Is it possible to generate a circle with a Bezier curve?

Question

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?

Closely related to these two questions: http://math.stackexchange.com/questions/271319/polynomial-approximation-of-circle-or-ellipse and http://math.stackexchange.com/questions/449035/is-it-possible-to-build-a-circle-with-quadratic-bezier-curves?rq=1 — bubba, Apr 17 '14 at 06:28

score 5 · Accepted Answer · edited May 23 '17 at 12:39

5

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.

edited May 23 '17 at 12:39

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answered Apr 15 '14 at 06:25

Martín-Blas Pérez Pinilla

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Well, isn't surprising because the circle has a rational parametrization: http://mathnow.wordpress.com/2009/11/06/a-rational-parameterization-of-the-unit-circle/. – Martín-Blas Pérez Pinilla Apr 15 '14 at 07:18

score 4 · Answer 2 · answered Apr 15 '14 at 07:08

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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.

answered Apr 15 '14 at 07:08

5xum

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do you still remember the proof or a sketch of it? – whiterock Oct 15 '22 at 20:02

Is it possible to generate a circle with a Bezier curve?

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