Bezier curve information

How many segments, and which ones, do we need to cut a Bezier curve into, in order for us to create an offset-curve simply by scaling the sections? In order to answer that question, we need to know where the Bezier inflects -- i.e. where its curvature changes direction in some interesting way. This is actually fairly simply checked, by looking at the first (blue) and second (red) derivatives of the Bezier curve function:

fx(t) = (1-t)³·x0 + (1-t)²·t·x1 + (1-t)·t²·x2 + t³·x3
fy(t) = (1-t)³·y0 + (1-t)²·t·y1 + (1-t)·t²·y2 + t³·y3

Every root for f '(t) and f "(t) (for t inside the range <0,1>) indicates an inflection point in the original curve. Let's math: everwhere a blue or red line crosses the t-axis, that's a point of interest.