数学 - Python - JavaScript - 線型代数 - 複素数、複素ベクトル空間 - 複素平面(絶対値、円)

線型代数入門

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)
• Nebo(Windows アプリ)
• iPad Pro + Apple Pencil
• MyScript Nebo(iPad アプリ)
• 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

線型代数入門(松坂 和夫(著)、岩波書店)の第4章(複素数、複素ベクトル空間)、2(複素平面)、問題7.を取り組んでみる。

7. • 等号の場合は、αを中心とする半径rの円の円周上の点の集合。

   • αを中心とする半径rの円の円周上の点と内部の点の集合。

   • 直交座標で確認。

     $\begin{array}{}{\left|z-\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\right|}^2=r^2\\ \left(z-\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\right)\stackrel{-}{\left(z-\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\right)}=r^2\\ \left(z-\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\right)\left(\stackrel{-}{z}-\stackrel{-}{\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}}\right)=r^2\\ {\left|z\right|}^2+{\left|\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\right|}^2-\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\stackrel{-}{z}-\stackrel{-}{\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}}z=r^2\\ {\left|z\right|}^2+{\left|\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\right|}^2-2\mathrm{Re}\left(\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\stackrel{-}{z}\right)=r^2\\ \mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}=a+bi\\ z=x+yi\\ \mathrm{Re}\left(\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\stackrel{-}{z}\right)\\ =\mathrm{Re}\left(\left(a+bi\right)\left(x+yi\right)\right)\\ =ax+by\\ x^2+y^2+a^2+b^2-2ax-2by=r^2\\ {\left(x-a\right)}^2+{\left(y-b\right)}^2=r^2\end{array}$
     $y=\pm \; <!--\; plus-minus\; sign\; -->\sqrt{r^2-{\left(x-a\right)}^2}+b$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, I, solve, plot

a, b, x, y = symbols('a, b, x, y', real=True)
r = symbols('r', positive=True)
α = a + b * I
z = x + y * I
eq = abs(z - α) - r

ys = solve(eq, y)
for t in ys:
    for s in [t, t.expand(), t.factor()]:
        pprint(s)
        print()
    print()

p = plot(
    *map(lambda y0: y0.subs({a: 1, b: 2, r: 3}), ys), show=False, legend=True)
p.save('sample7.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample7.py
      __________________________
b - ╲╱ (-a + r + x)⋅(a + r - x) 

       ________________________
      ╱    2            2    2 
b - ╲╱  - a  + 2⋅a⋅x + r  - x  

       ________________________
      ╱    2            2    2 
b - ╲╱  - a  + 2⋅a⋅x + r  - x  


      __________________________
b + ╲╱ (-a + r + x)⋅(a + r - x) 

       ________________________
      ╱    2            2    2 
b + ╲╱  - a  + 2⋅a⋅x + r  - x  

       ________________________
      ╱    2            2    2 
b + ╲╱  - a  + 2⋅a⋅x + r  - x  


$

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="-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">
<br>
<label for="a0">a0 = </label>
<input id="a0" type="number" step="1" value="1">
<label for="b0">b0 = </label>
<input id="b0" type="number" step="1" value="2">
<label for="r1">r1 = </label>
<input id="r1" type="number" step="1" min="0" value="3">

<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="sample7.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_a0 = document.querySelector('#a0'),
    input_b0 = document.querySelector('#b0'),
    input_r1 = document.querySelector('#r1'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_a0, input_b0, 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 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),
        a0 = parseFloat(input_a0.value),
        b0 = parseFloat(input_b0.value),
        r1 = parseFloat(input_r1.value);

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

    let points = [],
        lines = [[a0, y1, a0, y2, 'red'],
                 [x1, b0, x2, b0, 'red']],
        f1 = (x) => Math.sqrt(r1 ** 2 - (x - a0) ** 2) + b0,
        f2 = (x) => -Math.sqrt(r1 ** 2 - (x - a0) ** 2) + b0,
        fns = [[f1, 'green'],
               [f2, 'blue']],
        fns1 = [],
        fns2 = [];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                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 = dx =
x1 = x2 =
y1 = y2 =
a0 = b0 = r1 =