学習環境
- Surface Go、タイプ カバー、ペン(端末)
- Windows 10 Pro (OS)
- Nebo(Windows アプリ)
- iPad Pro + Apple Pencil
- MyScript Nebo - MyScript(iPad アプリ(iOS))
- 参考書籍
解析入門(下) (松坂和夫 数学入門シリーズ 6) (松坂 和夫(著)、岩波書店)の第24章(重積分の変数変換)、24.2(変数変換定理)、問題5-(b).を取り組んでみる。
極座標と曲線の長さと微分について。
よって、 求める曲線 カルジオイドの長さは、
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()
入出力結果(Terminal, Jupyter(IPython))
$ ./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,
padding = 50,
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])
.range([padding, width - padding]);
let yscale = d3.scaleLinear()
.domain([y1, y2])
.range([height - padding, padding]);
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();
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