Finding,for a planar cubic Bezier spline, the parameter value where curvature is zero (a point of inflection).
Let P(t) be a cubic bezier spline, b_(i) its control points. Then:
The curvature as a functions of the parameter t is:
Therefore, zero curvature is found when:
Let A,B,C be:
Then:
After some rearanging the cubic term is canceled and we have:
Which is a simple quadratic equation, with roots candidates for the parameter value where the curvature of the spline is zero. If we have a root in the range [0,1], then we have found an inflection point for that parameter value.