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    <p>So I assume that the endpoints are fixed, and then you have a number of (x,y) sample points that you want to fit with a cubic Bezier.</p> <p>The number of sample points that you have will determine what approach to take. Let's look through a few cases:</p> <p><strong>2 points</strong></p> <p>2 sample points is the simplest case. That gives you a total of 4 points, if you count the end points. This is the number of CVs in a cubic Bezier. To solve this, you need a parameter (t) value for both of the sample points. Then you have a system of 2 equations and 2 points that you need to solve, where the equation is the parametric equation of a Bezier curve at the t values you've chosen. </p> <p>The t values can be whatever you like, but you will get better results by using either 1/3 and 2/3, or looking at relative distances, or relative distances along a baseline, depending on your data.</p> <p><strong>1 point</strong></p> <p>This is similar to 2 points, except that you have insufficient information to uniquely determine all your degrees of freedom. What I would suggest is to fit a quadratic Bezier, and then degree elevate. I wrote up a detailed example of quadratic fitting in <a href="https://stackoverflow.com/questions/18476308/non-parametric-quadratic-bezier-curve-through-3-points-in-2d-in-r/18485522#18485522">this question</a>.</p> <p><strong>More than 2 points</strong></p> <p>In this case, there isn't a unique solution. I have used least-squares approximation with good results. The steps are:</p> <ul> <li>Pick t values for each sample</li> <li>Build your system of equations as a matrix</li> <li>Optionally add fairing or some other smoothing function</li> <li>Solve the matrix with a least-squares solver</li> </ul> <p>There is a good description of these steps in this <a href="http://tom.cs.byu.edu/~557/text/cagd.pdf" rel="nofollow noreferrer">free cagd textbook</a>, chapter 11. It talks about fitting b-splines, but a cubic bezier is a type of b-spline (knot vector is 0,0,0,1,1,1 and has 4 points).</p>
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    1. POAlgorithm for deriving control points of a bezier curve from points along that curve?
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    1. POAlgorithm for deriving control points of a bezier curve from points along that curve?
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    1. COThanks for the quick answer. I thought algebra would probably be required but that might get a bit obscene given a curve could have thousands of points. I'll give a try to breaking large curves up into smaller curves and applying the two point method. In case that doesn't work, I'll give the greater than two point try an attempt!
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    2. COIf the curve exactness doesn't matter too much, you could try just picking two points and doing the 2 point method.. :)
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    3. COI'm aiming for a decent recreation of edges isolated with the sobel operator so exactness would be very much preferred.
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