iOS Restrict movement along curved path

[noahgolm]

I have a player that I would like to be moved by the player dragging (this works just fine). Really, though, I would like to restrict the movement of the object along a locked path, maybe by determining the vector of the drag if necessary. Is there any easy way to accomplish this? I've looked into implementing a bezier curve, but making a regular arc, along with calculating the control points and such, seems a bit difficult for me.

Oh, and if it wasn't already made clear, the locked path is set to be an arc (the same shape as a quarter of a circle).

Thanks for any suggestions!