The condition for an inflection point

[jamadagni]

Hello people. I'm actually a humanities scholar but who has retained his interest in maths from high school.

Well curiously, in relation to one of my projects I'm investigating the properties of third order Beziers. Given the two nodes and control points of a third order Bezier, I needed to determine analytically any inflection points it may contain. I came across this web page which asserts that when the cross product of the first and second derivatives of the parametric representation of the curve is zero, then the curve has an inflection point.

Now I wonder whether this is just a well known textbook fact or some original formula of the website author. Sorry if the question is silly, but basically I need to incorporate this method of determining inflection points in a GPLed piece of software but the website says "material here is copyright". So I'd like to ensure that it is just a well known mathematical fact which is simply well explained on that website.

Thanks.

 

[fresh_42]

Look up "curl of a gradient field". Roughly speaking it's a measure for the tendency to rotate, and this is zero at inflection points: the function cannot decide between "clockwise and counterclockwise".