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I need to determine the first point of inflection on a Bézier curve, if it exists, for a computer graphics application. My original idea was to iteratively walk the curve, evaluating 2nd derivatives until it reached the end, or the magnitude of the 2nd derivative was at a maximum, but someone advised me that I could calculate the inflection points directly.

Is there a closed mathematical solution to find the inflection point?

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    If the degree is arbitrary, then in general there won't be a "closed form" for your inflection points; you will definitely have to use numerical methods.2012-07-13

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Hi I'm definitely no maths expert (and will probably be having lots of questions to ask here) but have been recently investigating this very same thing so I'll just re-post what I posted elsewhere:

To determine the points of inflection on a bezier, find the time or times in the interval (0, 1) [excluding the endpoints of course] for which the cross product of the first and second derivatives of the parametric equation of the bezier is zero i.e. $f' \times f'' = 0$.

This is noted in various sources like this page and p 4 of this paper.

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    Hey thanks for your clarification on the terminology. I agree -- cross product is a vector, so I should really call this a determinant.2013-04-09