The good pseudo-half-circle-like curves that quadratic bezier curves can create

Question

Since quadratic bezier curves can only approximate circles, I'm wondering what they can actually do well. That is, the sort of half-circle-like shapes that quadratic curves can make. For instance, in the letter u, maybe it's no good at creating a half-circle, but it might be really good at creating the U shape in some other fashion, maybe with just 1 or two curves that has some x contour. Wondering what that curve looks like and what is definition or properties are.

Answer 1

I can think of a few good things about quadratic Bézier curves:

1. They can represent parabolas exactly. Cubic curves can do this too, of course, but not as economically.
2. It's very easy to do computations with them. For example, intersection with a straight line requires only solution of a quadratic equation.
3. They are easy to convert to implicit form, so it's easy to check whether a given point lies on the curve.
4. Their shapes are very predictable. If the control points are $A$, $B$, $C$, then the curve passes through the point $\tfrac12\bigl(B + \tfrac12(A+C) \bigr)$, and it's tangent at this point to the line joining $\tfrac12(A+B)$ and $\tfrac12(B+C)$. So, there is an obvious and direct relationship between the shape of the control polygon and the shape of the curve.
5. Quadratics never have inflexions, which is important in some design scenarios.

If you want to approximate some complex shape, then using quadratic pieces will require you to use more (and shorter) segments than when using cubics. But each quadratic segment will require less data than a cubic one. Compare TrueType fonts (quadratic segments) with Postscript ones (cubic segments) to get a sense of this.