Finding a Control Point for a Bezier Curve?

I happened to go through the concept of Bezier Curves while trying to get the curl on a sphere based on a user swipe.

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Now, I would like to know if there is any way to calculate the point P1. Ibelieve P0 would be the starting point of touch while P2, the end point of touch.

This is a generic (non-Unity) math question. But I’m pretty sure that it’s impossible. Because:

Bezier curves start with any P0, P1, P2 and generate a particular quadratic(?) curve. If you start with the curve, it probably isn’t a quadratic(?), so there’s no possible P1 that would make it.