数学 - JavaScript - 解析学 - 距離空間の位相 - 位相の基礎的諸概念(丙級、球面、閉包(内部と境界)、閉集合)

解析入門〈3〉

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〈3〉(松坂 和夫(著)、岩波書店)の第12章(距離空間の位相)、12.1(位相の基礎的諸概念)、問題1.を取り組んでみる。

1. 
   $\begin{array}{l}B\text{'}\left(a;r\right)\\ =\left\{x\in X|d\left(a,x\right)\le r\right\}\\ =\left\{x\in X|d\left(a,x\right)<r\right\}\cup \left\{x\in X|d\left(a,x\right)=r\right\}\\ =B\left(a;r\right)\cup \left\{x\in X|d\left(a,x\right)=r\right\}\\ ={\left(B\text{'}\left(a;r\right)\right)}^i\cup {\left(B\text{'}\left(a;r\right)\right)}^f\\ ={\left(B\left(a;r\right)\right)}^a\end{array}$

   よって、閉球はその閉包(内部と境界)と等しいので閉集合である。

   $\begin{array}{l}S\left(a;r\right)\\ =\left\{x\in X|d\left(a,x\right)=r\right\}\\ =\varphi \cup \left\{x\in X|d\left(a,x\right)=r\right\}\\ ={\left(S\left(a;r\right)\right)}^i\cup {\left(S\left(a;r\right)\right)}^f\\ ={\left(S\left(a;r\right)\right)}^a\end{array}$

   よって、球面はその閉包(内部と境界の和集合)と等しいので、閉集合である。

コード(Emacs)

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="0.001">
<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">
<br>
<label for="ax">ax = </label>
<input id="ax" type="number" value="0">
<label for="ay">ay = </label>
<input id="ay" type="number" value="0">

<label for="r1">r1 = </label>
<input id="r1" type="number" min="0" value="2">

<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="sample1.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'),
    input_ax = document.querySelector('#ax'),
    input_ay = document.querySelector('#ay'),
    input_r1 = document.querySelector('#r1'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_ax, input_ay, input_r1],
    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 f1 = (x) => Math.exp(-x) * Math.cos(x) ** 2,
    f2 = (x) => 1 / Math.sqrt((x ** 2 + 1) ** 3),
    f3 = (x) => Math.log(x) ** 2 / x,
    f4 = (x) => Math.sqrt(Math.exp(x) - 1),
    f5 = (x) => Math.sin(Math.log(x)),
    f6 = (x) => 1 / (1 + 2 * Math.tan(x));

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),
        ax = parseFloat(input_ax.value),
        ay = parseFloat(input_ay.value),
        r1 = parseFloat(input_r1.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        f1 = (x) => Math.sqrt(r1 ** 2 - (x - ax) ** 2) + ay,
        f2 = (x) => -Math.sqrt(r1 ** 2 - (x - ax) ** 2) + ay,
        lines = [],
        fns = [[f1, 'green'],
               [f2, 'green']];

    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();

r = dx =
x1 = x2 =
y1 = y2 =
ax = ay = r1 =