Precisely (mathematically?) Define Splines

[glenn.mause]

I am currently trying to design a mechanism similar to the only used in Joysticks for example. (Reference, single axis from this)

However I am struggling to properly define the shape of the surface the bearing would ride on. The contact point of the spline to the bearing has to stay at specific angles during motion, while the retaining part and the upper part (joystick) move. 

I could not think of a decent way to define this at all points of the motion using regular constraints. While I can do a spline between the start and end point, it is difficult to precisely define the curve ion between those points.

What would be a good way to precisely define the surface? Is there a way to define a Spline according to a mathematical function?

For further explanation:

The solid part pivots around the upper circle, the three lower circles represent the bearing that is supposed to roll along the spline. The 20° and 10° represent the desired contact angles. The surface the bearing rides on is shown as the orange spline, rotating around the orange circle on the right.

[davebYYPCU]

Initial thoughts were not possible.

Requires constant radius to keep the bearing in contact with the surface.

Top down is

a 3 point arc, (binding)

3 point Spline tangent both sides, (not constant radius)

and Tangent Arc for the outside circles, (no constant contact)

I still think so.