Bezier curve fitting with known end points

Question

I want to fit a bezier curve with known end points (p0 and p3) to noisy 2d data. This seems like an easier problem than traditional 4-point bezier curve fitting but still too hard for me to figure out.

Can someone point me to existing code or an algorithm to find the best values for the control points p1 and p2?

edit: The points that I'm trying to fit with a bezier curve comes from curves drawn with a mouse (imagine drawing something with a brush in Paint, there could be hundreds of recorded points in one long stroke). The anchor points p0 and p3 are created in advance but the control points p1 and p2 should be calculated so that the bezier fits the shape of the curve sketched out with the mouse.

Answer 1

I stumbled on a paper called "Approximation of data using cubic Bezier curve least square fitting" by "M.Khan" which describes an algorithm to calculate the exact thing I'm looking for.

Implementation in javascript was easy. It works quite good and is fast but the resulting bezier curves are not perfect. Could be a bug in my code but I suspect that better curves could be obtained by iteratively adjusting the matching points on the bezier curve to better fit the data .

edit: It turns out you can use newton-raphson to optimize each individual t-value for the bezier curve. After doing that the curve fits great, atleast for curves with only few points that don't self intersect but I have to do some more testing.