1
$\begingroup$

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?

  • 1
    Well, the [Bernstein polynomials](http://en.wikipedia.org/wiki/Bernstein_polynomial) that are used in defining your Bézier curves admit simple expressions for their derivatives. If you're using cubic Bézier curves, you can use the fact that the second derivative of a cubic function is a linear function...2012-07-13
  • 0
    @J.M. unfortunately the curves can have an arbitrary number of control points and so can be of any degree.2012-07-13
  • 0
    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

1 Answers 1