TrueType uses 2nd order, and OpenType uses 3rd order. Why not any higher?
Is that because they have too much time/space complexity of rendering? In that case, I'd like to know their complexity in big O notation.
1 Answers
You can't design with higher order Beziers, and that's still the most important aspect of creating a typeface. Having floating control points that influence the curve indirectly makes the job harder, not easier. So, while there is objective value in using cubic rather than quadratic curves (much better arc approximation, terser curve expressions, etc), there is no clear benefit in using quartic (or above) over cubic. In fact, some properties that can be relied on for quadratic/cubic don't apply to quartic or above, necessitating slow and cpu-intensive algorithms during the design phase, making them objectively worse to work with. As such, no one has proposed a system that allows for them.
As for the complexity: the curve order is its complexity. A quadratic curve has complexity O(t²), a cubic curve O(t³), a quartic curve O(t⁴), and so forth (with "t" being the time parameter, because that's how Bezier curves are evaluated)
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