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 given points?

I tried least square method. I wrote this after I read the pdf document in http://www.mathworks.com/matlabcentral/fileexchange/15542-cubic-bezier-least-square-fitting. But I can’t find a good t(i) function.

using System;
using System.Collections.Generic;
using System.Linq;
using System.Windows;

namespace BezierFitting
{
    class CubicBezierFittingCalculator
    {
        private List<Point> data;

        public CubicBezierFittingCalculator(List<Point> data)
        {
            this.data = data;
        }

        private double t(int i)
        {
            return (double)(i - 1) / (data.Count - 1);
            // double s = 0.0, d = 0.0;
            // 
            // for (int j = 1; j < data.Count; j++)
            // {
            //     if (j < i)
            //     {
            //         s += (data[j] - data[j - 1]).Length;
            //     }
            //     d += (data[j] - data[j - 1]).Length;
            // }
            // return s / d;
        }

        public void Calc(ref Point p1, ref Point p2)
        {
            double n = data.Count;
            Vector p0 = (Vector)data.First();
            Vector p3 = (Vector)data.Last();

            double a1 = 0.0, a2 = 0.0, a12 = 0.0;
            Vector c1 = new Vector(0.0, 0.0), c2 = new Vector(0.0, 0.0);
            for (int i = 1; i <= n; i++)
            {
                double ti = t(i), qi = 1 - ti;
                double ti2 = ti * ti, qi2 = qi * qi;
                double ti3 = ti * ti2, qi3 = qi * qi2;
                double ti4 = ti * ti3, qi4 = qi * qi3;
                a1 += ti2 * qi4;
                a2 += ti4 * qi2;
                a12 += ti3 * qi3;

                Vector pi = (Vector)data[i - 1];
                Vector m = pi - qi3 * p0 - ti3 * p3;
                c1 += ti * qi2 * m;
                c2 += ti2 * qi * m;
            }
            a1 *= 9.0;
            a2 *= 9.0;
            a12 *= 9.0;
            c1 *= 3.0;
            c2 *= 3.0;

            double d = a1 * a2 - a12 * a12;
            p1 = (Point)((a2 * c1 - a12 * c2) / d);
            p2 = (Point)((a1 * c2 - a12 * c1) / d);
        }
    }
}

What’s the best way to get a cubic bezier curve closest to given points?

For example, here are 30 points:

22, 245
26, 240
39, 242
51, 231
127, 189
136, 185
140, 174
147, 171
163, 162
169, 155
179, 107
181, 147
189, 168
193, 187
196, 75
199, 76
200, 185
201, 68
204, 73
205, 68
208, 123
213, 118
216, 210
216, 211
218, 68
226, 65
227, 110
228, 102
229, 87
252, 247

Those points are distributed around the the cubic bezier curve controled by four points:

P0 (0, 256), P1 (512, 0), P2 (0, 0), P3 (256, 256).

Suppose the curve is from (0, 256) to (256, 256), how to get rest two control points close to the origional points?

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