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 - 直線を像にもつ図形(逆像)

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)、直線を像にもつ図形、問17.を取り組んでみる。

17. 1. 
       
       $(\begin{array}{cc}1& -2\\ 2& 3\end{array}\left)(\begin{array}{c}x\\ y\end{array}\right)=(\begin{array}{c}x-2y\\ 2x+3y\end{array})$
       $2x+3y=0$
    2. 
       
       $\begin{array}{}2x+3y=x-2y\\ x+5y=0\end{array}$
    3. 
       
       $\begin{array}{}2x+3y=2\left(x-2y\right)-5\\ 7y+5=0\end{array}$
    4. 
       
       $\begin{array}{}3\left(x-2y\right)+2x+3y-4=0\\ 5x-3y-4=0\end{array}$

コード(Emacs)

Python 3

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

x, y = symbols('x, y')
x0, y0 = symbols('x0, y0')
d = {x0: x - 2 * y, y0: 2 * x + 3 * y}
fs = [y0, y0 - x0, y0 - 2 * x0 + 5, 3 * x0 + y0 - 4]
for i, f in enumerate(fs, 1):
    print(f'({i})')
    pprint(f.subs(d))

gs = [solve(f.subs(d), y)[0] for f in fs]
hs = [solve(f.subs({x0: x, y0: y}), y)[0] for f in fs]
for t in zip(hs, gs):
    pprint(t)
p = plot(*hs, *gs, show=False, legend=True)
colors = ['red', 'green', 'blue', 'orange',
          'brown', 'pink', 'purple', 'yellow']
for i, color in enumerate(colors):
    p[i].line_color = color
p.save('sample17.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample17.py
(1)
2⋅x + 3⋅y
(2)
x + 5⋅y
(3)
7⋅y + 5
(4)
5⋅x - 3⋅y - 4
⎛   -2⋅x ⎞
⎜0, ─────⎟
⎝     3  ⎠
⎛   -x ⎞
⎜x, ───⎟
⎝    5 ⎠
(2⋅x - 5, -5/7)
⎛          5⋅x   4⎞
⎜-3⋅x + 4, ─── - ─⎟
⎝           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="-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">

<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="sample17.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 f1 = (x) => 0,
    g1 = (x) => -2 / 3 * x,
    f2 = (x) => x,
    g2 = (x) => -1 / 5 * x,
    f3 = (x) => 2 * x - 5,
    g3 = (x) => - 7 / 5,
    f4 = (x) => -3 * x + 4,
    g4 = (x) => 5 / 3 * x - 4;

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 = [],
        fns = [[f1, 'red'],
               [g1, 'green'],
               [f2, 'blue'],
               [g2, 'orange'],
               [f3, 'brown'],
               [g3, 'pink'],
               [f4, 'purple'],
               [g4, 'yellow']],
        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]);
            }
        });

    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 =