Yes, the article is wrong. It’s possible to exactly convert a quadratic curve to a cubic curve, but going the other way the best possible is just an approximation. (Though the integer control point locations required by font descriptions can make it not quite exact.)

Also, this conversion doesn’t seem like something anyone should do by hand (e.g. adding extra control points which don’t change the curve): some algorithm is going to be able to optimize curves to minimize points, convert between curve types, etc. much better than a human is, and much much faster. [Though for all I know some existing software has crappy implementations]

Personally, I’d recommend people use Raph Levien’s Spiro curves to design their fonts (and not use Béziers directly); they make it much much easier to make whatever shape the designer is actually trying to make. (Folks on Hacker News interested in the mathematics of curve drawing should also definitely check out his PhD thesis.) http://www.levien.com/spiro/

Hey!

Isn't it the other way around? I mean, isn't cubic a special case of quadratic?

Please correct me if I'm wrong (or my references), I want this article to be as accurate as possible!

Thanks!

A cubic is more general than a quadratic, so clearly a quadratic is going to be a special case of a cubic.

We can demonstrate this by looking at the equation for a quadratic Bézier:

     (1 - t)^2 P_0 + 2(1 - t)t P_1 + t^2 P_2

Now, let's increase the degree.

     = [ (1 - t)^2 P_0 + 2(1 - t)t P_1 + t^2 P_2 ] * [ 1 - t + t ]

clearly this equation is equal to the above one because 1-t+t == 1 and multiplying by one does nothing.

     = [ (1 - t)^2 P_0 + 2(1 - t)t P_1 + t^2 P_2 ] * [ (1 - t) + t ]
     = (1 - t)^3 P_0 + t(1 - t)^2 P_0 + 2(1 - t)^2t P_1 + 2(1 - t)t^2 P_1 + (1 - t)t^2 P_2 + t^3 P_2

Group by powers of t:

     = (1 - t)^3 P_0 + t(1 - t)^2 [P_0 + 2*P_1] + (1 - t)t^2 [2P_1 + P_2] + t^3 P_2

Which is exactly a cubic Bezier curve.

Thus, all quadratic Beziers can represented cubic ones, but the reverse does not hold, thus quadratics are specializations of cubics.

QED.

This is known as degree elevation.

Oh, I see my mistake! Thank you very much!

I'm not very math savvy and I was trying to get this right.

Anyway, thank you very much for making this clear!

Cheers!