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March 12th, 2017, 11:10 PM   #1
Joined: Nov 2016
From: Kansas

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Bezier Curve

Can anyone tell how to convert this polynomial to a bezier curve = 1/(4x^2 +1). I have to take 5 knots i.e from -1 to 1 with a 0.5 increment.
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March 13th, 2017, 09:01 AM   #2
Joined: Jan 2016
From: Athens, OH

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I'm not sure what your mean. Perhaps this?

Let $f(x)=1/(x^2+4)$, $x_i=-1+(1/2)i$ and $P_i=(x_i,f(x_i)),\,i=0\cdots 4$

Now form a Bezier curve with the 5 control points $P_i$. The obvious choice of degree of the Bezier curve is 4. Just use the standard formula for the curve. Here's a drawing with f(x) dotted and the Bezier curve in red:

Another option is to paste together Bezier curves of smaller degree, say quadratic curves. As you know 3 control points are needed for a quadratic Bezier. To paste these together so that the resultant curve is $C^1$ continuous at the joints (points where the curves are pasted together), you can add control points to make this happen. In the following, the red circles indicate the added control points:

As you know, Bezier curves do not interpolate the control points. But the technique of pasting together Bezier curves can be used to approximate a curve. In this last drawing there were 31 "equally spaced" control points with the pieces quartic Bezier curves:

Last edited by johng40; March 13th, 2017 at 09:15 AM.
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