数学 - Python - JavaScript - 線型代数 - 線型写像 - 線型写像の定義と例(平面の点、回転、線型変換)

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)
• 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

線型代数入門(松坂 和夫(著)、岩波書店)の第3章(線型写像)、3(線型写像の定義と例)、問題5.を取り組んでみる。

5. • 60°回転した点。

     $\begin{array}{l}\phi \left(e_1\right)=\left(\cos \theta ,\sin \theta \right)\\ \phi \left(e_2\right)=\left(-\sin \theta ,\cos \theta \right)\\ \\ \phi \left(\left(x,y\right)\right)\\ =x\phi \left(e_1\right)+y\phi \left(e_2\right)\\ =\left(x\cos \theta ,x\sin \theta \right)+\left(-y\sin \theta ,y\cos \theta \right)\\ =\left(x\cos \theta -y\sin \theta ,x\sin \theta +y\cos \theta \right)\\ \\ \left(3\cos \frac{\pi }{3}-4\sin \frac{\pi }{3},3\sin \frac{\pi }{3}+4\cos \frac{\pi }{3}\right)\\ =\left(3·\frac{1}{2}-4·\frac{\sqrt{3}}{2},3·\frac{\sqrt{3}}{2}+4·\frac{1}{2}\right)\\ =\left(\frac{3}{2}-2\sqrt{3},\frac{3\sqrt{3}}{2}+2\right)\end{array}$

   • -135°回転した点。

     $\begin{array}{l}\left(3\cos \left(-\frac{3}{4}\pi \right)-4\sin \left(-\frac{3}{4}\pi \right),3\sin \left(-\frac{3}{4}\pi \right)+4\cos \left(-\frac{3}{4}\pi \right)\right)\\ =\left(3·\left(-\frac{1}{\sqrt{2}}\right)-4·\left(-\frac{1}{\sqrt{2}}\right),3·\left(-\frac{1}{\sqrt{2}}\right)+4·\left(-\frac{1}{\sqrt{2}}\right)\right)\\ =\left(\frac{1}{\sqrt{2}},-\frac{7}{\sqrt{2}}\right)\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from sympy import pprint, symbols, Matrix, sin, cos, pi


def f(θ):
    return Matrix([3 * cos(θ) - 4 * sin(θ), 3 * cos(θ) + 4 * sin(θ)]).T


for θ0 in [pi / 3, -3 * pi / 4]:
    pprint(f(θ0))
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample5.py
[-2⋅√3 + 3/2  3/2 + 2⋅√3]

⎡√2  -7⋅√2 ⎤
⎢──  ──────⎥
⎣2     2   ⎦

$

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">

<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="sample5.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 f = (x) => 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);

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

    let points = [],
        lines = [[0, 0, 3, 4, 'red'],
                 [0, 0, 3 / 2 - 2 * Math.sqrt(3),
                  3 * Math.sqrt(3) / 2 + 2, 'green'],
                 [0, 0, 1 / Math.sqrt(2), -7 / Math.sqrt(2), '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')));
};

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

r = dx =
x1 = x2 =
y1 = y2 =