On the segments $P_0P_1$, $P_1P_2$ and $P_2P_3$ the points $Q_0$,$Q_1$ and $Q_2$ are drawn such that $\frac{P_0Q_0}{P_0P_1} = \frac{P_1Q_1}{P_1P_2} = \frac{P_2Q_2}{P_2P_3} = t$, with $t$ in [0,1].
On the segments $Q_0Q_1$ and $Q_1Q_2$ the points $R_0$ and $R_1$ are drawn such that $\frac{Q_0R_0}{Q_0Q_1} = \frac{Q_1R_1}{Q_1Q_2} = t$ for the same value of $t$.
On the segment $R_0R_1$ the point $B$ is drawn such that $\frac{R_0B}{R_0R_1} = t$ for the same value of $t$.
The Bézier curve defined by the points $P_0$, $P_1$, $P_2$, $P_3$ is the locus of $B$ as $Q_0$ takes all possible positions on the segment $P_0P_1$. (Please note that for all points the value of $t$ is the same.)
From the construction it is clear that the Bézier curve will be tangent to the segments $P_0P_1$ in $P_0$ and $P_2P_3$ in $P_3$.
A cubic Bézier curve with $P_0 = (1, 0)$, $P_1 = (1, v)$, $P_2 = (v, 1)$ and $P_3 = (0, 1)$ is used to approximate a quarter circle. The value $v > 0$ is chosen such that the area enclosed by the lines $OP_0$, $OP_3$ and the curve is equal to $\frac{π}{4}$ (the area of the quarter circle).
By how many percent does the length of the curve differ from the length of the quarter circle? That is, if $L$ is the length of the curve, calculate $100 × \displaystyle\frac{L − \frac{π}{2}}{\frac{π}{2}}$. Give your answer rounded to 10 digits behind the decimal point.
