数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 曲線をえがくこと - パラメーター表示による曲線(極座標表示、直交座標に変換、グラフの描画、渦巻き)

解析入門 原書第3版
原著

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
  • Pythonからはじめる数学入門
  • JavaScriptリファレンス 第6版
  • インタラクティブ・データビジュアライゼーション

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)、5(パラメーター表示による曲線)、練習問題6、7.を取り組んでみる。

6. 
   $\begin{array}{l}x=\sqrt{x^2+y^2}\cos \left(\pi t^2\right)\\ t=r^{\frac{1}{3}}={\left(x^2+y^2\right)}^{\frac{1}{6}}\\ x=\sqrt{x^2+y^2}\cos \left(\pi {\left(x^2+y^2\right)}^{\frac{1}{3}}\right)\\ \sqrt{x^2+y^2}\cos \left(\pi {\left(x^2+y^2\right)}^{\frac{1}{3}}\right)-x=0\end{array}$
7. 
   $\begin{array}{l}\sqrt{x^2+y^2}=t\\ \theta =x^2+y^2\\ x=r\cos \left(x^2+y^2\right)\\ x=\sqrt{x^2+y^2}\cos \left(x^2+y^2\right)\\ \sqrt{x^2+y^2}\cos \left(x^2+y^2\right)-x=0\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from sympy import pprint, symbols, plot, solve, Rational, sqrt, cos, pi

x, y = symbols('x y', real=True)
eqs = [sqrt(x ** 2 + y ** 2) *
       cos(pi * (x ** 2 + y ** 2) ** Rational(1, 3)) - x,
       sqrt(x ** 2 + y ** 2) * cos(x ** 2 + y ** 2) - x]


for i, eq in enumerate(eqs, 6):
    try:
        print(f'{i}.')
        s = solve(eq, y)
        pprint(s)
        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)

$ ./sample6.py
6.
<class 'NotImplementedError'> multiple generators [cos(pi*(x**2 + y**2)**(1/3)), sqrt(x**2 + y**2)]
No algorithms are implemented to solve equation -x + sqrt(x**2 + y**2)*cos(pi*(x**2 + y**2)**(1/3))

7.
<class 'NotImplementedError'> multiple generators [cos(x**2 + y**2), sqrt(x**2 + y**2)]
No algorithms are implemented to solve equation -x + sqrt(x**2 + y**2)*cos(x**2 + y**2)

$

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="dt">dt = </label>
<input id="dt" type="number" min="0" step="0.0001" value="0.001">
<br>
<label for="t1">t1 = </label>
<input id="t1" type="number" value="-10">
<label for="t2">t2 = </label>
<input id="t2" type="number" value="10">
<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="sample6.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_dt = document.querySelector('#dt'),
    input_t1 = document.querySelector('#t1'),
    input_t2 = document.querySelector('#t2'),
    input_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    input_y1 = document.querySelector('#y1'),
    input_y2 = document.querySelector('#y2'),
    inputs = [input_r, input_dt, 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 fr4 = (t) => t ** 3,
    fΘ4 = (t) => Math.PI * t ** 2,
    fx4 = (t) => fr4(t) * Math.cos(fΘ4(t)),
    fy4 = (t) => fr4(t) * Math.sin(fΘ4(t)),
    fr5 = (t) => t,
    fΘ5 = (t) => t ** 2,
    fx5 = (t) => fr5(t) * Math.cos(fΘ5(t)),
    fy5 = (t) => fr5(t) * Math.sin(fΘ5(t));

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

    let r = parseFloat(input_r.value),
        dt = parseFloat(input_dt.value),
        t1 = parseFloat(input_t1.value),
        t2 = parseFloat(input_t2.value),
        x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value);

    if (r === 0 || dt === 0 || t1 > t2 || x1 > x2 || y1 > y2) {
        return;
    }    

    let points = [],
        lines = [],
        fns = [[fx4, fy4, 'red'],
               [fx5, fy5, 'blue']],
        fns1 = [],
        fns2 = [];

    fns
        .forEach((o) => {
            let [fx, fy, color] = o;
            for (let t = t1; t <= t2; t += dt) {
                let x = fx(t),
                    y = fy(t);

                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();

r = dt =
t1 = t2 =
x1 = x2 =
y1 = y2 =