Make object face center of circle without flipping in 2D space.

Simple concept:

I have a sling in a 2D game.

The cup of the sling is a game object.

When pulled back, the cup of the sling should always face towards a center point, just like the cup of a real sling would when under tension.

Seems simple, but I have not been able to find a solution.

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Things that did not work -

Using lookat does not work because the cup will flip halfway around the circle.

This solution:

                 var relativePos = center.transform.position - cup.transform.position;                       
                 var rotation = Quaternion.LookRotation(relativePos,cup.transform.up);                    
                 cup.transform.rotation = rotation;

                     

almost works at the edges, but introduces rotation around y and z axes anywhere near the center. (this is a 2d game, so any rotation along those axes is no good)

Clamping those axes seems to break the solution as a whole.

This solution, found in another answer:

 Vector3 oldV = targetPos-oldPos, newV = targetPos-curPos;
 Quaternion changeRot = Quaternion.LookRotation(oldV, newV);
 // the order of *'s matters, can never remember which one:
 transform.rotation = changeRot * transform.rotation;

seems to just result in wild spinning.

edit: Sorry for not being able to get the code window to look right. It is not for lack of trying.