unity cylinder), then you're looking for a quaternion that rotates (0, 1, 0) to lie parallel to (target.position - pivot.position).

(If that's not how your axes are set up for this object, be sure to edit your question so that critical details like this are made explicit)

Unity has a a couple of great convenience methods to do just this. You could try:

transform.rotation = Quaternion.FromToRotation(
                         Vector3.up,
                         target.position - pivot.position
                     );

or

Quaternion zToTarget = Quaternion.LookRotation(target.position - pivot.position);
Quaternion yToZ = Quaternion.Euler(90, 0, 0);
transform.rotation = zToTarget * yToZ;