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I have a curve that I know is a (non-periodic) Cubic Bézier Curve (because I constructed it as such). I stored each ordered pair in the curve, but not the control points. Is it mathematically possible, knowing these constraints, to reconstruct the control points that created this curve? If so, how would I do so?

Thank you!

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    $\begingroup$ If you know four points from your (non-composite!) Bézier curve, then it is a simple matter of solving four equations in four unknowns, since he endpoints of your curve are necessarily two of the control points. $\endgroup$ – J. M. is a poor mathematician Aug 17 '12 at 0:42
  • $\begingroup$ I know this seems like a "Can you do it for me" question - but I generated the curve using a computer program - I don't actually know the equations. I am assuming you are telling me to use the original equations for the cubic bezier curve, and just plug in the values I know, and leave out the values I don't, and solve, right? $\endgroup$ – TheNerd Aug 17 '12 at 1:13
  • $\begingroup$ I'm actually telling you how to get the equations for your Bézier curve. You're familiar with the Bernstein polynomials, I presume? You can set up the explicit expression in terms of the Bernstein polynomials for each component ($x$, $y$) and match up values. Two equations for each component... $\endgroup$ – J. M. is a poor mathematician Aug 17 '12 at 1:20
  • $\begingroup$ Unfortunately I'm sort of a math idiot/newbie, but I'm willing to do research and learn. I'm off to Wikipedia to gain an understanding of Bernstein Polynomials, so I can better understand your answer. You posted it as a comment, so I can't mark it as an answer though. $\endgroup$ – TheNerd Aug 17 '12 at 1:29
  • $\begingroup$ I only posted a comment since what I've written isn't sufficiently elaborate to be an answer, and writing out the solution for this is somewhat of a pain for now. If nobody writes an answer later, I'll try to write one. $\endgroup$ – J. M. is a poor mathematician Aug 17 '12 at 1:32
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I am assuming that you saved the original interpolation points and not some sampled points. The accuracy of the reconstruction would depend on matching that computer program's choice of parameterization (uniform/chordal) and end point conditions (usually natural end condition).

This paper describes a popular cubic bezier interpolation alorithm, http://www-hagen.informatik.uni-kl.de/~alggeom/pdf/alggeom_script_ws11_03.pdf

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