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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1$\begingroup$ Closely related to these two questions: math.stackexchange.com/questions/271319/… and math.stackexchange.com/questions/449035/… $\endgroup$– bubbaCommented Apr 17, 2014 at 6:28
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2 Answers
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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.
answered Apr 15, 2014 at 6:25
Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla
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$\begingroup$ Well, isn't surprising because the circle has a rational parametrization: mathnow.wordpress.com/2009/11/06/…. $\endgroup$ Commented Apr 15, 2014 at 7:18
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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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$\begingroup$ do you still remember the proof or a sketch of it? $\endgroup$ Commented Oct 15, 2022 at 20:02