7d9496e52c
* fix: clean-up Project Euler 361-380 * fix: improve wording Co-authored-by: Sem Bauke <46919888+Sembauke@users.noreply.github.com> * fix: remove unnecessary paragraph * fix: corrections from review Co-authored-by: Tom <20648924+moT01@users.noreply.github.com> Co-authored-by: Sem Bauke <46919888+Sembauke@users.noreply.github.com> Co-authored-by: Tom <20648924+moT01@users.noreply.github.com>
57 lines
2.1 KiB
Markdown
57 lines
2.1 KiB
Markdown
---
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id: 5900f4d91000cf542c50ffeb
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title: 'Problem 363: Bézier Curves'
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challengeType: 5
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forumTopicId: 302024
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dashedName: problem-363-bzier-curves
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---
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# --description--
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A cubic Bézier curve is defined by four points: $P_0$, $P_1$, $P_2$ and $P_3$.
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The curve is constructed as follows:
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<img class="img-responsive center-block" alt="construction of Bézier curve" src="https://cdn.freecodecamp.org/curriculum/project-euler/bzier-curves.png" style="background-color: white; padding: 10px;">
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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].
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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 $t$.
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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 $t$.
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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.)
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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 $P_3$.
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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).
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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.
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# --hints--
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`bezierCurves()` should return `0.0000372091`.
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```js
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assert.strictEqual(bezierCurves(), 0.0000372091);
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```
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# --seed--
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## --seed-contents--
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```js
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function bezierCurves() {
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return true;
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}
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bezierCurves();
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```
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# --solutions--
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```js
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// solution required
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```
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