## 2018年12月11日火曜日

### 数学 - Python - JavaScript - 解析学 - 重積分の変数変換 - 変数変換定理(カルジオイド(cardioid)、曲線の長さ、極座標、積分、三角関数(余弦、正弦)、累乗(べき乗))

1. 極座標と曲線の長さと微分について。

$\begin{array}{}x=r\mathrm{cos}\theta \\ y=r\mathrm{sin}\theta \\ {\left(\frac{\mathrm{dx}}{d\theta }\right)}^{2}+{\left(\frac{\mathrm{dy}}{d\theta }\right)}^{2}\\ ={\left(\frac{dr}{d\theta }\mathrm{cos}\theta -r\mathrm{sin}\theta \right)}^{2}+{\left(\frac{dr}{d\theta }\mathrm{sin}\theta +r\mathrm{cos}\theta \right)}^{2}\\ ={\left(\frac{dr}{d\theta }\right)}^{2}{\mathrm{cos}}^{2}\theta +{r}^{2}{\mathrm{sin}}^{2}\theta -2r\mathrm{sin}\theta \mathrm{cos}\theta \frac{dr}{d\theta }\\ +{\left(\frac{dr}{d\theta }\right)}^{2}{\mathrm{sin}}^{2}\theta +{r}^{2}{\mathrm{cos}}^{2}\theta +2r\mathrm{sin}\theta \mathrm{cos}\theta \frac{dr}{d\theta }\\ ={r}^{2}\left({\mathrm{sin}}^{2}\theta +{\mathrm{cos}}^{2}\theta \right)+{\left(\frac{dr}{d\theta }\right)}^{2}\left({\mathrm{sin}}^{2}\theta +{\mathrm{cos}}^{2}\theta \right)\\ ={r}^{2}+{\left(\frac{dr}{d\theta }\right)}^{2}\\ ={a}^{2}{\left(1+\mathrm{cos}\theta \right)}^{2}+{\left(-\mathrm{arcsin}\theta \right)}^{2}\\ ={a}^{2}{\left(1+\mathrm{cos}\theta \right)}^{2}+{a}^{2}{\mathrm{sin}}^{2}\theta \\ ={a}^{2}\left(1+2\mathrm{cos}\theta +{\mathrm{cos}}^{2}\theta +{\mathrm{sin}}^{2}\theta \right)\\ ={a}^{2}\left(2+2\mathrm{cos}\theta \right)\\ =2{a}^{2}\left(1+\mathrm{cos}\theta \right)\\ =2{a}^{2}·2{\mathrm{cos}}^{2}\left(\frac{\theta }{2}\right)\\ ={\left(\left(2a\right)\mathrm{cos}\left(\frac{\theta }{2}\right)\right)}^{2}\end{array}$

よって、 求める曲線 カルジオイドの長さは、

$\begin{array}{}2{\int }_{0}^{\pi }\sqrt{{\left(\left(2a\right)\mathrm{cos}\left(\frac{\theta }{2}\right)\right)}^{2}}d\theta \\ =4a{\int }_{0}^{\pi }\mathrm{cos}\left(\frac{\theta }{2}\right)d\theta \\ =4a{\left[2\mathrm{sin}\frac{\theta }{2}\right]}_{0}^{\pi }\\ =8a\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Integral, pi, cos, sin, sqrt, Derivative

print('5-(b).')

theta = symbols('θ', real=True)
a = symbols('a', positive=True)
r = a * (1 + cos(theta))
x = r * cos(theta)
y = r * sin(theta)
f = sqrt(Derivative(x, theta, 1) ** 2 + Derivative(y, theta, 1) ** 2)

I = 2 * Integral(f, (theta, 0, pi))

for t in [I, I.doit(), I.doit().simplify()]:
pprint(t)
print()

f = sqrt(Derivative(x, theta, 1).doit() ** 2 +
Derivative(y, theta, 1).doit() ** 2)
I = 2 * Integral(f, (theta, 0, pi))

for t in [I, I.doit(), I.doit().simplify()]:
pprint(t)
print()

I = 2 * Integral(sqrt((2 * a * cos(theta / 2)) ** 2), (theta, 0, pi))
for t in [I, I.doit()]:
pprint(t)
print()


$./sample5.py 5-(b). π ⌠ ⎮ _____________________________________________________________ ⎮ ╱ 2 2 ⎮ ╱ ⎛∂ ⎞ ⎛∂ ⎞ 2⋅⎮ ╱ ⎜──(a⋅(cos(θ) + 1)⋅sin(θ))⎟ + ⎜──(a⋅(cos(θ) + 1)⋅cos(θ))⎟ dθ ⎮ ╲╱ ⎝∂θ ⎠ ⎝∂θ ⎠ ⌡ 0 π ⌠ ⎮ ___________________ __________________________________ ⎮ ╱ 2 2 ╱ 2 2 2⋅a⋅⎮ ╲╱ sin (θ) + cos (θ) ⋅╲╱ sin (θ) + cos (θ) + 2⋅cos(θ) + 1 dθ ⌡ 0 π ⌠ ⎮ ____________ 2⋅a⋅⎮ √2⋅╲╱ cos(θ) + 1 dθ ⌡ 0 π ⌠ ⎮ ______________________________________________________________________ ⎮ ╱ ⎮ ╱ 2 ⎛ 2⋅⎮ ╲╱ (-a⋅(cos(θ) + 1)⋅sin(θ) - a⋅sin(θ)⋅cos(θ)) + ⎝a⋅(cos(θ) + 1)⋅cos(θ) ⌡ 0 ______________ 2 2 ⎞ - a⋅sin (θ)⎠ dθ π ⌠ ⎮ ___________________ __________________________________ ⎮ ╱ 2 2 ╱ 2 2 2⋅a⋅⎮ ╲╱ sin (θ) + cos (θ) ⋅╲╱ sin (θ) + cos (θ) + 2⋅cos(θ) + 1 dθ ⌡ 0 π ⌠ ⎮ ____________ 2⋅a⋅⎮ √2⋅╲╱ cos(θ) + 1 dθ ⌡ 0 π ⌠ ⎮ │ ⎛θ⎞│ 2⋅⎮ 2⋅a⋅│cos⎜─⎟│ dθ ⎮ │ ⎝2⎠│ ⌡ 0 8⋅a$


HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="0.5">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.0001" value="0.005">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5">

<button id="draw0">draw</button>
<button id="clear0">clear</button>

<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>

<script src="sample5.js"></script>


JavaScript

let div0 = document.querySelector('#graph0'),
pre0 = document.querySelector('#output0'),
width = 600,
height = 600,
btn0 = document.querySelector('#draw0'),
btn1 = document.querySelector('#clear0'),
input_r = document.querySelector('#r0'),
input_dx = document.querySelector('#dx'),
input_x1 = document.querySelector('#x1'),
input_x2 = document.querySelector('#x2'),
input_y1 = document.querySelector('#y1'),
input_y2 = document.querySelector('#y2'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
p = (x) => pre0.textContent += x + '\n',
range = (start, end, step=1) => {
let res = [];
for (let i = start; i < end; i += step) {
res.push(i);
}
return res;
};

let a = 2,
fr = theta => a * (1 + Math.cos(theta)),
fns = [];

let draw = () => {
pre0.textContent = '';

let r = parseFloat(input_r.value),
dx = parseFloat(input_dx.value),
x1 = parseFloat(input_x1.value),
x2 = parseFloat(input_x2.value),
y1 = parseFloat(input_y1.value),
y2 = parseFloat(input_y2.value);

if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
return;
}

let points = [],
lines = [];

for (let theta = 0; theta < Math.PI / 2; theta += dx) {
let r0 = fr(theta);

points.push([r0 * Math.cos(theta), r0 * Math.sin(theta), 'red']);
}
for (let theta = Math.PI / 2; theta < Math.PI; theta += dx) {
let r0 = fr(theta);

points.push([r0 * Math.cos(theta), r0 * Math.sin(theta), 'green']);
}
for (let theta = Math.PI; theta < 3 / 2 * Math.PI; theta += dx) {
let r0 = fr(theta);

points.push([r0 * Math.cos(theta), r0 * Math.sin(theta), 'blue']);
}
for (let theta = 3 / 2 * Math.PI; theta < 2 * Math.PI; theta += dx) {
let r0 = fr(theta);

points.push([r0 * Math.cos(theta), r0 * Math.sin(theta), 'orange']);
}

fns
.forEach((o) => {
let [fn, color] = o;

for (let x = x1; x <= x2; x += dx) {
let y = fn(x);

if (Math.abs(y) < Infinity) {
points.push([x, y, color]);
}
}
});

let xscale = d3.scaleLinear()
.domain([x1, x2])

let yscale = d3.scaleLinear()
.domain([y1, y2])

let xaxis = d3.axisBottom().scale(xscale);
let yaxis = d3.axisLeft().scale(yscale);
div0.innerHTML = '';
let svg = d3.select('#graph0')
.append('svg')
.attr('width', width)
.attr('height', height);

svg.selectAll('circle')
.data(points)
.enter()
.append('circle')
.attr('cx', (d) => xscale(d[0]))
.attr('cy', (d) => yscale(d[1]))
.attr('r', r)
.attr('fill', (d) => d[2] || 'green');

svg.selectAll('line')
.data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
.enter()
.append('line')
.attr('x1', (d) => xscale(d[0]))
.attr('y1', (d) => yscale(d[1]))
.attr('x2', (d) => xscale(d[2]))
.attr('y2', (d) => yscale(d[3]))
.attr('stroke', (d) => d[4] || 'black');

svg.append('g')
.attr('transform', translate(0, ${height - padding})) .call(xaxis); svg.append('g') .attr('transform', translate(${padding}, 0))
.call(yaxis);
p(fns.join('\n'));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();