What is the value of my Bézier curve (with known control points), at $t = \frac12$?

Question

A Bézier curve $Q$ has control points $P_0 = (0,0,0)$, $P_1 = (0,1,0)$, $P_2 = (1,1,0)$ and $P_3 = (2,0,0)$. What point is $Q(\frac12)$?

Answer 1

The equation of the Bézier curve is $$ Q(t) = (1-t)^3P_0 + 3t(1-t)^2P_1 + 3t^2(1-t)P_2 + t^3P_3 $$ Setting $t=\tfrac12$ gives $$ Q(\tfrac12) = \tfrac18 P_0 + \tfrac38 P_1 + \tfrac38 P_2 + \tfrac18 P_3 $$ Then substitute your known control points.