How can I get a cubic bezier curve closest to given points?

Question

Given n points:

p0, p1, p2, ..., pn;

How can I get the point c1, c2 so that the cubic bezier curve defined by

p0, c1, c2, pn

closest to the g

Answer 1

Your problem is very hard if you want to create curves with cusps. I can think of a heuristic to create an initial set of control points. For the first control point, try taking the first 1/3 of the points you have available, when sorted from the distance to the first anchor point. The sorting is necessary, otherwise, you may be jumping all over. Take that 1/3 of your points and do a linear least squares fit, which is has linear time complexity. That gives you the direction your curve needs to take off. Do the same thing with the last 1/3, and you have the "landing" direction.

Use those linear solutions to create vectors pointing away from the anchor points, then try making those vectors longer and shorter, trying to minimize the error. The control points would be along those vectors from the anchor points.

Here are some other ideas (I can only post two links!): physics forum question bezier curve fitting thesis