数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 曲線をえがくこと - 極座標(直交座標に変換、グラフの描画、三角関数(余弦)、絶対値)

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)、4(極座標)、練習問題19、20、21.を取り組んでみる。

19. 
    $\begin{array}{l}\cos \theta =\frac{x}{r}\\ =\frac{x}{\sqrt{x^2+y^2}}\\ \sqrt{x^2+y^2}=\frac{3\sqrt{x^2+y^2}}{x}\\ 1=\frac{3}{x}\\ x=3\end{array}$
20. 
    $\begin{array}{l}\cos \theta =\frac{x}{r}\\ =\frac{x}{\sqrt{x^2+y^2}}\\ \sqrt{x^2+y^2}=\frac{2}{2-\frac{x}{\sqrt{x^2+y^2}}}\\ \sqrt{x^2+y^2}=\frac{2\sqrt{x^2+y^2}}{2\sqrt{x^2+y^2}-x}\\ 2\sqrt{x^2+y^2}-x=2\\ 2\sqrt{x^2+y^2}-x-2=0\end{array}$
21. 
    $\begin{array}{l}\cos \theta =\frac{x}{r}\\ =\frac{x}{\sqrt{x^2+y^2}}\\ \sqrt{x^2+y^2}=\left|1+\frac{2x}{\sqrt{x^2+y^2}}\right|\\ x^2+y^2=\left|\sqrt{x^2+y^2}+2x\right|\\ x^2+y^2-\left|\sqrt{x^2+y^2}+2x\right|=0\end{array}$

コード(Emacs)

Python 3

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

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

x, y = symbols('x y', real=True)
a = symbols('a', positive=True)
eqs = [2 * sqrt(x ** 2 + y ** 2) - x - 2,
       x ** 2 + y ** 2 - abs(sqrt(x ** 2 + y ** 2) + 2 * x)]


for i, eq in enumerate(eqs, 20):
    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)

$ ./sample19.py
20.
⎡   ____________________     ____________________⎤
⎢-╲╱ -(x - 2)⋅(3⋅x + 2)    ╲╱ -(x - 2)⋅(3⋅x + 2) ⎥
⎢────────────────────────, ──────────────────────⎥
⎣           2                        2           ⎦

21.
⎡    ___________________________________      ________________________________
⎢   ╱      2             __________          ╱      2             __________  
⎢-╲╱  - 4⋅x  - 8⋅x - 2⋅╲╱ -8⋅x + 1  + 2    ╲╱  - 4⋅x  - 8⋅x - 2⋅╲╱ -8⋅x + 1  +
⎢────────────────────────────────────────, ───────────────────────────────────
⎢                   2                                        2                
⎢                                                                             
⎣                                                                             

___      ___________________________________      ____________________________
        ╱      2             __________          ╱      2             ________
 2   -╲╱  - 4⋅x  - 8⋅x + 2⋅╲╱ -8⋅x + 1  + 2    ╲╱  - 4⋅x  - 8⋅x + 2⋅╲╱ -8⋅x + 
───, ────────────────────────────────────────, ───────────────────────────────
                        2                                        2            
                                                                              
                                                                              

_______  ⎧    __________________________________                ______________
__       ⎪   ╱      2             _________                    ╱           ___
1  + 2   ⎪-╲╱  - 4⋅x  + 8⋅x - 2⋅╲╱ 8⋅x + 1  + 2              ╲╱  8⋅x - 2⋅╲╱ 8⋅
───────, ⎨───────────────────────────────────────  for 2⋅x + ─────────────────
         ⎪                   2                                            2   
         ⎪                                                                    
         ⎩                  nan                                    otherwise  

___________      ⎧   __________________________________               ________
______           ⎪  ╱      2             _________                   ╱        
x + 1  + 2       ⎪╲╱  - 4⋅x  + 8⋅x - 2⋅╲╱ 8⋅x + 1  + 2             ╲╱  8⋅x - 2
─────────── ≥ 0, ⎨─────────────────────────────────────  for 2⋅x + ───────────
                 ⎪                  2                                         
                 ⎪                                                            
                 ⎩                 nan                                   other

_________________      ⎧    __________________________________                
   _________           ⎪   ╱      2             _________                    ╱
⋅╲╱ 8⋅x + 1  + 2       ⎪-╲╱  - 4⋅x  + 8⋅x + 2⋅╲╱ 8⋅x + 1  + 2              ╲╱ 
───────────────── ≥ 0, ⎨───────────────────────────────────────  for 2⋅x + ───
  2                    ⎪                   2                                  
                       ⎪                                                      
wise                   ⎩                  nan                                 

_________________________      ⎧   __________________________________         
           _________           ⎪  ╱      2             _________              
 8⋅x + 2⋅╲╱ 8⋅x + 1  + 2       ⎪╲╱  - 4⋅x  + 8⋅x + 2⋅╲╱ 8⋅x + 1  + 2          
───────────────────────── ≥ 0, ⎨─────────────────────────────────────  for 2⋅x
          2                    ⎪                  2                           
                               ⎪                                              
   otherwise                   ⎩                 nan                          

      _________________________    ⎤
     ╱           _________         ⎥
   ╲╱  8⋅x + 2⋅╲╱ 8⋅x + 1  + 2     ⎥
 + ──────────────────────────── ≥ 0⎥
                2                  ⎥
                                   ⎥
         otherwise                 ⎦
<class 'TypeError'> Invalid comparison of complex -19.5 - 4.44409720865779*I

$

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="-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="sample19.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 f19 = (Θ) => 3 / Math.cos(Θ),
    f20 = (Θ) => 2 / (2 - Math.cos(Θ)),
    f21 = (Θ) => Math.abs(1 + 2 * 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 = [],
        f16_1 = (Θ) => Math.sqrt(2 * a0 ** 2 * Math.cos(2 * Θ)),
        f16_2 = (Θ) => -f16_1(Θ),
        fns = [[f19, 'red'],
               [f20, 'green'],
               [f21, 'blue']],
        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();

r = dΘ =
x1 = x2 =
y1 = y2 =