quadratic Bézier curves

For quadratic Bézier curves, there are three control points. The first two control points are the two endpoints of the curve, $P_1$ and $P_2$. The last control point is an intermediate point $Q$ which controls the direction of the tangents of the curve at both ends. This point is generally away from the curve itself. The Bernshtein polynomial and its first derivatives are:

$$\left\{\begin{aligned}B_2(t) &= (1-t)^2 P_1 + 2t(1-t) Q + t^2 P... ...\ B_2''(t) &= 2\left[P_1 - 2 Q + P_2\right] \end{aligned}\right.$$

Figure 3: quadratic Bézier curve