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I am designing an algorithm that generates shapes of bezier curves. Each output are control points for a single curve.

In some cases, it should return a circle. Which control points does the algorithm have to output for the shape to become a circle?

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Is not possible draw a perfect circle with Bézier curves, but the approximation is good enough. See How to create circle with Bézier curves? and Approximate a circle with cubic Bézier curves.

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Without using rational Bezier curves, drawing a circle is impossible. I remember I had to prove that once for some class I had, and the proof isn't pretty.

However, drawing the circle using rational bezier curves is quite easy, as I recall. Take the points $(1,0)$, $(1,1)$ and $(0,1)$ and weights $1,\frac{\sqrt2}{2}, 1$ (I'm not sure, the middle weight may also be $\sqrt2$) and you get a circle.

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