学習環境
- Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
- Windows 10 Pro (OS)
- 数式入力ソフト(TeX, MathML): MathType
- MathML対応ブラウザ: Firefox、Safari
- MathML非対応ブラウザ(Internet Explorer, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
- 参考書籍
解析入門 原書第3版 (S.ラング(著)、松坂 和夫(翻訳)、片山 孝次(翻訳)、岩波書店)の第2部(微分と基本的な関数)、第6章(曲線をえがくこと)、4(極座標)、練習問題10、11、12.を取り組んでみる。
中心(3, 0)、半径3の円。
コード(Emacs)
Python 3
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
from sympy import pprint, symbols, solve, sqrt, plot
x, y = symbols('x y', real=True)
eqs = [x ** 4 - x ** 3 + 4 * x ** 2 * y ** 2 + 3 * x * y ** 2 + y ** 4,
(x ** 2 + y ** 2) ** 2 - abs(4 * x ** 3 - 3 * x * (x ** 2 + y ** 2)),
(x - 3) ** 2 + y ** 2 - 3 ** 2]
for i, eq in enumerate(eqs, 10):
print(f'{i}.')
s = solve(eq, y)
pprint(s)
try:
p = plot(*s, show=False, legend=True)
p.save(f'sample{i}.svg')
except Exception as err:
print(type(err), err)
print()
入出力結果(Terminal, IPython)
$ ./sample10.py
10.
⎡ ________________________________________ ______________________
⎢ ╱ __________________ ╱ _______
⎢ ╱ ╱ 2 ╱ ╱ 2
⎢ ╱ 2 x⋅╲╱ 12⋅x + 28⋅x + 9 3⋅x ╱ 2 x⋅╲╱ 12⋅x
⎢- ╱ - 2⋅x - ─────────────────────── - ─── , ╱ - 2⋅x - ────────────
⎣ ╲╱ 2 2 ╲╱ 2
__________________ ________________________________________ ___
___________ ╱ __________________ ╱
╱ ╱ 2 ╱
+ 28⋅x + 9 3⋅x ╱ 2 x⋅╲╱ 12⋅x + 28⋅x + 9 3⋅x ╱
─────────── - ─── , - ╱ - 2⋅x + ─────────────────────── - ─── , ╱ -
2 ╲╱ 2 2 ╲╱
_____________________________________⎤
__________________ ⎥
╱ 2 ⎥
2 x⋅╲╱ 12⋅x + 28⋅x + 9 3⋅x ⎥
2⋅x + ─────────────────────── - ─── ⎥
2 2 ⎦
11.
⎡ ______________________________ ______________________________
⎢ ╱ ⎛ ___________ ⎞ ╱ ⎛ ___________ ⎞
⎢√2⋅╲╱ x⋅⎝-2⋅x - ╲╱ -16⋅x + 9 + 3⎠ -√2⋅╲╱ x⋅⎝-2⋅x + ╲╱ -16⋅x + 9 + 3⎠
⎢────────────────────────────────────, ──────────────────────────────────────,
⎢ 2 2
⎢
⎣
______________________________ ______________________________
╱ ⎛ ___________ ⎞ ╱ ⎛ ___________ ⎞
√2⋅╲╱ x⋅⎝-2⋅x + ╲╱ -16⋅x + 9 + 3⎠ -√2⋅╲╱ -x⋅⎝2⋅x + ╲╱ -16⋅x + 9 - 3⎠
────────────────────────────────────, ──────────────────────────────────────,
2 2
⎧ _____________________________
⎪ ╱ ⎛ __________ ⎞ 2 ⎛ __________ ⎞
⎪√2⋅╲╱ x⋅⎝-2⋅x - ╲╱ 16⋅x + 9 - 3⎠ x ⋅⎝8⋅x + 3⋅╲╱ 16⋅x + 9 + 9⎠
⎨─────────────────────────────────── for ───────────────────────────── ≥ 0,
⎪ 2 2
⎪
⎩ nan otherwise
⎧ _____________________________ ⎧
⎪ ╱ ⎛ __________ ⎞ 2 ⎛ __________ ⎞ ⎪
⎪√2⋅╲╱ x⋅⎝-2⋅x + ╲╱ 16⋅x + 9 - 3⎠ x ⋅⎝8⋅x - 3⋅╲╱ 16⋅x + 9 + 9⎠ ⎪
⎨─────────────────────────────────── for ───────────────────────────── ≥ 0, ⎨
⎪ 2 2 ⎪
⎪ ⎪
⎩ nan otherwise ⎩
_____________________________
╱ ⎛ __________ ⎞ 2 ⎛ __________ ⎞
-√2⋅╲╱ -x⋅⎝2⋅x - ╲╱ 16⋅x + 9 + 3⎠ x ⋅⎝8⋅x - 3⋅╲╱ 16⋅x + 9 + 9⎠
───────────────────────────────────── for ───────────────────────────── ≥ 0,
2 2
nan otherwise
⎧ _____________________________ ⎤
⎪ ╱ ⎛ __________ ⎞ 2 ⎛ __________ ⎞ ⎥
⎪-√2⋅╲╱ -x⋅⎝2⋅x + ╲╱ 16⋅x + 9 + 3⎠ x ⋅⎝8⋅x + 3⋅╲╱ 16⋅x + 9 + 9⎠ ⎥
⎨───────────────────────────────────── for ───────────────────────────── ≥ 0⎥
⎪ 2 2 ⎥
⎪ ⎥
⎩ nan otherwise ⎦
<class 'TypeError'> Invalid comparison of complex -3550.0 + 1843.23085911668*I
12.
⎡ ____________ ____________⎤
⎣╲╱ x⋅(-x + 6) , -╲╱ -x⋅(x - 6) ⎦
$
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="dΘ">dΘ = </label> <input id="dΘ" type="number" min="0" step="0.0001" value="0.001"> <br> <label for="x1">x1 = </label> <input id="x1" type="number" value="-10"> <label for="x2">x2 = </label> <input id="x2" type="number" value="10"> <br> <label for="y1">y1 = </label> <input id="y1" type="number" value="-10"> <label for="y2">y2 = </label> <input id="y2" type="number" value="10"> <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="sample10.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_dΘ = document.querySelector('#dΘ'),
input_x1 = document.querySelector('#x1'),
input_x2 = document.querySelector('#x2'),
input_y1 = document.querySelector('#y1'),
input_y2 = document.querySelector('#y2'),
inputs = [input_r, input_dΘ, 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 f10 = (Θ) => Math.cos(3 * Θ),
f11 = (Θ) => Math.abs(Math.cos(3 * Θ)),
f12 = (Θ) => 6 * Math.cos(Θ);
let draw = () => {
pre0.textContent = '';
let r = parseFloat(input_r.value),
dΘ = parseFloat(input_dΘ.value),
x1 = parseFloat(input_x1.value),
x2 = parseFloat(input_x2.value),
y1 = parseFloat(input_y1.value),
y2 = parseFloat(input_y2.value);
if (r === 0 || dΘ === 0 || x1 > x2 || y1 > y2) {
return;
}
let points = [],
lines = [],
fns = [[f10, 'rgba(255, 0, 0, 0.1)'],
[f11, 'rgba(0, 255, 0, 0.1)'],
[f12, 'rgba(0, 0, 255, 0.1)']],
fns1 = [],
fns2 = [];
fns
.forEach((o) => {
let [f, color] = o;
for (let Θ = 0; Θ <= 2 * Math.PI; Θ += dΘ) {
let r = f(Θ),
x = r * Math.cos(Θ),
y = r * Math.sin(Θ);
points.push([x, y, color]);
}
});
fns2
.forEach((o) => {
let [f, color] = o;
for (let x = x1; x <= x2; x += dx0) {
let g = f(x);
lines.push([x1, g(x1), x2, g(x2), 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('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.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.append('g')
.attr('transform', `translate(0, ${height - padding})`)
.call(xaxis);
svg.append('g')
.attr('transform', `translate(${padding}, 0)`)
.call(yaxis);
[fns, fns1, fns2].forEach((fs) => p(fs.join('\n')));
};
inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();
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