I cannot offer a mathematical proof of the matter, yet my understanding of quadratic Bézier curves is that it is, with absolute mathematical precision, impossible to use them to produce a circle or an ellipse.
One can, however, use a number of quadratic Bézier curves joined together to produce an approximation of a circle or an ellipse.
It is a convenient analogy to think of the way that a circle can be approximated using straight lines. Using three straight lines produces a triangle which is very unlike a circle, using four straight lines is not much better, yet once one uses, say, twelve or sixteen straight lines the approximation is better. With a thousand straight lines the approximation would be much better.
It is much the same with using Bézier curves; however, with Bézier curves it is possible to get a much better approximation using a smaller number of items than with straight lines.
Since using Bézier curves joined together to produce an approximation of a circle or an ellipse produces an approximation to a circle or an ellipse rather than an exact mathematical circle or ellipse, there is the matter of deciding how many quadratic Bézier curves to use in making the approximation. I usually use an approximation to a circle that I produced for my Stardisc font in 2007.
I have the font installed on the computer that I use, so it is just a matter of opening the font file in FontCreator and copying the glyph for the disc into a new font whenever I need a contour that is a circle. Often I scale the glyph in the new font. That circle has eight quadratic Bézier curves. It is a very regular design with the eight curves being, within the limits of the approximation, each the same size. If I need an ellipse, I scale the circle.
I have tried drawing a few filled circles and ellipses in Serif PagePlus X4 and then using copy and then pasting into FontCreator. The results look alright: close inspection shows mostly curves, of various lengths, with sometimes a very short straight line amongst them.
It is a matter of judgement at the time, by a person or by an automated process, as to how many Bézier curve pieces are needed to approximate to a circle or an ellipse.
William Overington
8 June 2012