数学 - Python - JavaScript - 図形の変換の方法 - 線形写像・1次変換 – 1次変換による色々な図形の像 - 領域の像(3つの直線、三角形の周)

数学読本〈6〉

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

数学読本〈6〉線形写像・1次変換/数論へのプレリュード/集合論へのプレリュード/εとδ/落ち穂拾い など(松坂 和夫(著)、岩波書店)の第22章(図形の変換の方法 - 線形写像・1次変換)、22.3(1次変換による色々な図形の像)、領域の像、問18-(2).を取り組んでみる。

18. 2. 
       
       $\frac{1}{-5+6}(\begin{array}{cc}5& 2\\ -3& -1\end{array}\left)(\begin{array}{c}x\\ y\end{array}\right)=(\begin{array}{c}5x+2y\\ -3x-y\end{array})$
       $\begin{array}{}25x+10y-3x-y+2=0\\ 22x+9y+2=0\\ y=-\frac{22}{9}x-\frac{2}{9}\end{array}$
       $\begin{array}{}-3x-y=5x+2y-2\\ 3y=-8x+2\\ y=-\frac{8}{3}x+\frac{2}{3}\end{array}$
       $\begin{array}{}5x+2y-6x-2y-5=0\\ -x-5=0\\ x=-5\end{array}$
       $\begin{array}{}-\frac{22}{9}x-\frac{2}{9}=-\frac{8}{3}x+\frac{2}{3}\\ -22x-2=-24x+6\\ 2x=8\\ x=4\\ y=-\frac{32}{3}+\frac{2}{3}=-10\end{array}$
       $y=\frac{40}{3}+\frac{2}{3}=\frac{42}{3}=14$
       $y=\frac{110}{9}-\frac{2}{9}=\frac{108}{9}=12$

       よって、求める三角形の周は、

       $\left(4,-10\right),\left(-5,14\right),\left(-5,12\right)$

       の3点を頂点とする三角形の周。

コード(Emacs)

Python 3

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

x = symbols('x')
fs = [-5 * x + -2,
      x - 2,
      -x / 2 + Rational(5, 2),
      -22 * x / 9 - Rational(2, 9),
      -8 * x / 3 + Rational(2, 3)]

for f in fs:
    pprint(f)
    print()

p = plot(*fs, (x, -5, 5), show=False, legend=True)
colors = ['red', 'green', 'blue', 'orange', 'brown']
for i, color in enumerate(colors):
    p[i].line_color = color
p.save('sample18.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample18.py
-5⋅x - 2

x - 2

  x   5
- ─ + ─
  2   2

  22⋅x   2
- ──── - ─
   9     9

  8⋅x   2
- ─── + ─
   3    3

$

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="0.001" value="0.005">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-15">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="15">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-15">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="15">

<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="sample18.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'),
    inputs = [input_r, input_dx, 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 fns1 = [[(x) => 5 * x - 2, 'red'],
           [(x) => x - 2, 'red'],
           [(x) => -x / 2 + 5 / 2, 'red'],
           [(x) => -22 / 9 * x - 2 / 9, 'green'],
           [(x) => - 8 / 3 * x + 2 / 3, 'green']];

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);

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

    let points = [],
        lines = [[-5, y1, -5, y2, 'green']],
        fns = [],
        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]);
            }
        });

    fns1
        .forEach((o) => {
            let [f, color] = o;
            
            lines.push([x1, f(x1), x2, f(x2), 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 =