数学 - Python - JavaScript - 線型代数 - 複素数、複素ベクトル空間 - 複素数と平面幾何学(正三角形の頂点となるための必要十分条件)

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)、5(複素数と平面幾何学)、問題1.を取り組んでみる。

1. 
   

   正三角形の各2つの辺のなす角を考えれば、必要十分条件は

   $\begin{array}{}\frac{\mathrm{\gamma \; <!--\; greek\; small\; letter\; gamma\; -->}-\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}}{\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}-\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}}=\frac{\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}-\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}}{\mathrm{\gamma \; <!--\; greek\; small\; letter\; gamma\; -->}-\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}}\\ \left(\mathrm{\gamma \; <!--\; greek\; small\; letter\; gamma\; -->}-\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\right)\left(\mathrm{\gamma \; <!--\; greek\; small\; letter\; gamma\; -->}-\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}\right)=-{\left(\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}-\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}\right)}^2\\ {\mathrm{\gamma \; <!--\; greek\; small\; letter\; gamma\; -->}}^2-\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\mathrm{\gamma \; <!--\; greek\; small\; letter\; gamma\; -->}-\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}\mathrm{\gamma \; <!--\; greek\; small\; letter\; gamma\; -->}+\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}=-\left({\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}}^2+{\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}}^2-2\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}\right)\\ {\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}}^2+{\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}}^2+{\mathrm{\gamma \; <!--\; greek\; small\; letter\; gamma\; -->}}^2-\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}-\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}\mathrm{\gamma \; <!--\; greek\; small\; letter\; gamma\; -->}-\mathrm{\gamma \; <!--\; greek\; small\; letter\; gamma\; -->}\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}=0\end{array}$

コード(Emacs)

Python 3

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

a, b, c = symbols('a, b, c', imag=True)

eq = (c - a) / (b - a) - (a - b) / (c - b)

for t in [eq, eq.expand(), eq.factor()]:
    pprint(t)
    print()

a1, b1, a2, b2, a3, b3 = symbols('a1, b1, a2, b2, a3, b3', real=True)
a = a1 + b1 * I
b = a2 + b2 * I
c = a3 + b3 * I
eq = a ** 2 + b ** 2 + c ** 2 - b * c - c * a - a * b
for t in [eq, eq.expand(), eq.factor()]:
    pprint(t)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample1.py
  a - b    -a + c
- ────── + ──────
  -b + c   -a + b

    a        a        b        c   
- ────── - ────── + ────── + ──────
  -b + c   -a + b   -b + c   -a + b

 2                2          2
a  - a⋅b - a⋅c + b  - b⋅c + c 
──────────────────────────────
       (a - b)⋅(b - c)        

           2                                                                  
(a₁ + ⅈ⋅b₁)  - (a₁ + ⅈ⋅b₁)⋅(a₂ + ⅈ⋅b₂) - (a₁ + ⅈ⋅b₁)⋅(a₃ + ⅈ⋅b₃) + (a₂ + ⅈ⋅b₂)

2                                        2
  - (a₂ + ⅈ⋅b₂)⋅(a₃ + ⅈ⋅b₃) + (a₃ + ⅈ⋅b₃) 

  2                                                     2                     
a₁  - a₁⋅a₂ - a₁⋅a₃ + 2⋅ⅈ⋅a₁⋅b₁ - ⅈ⋅a₁⋅b₂ - ⅈ⋅a₁⋅b₃ + a₂  - a₂⋅a₃ - ⅈ⋅a₂⋅b₁ + 

                        2                                     2               
2⋅ⅈ⋅a₂⋅b₂ - ⅈ⋅a₂⋅b₃ + a₃  - ⅈ⋅a₃⋅b₁ - ⅈ⋅a₃⋅b₂ + 2⋅ⅈ⋅a₃⋅b₃ - b₁  + b₁⋅b₂ + b₁⋅b

      2             2
₃ - b₂  + b₂⋅b₃ - b₃ 

  2                                                     2                     
a₁  - a₁⋅a₂ - a₁⋅a₃ + 2⋅ⅈ⋅a₁⋅b₁ - ⅈ⋅a₁⋅b₂ - ⅈ⋅a₁⋅b₃ + a₂  - a₂⋅a₃ - ⅈ⋅a₂⋅b₁ + 

                        2                                     2               
2⋅ⅈ⋅a₂⋅b₂ - ⅈ⋅a₂⋅b₃ + a₃  - ⅈ⋅a₃⋅b₁ - ⅈ⋅a₃⋅b₂ + 2⋅ⅈ⋅a₃⋅b₃ - b₁  + b₁⋅b₂ + b₁⋅b

      2             2
₃ - b₂  + b₂⋅b₃ - b₃ 

$

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="-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>
<input id="a1" type="number" step="0.01" value="0">
+
<input id="b1" type="number" step="0.01" value="0">i
<br>
<input id="a2" type="number" step="0.01" value="5">
+
<input id="b2" type="number" step="0.01" value="0">i
<br>
<input id="a3" type="number" step="0.01" value="2.5">
+
<input id="b3" type="number" step="0.01" value="4.3301">i


<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_a1 = document.querySelector('#a1'),
    input_b1 = document.querySelector('#b1'),
    input_a2 = document.querySelector('#a2'),
    input_b2 = document.querySelector('#b2'),
    input_a3 = document.querySelector('#a3'),
    input_b3 = document.querySelector('#b3'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_a1, input_b1,input_a2, input_b2,input_a3, input_b3],
    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 f = (x) => Math.sqrt(1 - x ** 2),
    g = (x) => -f(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),
        a1 = parseFloat(input_a1.value),
        b1 = parseFloat(input_b1.value),
        a2 = parseFloat(input_a2.value),
        b2 = parseFloat(input_b2.value),
        a3 = parseFloat(input_a3.value),
        b3 = parseFloat(input_b3.value),
        real = a1 ** 2 - a1 * a2 - a1 * a3 + a2 ** 2 - a2 * a3 + a3 ** 2 - b1 ** 2 + b1 * b2 + b1 * b3 - b2 ** 2 + b2 * b3 - b3 ** 2,
        imag = 2 * a1 * b1 - a1 * b2 - a1 * b3 - a2 * b1 + 2 * a2 * b2 - a2 * b3 - a3 * b1 - a3 * b2 + 2 * a3 * b3;

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

    let points = [],
        lines = [[a1, b1, a2, b2, 'red'],
                 [a2, b2, a3, b3, 'green'],
                 [a3, b3, a1, b1, 'blue']],
        fns = [],
        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')));
    p(`${real} + ${imag}i`);
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

r = dx =
x1 = x2 =
y1 = y2 =
+ i
+ i
+ i