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James Puckett
Posts: **1,970**

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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## Comments

1,654185`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 lickety-split. 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.`

1,970