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I would like to approximate one cubic Bezier curve with two quadratic ones. In other words, I would like to split a cubic curve at some parameter t and approximate the resulting halves with a quadratic Bezier each.

(cubic -> split at t -> two cubics -> two quadratics)

Obviously, it is possible to split the original curve very badly, say by choosing t closely to 0 or 1, if the curve looks like a simple arch.

Please, I have two questions:

  • how can I formalize this "badness"? How to decide if one split is better than another? Is there any way to tell how "different" one approximation is that the other? Related, how to tell how well the result approximates the original curve?

  • how to choose good t? With regards to the original cubic curve coordinates? If I knew what "badness" is, would it be possible to choose such t that the resulting quadratic splits would be equally "bad", i.e. to how to fairly split the original curve?

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Perhaps the absolute value of the difference of slopes at the "splitting point" would be a good measure of badness. – Emily Oct 26 '12 at 20:09
Or minimise $\int_0^1 (B_{cubic}-B_{quad})^2\,d\,t$. – Daryl Oct 26 '12 at 20:49

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