data/problems/363.yml
---
:id: 363
:name: Bézier Curves
:url: https://projecteuler.net/problem=363
:content: "A cubic Bézier curve is defined by four points: P<sub>0</sub>, P<sub>1</sub>,
P<sub>2</sub> and P<sub>3</sub>.\n\n \n\nThe
curve is constructed as follows: \nOn the segments P<sub>0</sub>P<sub>1</sub>,
P<sub>1</sub>P<sub>2</sub> and P<sub>2</sub>P<sub>3</sub> the points Q<sub>0</sub>,Q<sub>1</sub>
and Q<sub>2</sub> are drawn such that \nP<sub>0</sub>Q<sub>0</sub> / P<sub>0</sub>P<sub>1</sub>
= P<sub>1</sub>Q<sub>1</sub> / P<sub>1</sub>P<sub>2</sub> = P<sub>2</sub>Q<sub>2</sub>
/ P<sub>2</sub>P<sub>3</sub> = t (t in [0,1]). \nOn the segments Q<sub>0</sub>Q<sub>1</sub>
and Q<sub>1</sub>Q<sub>2</sub> the points R<sub>0</sub> and R<sub>1</sub> are drawn
such that \nQ<sub>0</sub>R<sub>0</sub> / Q<sub>0</sub>Q<sub>1</sub> = Q<sub>1</sub>R<sub>1</sub>
/ Q<sub>1</sub>Q<sub>2</sub> = t for the same value of t. \nOn the segment R<sub>0</sub>R<sub>1</sub>
the point B is drawn such that R<sub>0</sub>B / R<sub>0</sub>R<sub>1</sub> = t for
the same value of t. \nThe Bézier curve defined by the points P<sub>0</sub>, P<sub>1</sub>,
P<sub>2</sub>, P<sub>3</sub> is the locus of B as Q<sub>0</sub> takes all possible
positions on the segment P<sub>0</sub>P<sub>1</sub>. \n(Please note that for all
points the value of t is the same.)\n\nAt [this (external) web address](http://home.kpn.nl/hklein/bezier/bezier.html)
you will find an applet that allows you to drag the points P<sub>0</sub>, P<sub>1</sub>,
P<sub>2</sub> and P<sub>3</sub> to see what the Bézier curve (green curve) defined
by those points looks like. You can also drag the point Q<sub>0</sub> along the
segment P<sub>0</sub>P<sub>1</sub>.\n\nFrom the construction it is clear that the
Bézier curve will be tangent to the segments P<sub>0</sub>P<sub>1</sub> in P<sub>0</sub>
and P<sub>2</sub>P<sub>3</sub> in P<sub>3</sub>.\n\nA cubic Bézier curve with P<sub>0</sub>=(1,0),
P<sub>1</sub>=(1,<var>v</var>), P<sub>2</sub>=(<var>v</var>,1) and P<sub>3</sub>=(0,1)
is used to approximate a quarter circle. \nThe value <var>v</var> \\> 0 is chosen
such that the area enclosed by the lines OP<sub>0</sub>, OP<sub>3</sub> and the
curve is equal to <sup>π</sup>/<sub>4</sub> (the area of the quarter circle).\n\nBy
how many percent does the length of the curve differ from the length of the quarter
circle?\n\n| That is, if L is the length of the curve, calculate 100 × | \n\nL −
π/2\nπ/2\n |\n\nGive your answer rounded to 10 digits behind the decimal point.\n\n"