cubic Bézier curves

Cubic Bézier curves are the classical Bézier curves, the lower degree curves presented before should be seen as extensions of classical Bézier curves to low degrees.

For cubic Bézier curves, there are four control points. The first two are the two endpoints of the curve, $P_1$ and $P_2$. The two remaining ones are intermediate points $Q_1$ and $Q_2$. These intermediate points control the tangent and the curvature at both ends. The Bernshtein polynomial and its first derivatives are:

$$\left\{\begin{aligned}B_3(t) &= (1-t)^3 P_1 + 3t(1-t)^2 Q_1 + 3... ...P_1 - (2-3t) Q_1 + (1-3t) Q_2 + t P_2\right] \end{aligned}\right.$$

Figure 4: cubic Bézier curve