Why does TrueType use quadratic splines?
James Puckett
Posts: 1,992
Why does TrueType use quadratic splines instead of cubic béziers? Did Apple think that people would actually want to draw that way?
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My understanding is that, although the paths were more difficult to edit, it was computationally more efficient than cubic béziers. Remember, at the time, the use of outline fonts for screen display was not really a thing, and that's how TrueType was going to work. PostScript fonts relied on bitmap screen fonts partly because of this (ATM didn't exist yet). On top of that, it had to have reasonable performance even on relatively lowpowered 68000 chips, such as the 16 MHz Powerbook 100 or the Macintosh SE. It was probably a reasonable trade off. You only had to make a font once, but it had to be displayed on user's machines continuously.5

I second what Mark Simonson said and would like to quote Donald E. Knuth:My impression is that licenses and patents could have played a role, too. It's probably jurisdictionally safer to implement a "new technology" relying on quadratic Bézier curves instead of cubic Bézier curves.
The main thing is that TrueType uses only quadratic splines, and that Type1 fonts use cubic splines, which allow you to get by with a lot fewer points where you have to specify things. The quadratic has the great advantage that there's a real cheap way to render them. You can make hardware to draw a quadratic spline licketysplit. It's all Greek mathematics, the conic sections. You can describe a quadratic spline by a quadratic equation (x, y) so that the value of f(x, y) is positive on one side of the curve and negative on the other side. And then you can just follow along pixel by pixel, and when x changes by one and y changes by one, you can see which way to move to draw the curve in the optimal way. And the mathematics is really simple for a quadratic. The corresponding thing for a cubic is six times as complicated, and it has extra very strange effects in it because cubic curves can have cusps in them that are hidden. They can have places where the function will be plus on both sides of the cubic, instead of plus on one side and minus on the other. The algorithm that's like the quadratic one, but for cubics, turns out that you can be in something that looks like a very innocuous curve, but mathematically you're passing a singular point. That's sort of like a dividing by zero even though it doesn't look like there's any reason to do so. The bottom line is that the quadratic curves that TrueType uses allow extremely fast hardware implementations, in parallel.
9 
Thanks, Mark and Linus.
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