Maximum distance of Bezier curve

Question

I have a Bezier curve given by four points: a start point ($P_0$), an end point ($P_3$), and two control points ($P_1$ and $P_2$). The points lie in a certain way. $P_1$ and $P_3$ lie on the x axis. $P_1$ lies on a vertical line with $P_0$ and $P_2$ lies on a vertical line with $P_3$. $P_1$ and $P_2$ also lie on the same horizontal line. Let's say the following

\begin{aligned} P_0 &= (0,0), & P_1 &= (x_1,y_1), & P_2 &= (x_2,y_1), & P_4 &= (x_2,0) \end{aligned}

How do I get the maximum distance between the baseline (line from $P_0$ and $P_3$) and the Bezier curve? In other words, how do I get the top of the bezier curve?

Answer 1

If I understand correctly, your points form a rectangle aligned with the coordinate axes. I think your formula for $P_1$ has a typo — it should be $P_1 = (0,y_1)$.

In that case, the height of the Bézier curve is $3/4$ the height of the rectangle.

Why? Well, by symmetry, the highest point must occur where $t = 1/2$. If you plug $t=1/2$ into the equation of the Bézier curve, you’ll get the coordinates of the highest point. The $y$ coordinate of this point is the value you want.