Algorithm for deriving control points of a bezier curve from points along that curve? -


i've been looking for, not finding, algorithm allow me plug in list of x,y coordinates known along curve 4 control points cubic bezier curve spit out.

to more precise, i'm looking algorithm give me 2 control points required shape curve while inputting series of discrete points including 2 control points determine start , end of curve.

thanks!

edit: okay, due math, old foe, need ask bezier curve of best fit polynomial function.

so assume endpoints fixed, , have number of (x,y) sample points want fit cubic bezier.

the number of sample points have determine approach take. let's through few cases:

2 points

2 sample points simplest case. gives total of 4 points, if count end points. number of cvs in cubic bezier. solve this, need parameter (t) value both of sample points. have system of 2 equations , 2 points need solve, equation parametric equation of bezier curve @ t values you've chosen.

the t values can whatever like, better results using either 1/3 , 2/3, or looking @ relative distances, or relative distances along baseline, depending on data.

1 point

this similar 2 points, except have insufficient information uniquely determine degrees of freedom. suggest fit quadratic bezier, , degree elevate. wrote detailed example of quadratic fitting in this question.

more 2 points

in case, there isn't unique solution. have used least-squares approximation results. steps are:

  • pick t values each sample
  • build system of equations matrix
  • optionally add fairing or other smoothing function
  • solve matrix least-squares solver

there description of these steps in free cagd textbook, chapter 11. talks fitting b-splines, cubic bezier type of b-spline (knot vector 0,0,0,1,1,1 , has 4 points).


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