1. Bézier curves with zero second derivative at one end

In order to handle the relaxed end conditions, we shall need to be able to tell when a Bézier curve has zero second derivative at one end. Recall that for a cubic Bézier curve $P(t)$ with control points $P_0$, $P_1$, $P_2$, $P_3$,

$P''(0) = 6(P_0 - 2 P_1 + P_2)$.

This quantity is zero when $2 P_1 = P_0 + P_2$, or equivalently, when

$P_1 = {\frac 1 2} P_0 + {\frac 1 2} P_2$.

A similar relation holds in case $P''(1) = 0$. Even more simply:

Observation. $P''(0) = 0$ if and only if $P_1$ is the midpoint of the segment $\overline {P_0 P_2}$; $P''(1) = 0$ if and only if $P_2$ is the midpoint of the segment $\overline {P_1 P_3}$. Some examples are shown in Figure .

Figure: Examples of second derivatives