Problem 363
Bézier Curves
A cubic Bézier curve is defined by four points: P0, P1, P2 and P3.
The curve is constructed as follows:
On the segments P0P1, P1P2 and P2P3 the points Q0,Q1 and Q2 are drawn such that
P0Q0 / P0P1 = P1Q1 / P1P2 = P2Q2 / P2P3 = t (t in [0,1]).
On the segments Q0Q1 and Q1Q2 the points R0 and R1 are drawn such that
Q0R0 / Q0Q1 = Q1R1 / Q1Q2 = t for the same value of t.
On the segment R0R1 the point B is drawn such that R0B / R0R1 = t for the same value of t.
The Bézier curve defined by the points P0, P1, P2, P3 is the locus of B as Q0 takes all possible positions on the segment P0P1.
(Please note that for all points the value of t is the same.)
At this (external) web address you will find an applet that allows you to drag the points P0, P1, P2 and P3 to see what the Bézier curve (green curve) defined by those points looks like. You can also drag the point Q0 along the segment P0P1.
From the construction it is clear that the Bézier curve will be tangent to the segments P0P1 in P0 and P2P3 in P3.
A cubic Bézier curve with P0=(1,0), P1=(1,v), P2=(v,1) and P3=(0,1) is used to approximate a quarter circle.
The value v > 0 is chosen such that the area enclosed by the lines OP0, OP3 and the curve is equal to 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 × $\frac{L − π/2}{π/2}$.
Give your answer rounded to 10 digits behind the decimal point.
贝塞尔曲线
一条立方贝塞尔曲线由四个点确定：P0、P1、P2和P3。
曲线的构造方式如下所述：
在线段P0P1、P1P2和P2P3上，取点Q0、Q1和Q2，满足
P0Q0 / P0P1 = P1Q1 / P1P2 = P2Q2 / P2P3 = t (t 属于 [0,1])。
在线段Q0Q1和Q1Q2上，取点R0和R1，满足
Q0R0 / Q0Q1 = Q1R1 / Q1Q2 = t，t的值同上。
在线段R0R1上，取点B，满足R0B / R0R1 = t，t的值同上。
由点P0、P1、P2、P3确定的贝塞尔曲线即为Q0在线段P0P1上取遍所有位置时点B的轨迹。
（请注意，对于所有的取点过程，t的值都是相同的。）
在这个（外部链接）网址，你会找到一个应用，允许你拖动点P0、P1、P2和P3，以观察由这些点确定的贝塞尔曲线（绿色曲线）的外观。你也可以在线段P0P1上拖动点Q0。
根据构造过程，显然贝塞尔曲线在P0处与线段P0P1相切，在P3处于P2P3相切。
由P0=(1,0)、P1=(1,v)、P2=(v,1)和P3=(0,1)确定的贝塞尔曲线被用于近似表示四分之一圆.
适当地选取v > 0，以使由线段OP0、OP3和该曲线包围的面积等于π/4（四分之一圆的面积）。
该曲线的长度和四分之一圆弧的长度相差百分之多少？
也就是说，如果该曲线的长度为L，试计算100 × $\frac{L − π/2}{π/2}$。
将你的答案保留小数点后10位数字。