数学 - Python - JavaScript - 線型代数 - 2次元と3次元の簡単な幾何学 - 直線の方程式(2)(媒介変数、平行、直交、交角、余弦、ベクトル方程式との比較、2等分線、一般化、推薦の足、点との距離)

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, 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1章(2次元と3次元の簡単な幾何学)、7(直線の方程式(2))、問1、2、3、4、5、6.を取り組んでみる。

1. 
   

   平行であるための必要十分条件。

   $\begin{array}{l}\left(x,y\right)=\left(x_0,y_0\right)+t\left(\alpha ,\beta \right)\\ \left(\alpha ,\beta \right)·\left(a,b\right)=0\\ a\alpha +b\beta =0\end{array}$

   直交するための必要十分条件。

   $a\beta -\alpha b=0$
2. 
   $\begin{array}{l}\left(3,-2\right)·\left(4,2\right)=\sqrt{9+4}·\sqrt{16+4}\cos \left(\pm \theta \right)\\ \pm \cos \theta =\frac{12-4}{\sqrt{13}\sqrt{20}}\\ =\pm \frac{8}{\sqrt{13}\sqrt{20}}\\ =\pm \frac{4}{\sqrt{13}\sqrt{5}}\\ =\pm \frac{4}{\sqrt{65}}\end{array}$
3. 
   $\begin{array}{l}\left(a-b\right)·x=\left(a-b\right)·t\left(a+b\right)\\ t\left(a-b\right)·\left(a+b\right)\\ =t\left({\left|a\right|}^2-{\left|b\right|}^2\right)\\ =0\\ \\ \left(a-b\right)·x=0\end{array}$
4. 
   $\begin{array}{l}\left|\frac{a}{\left|a\right|}\right|=\left|\frac{b}{\left|b\right|}\right|=1\ne 0\\ a\ne b\\ \frac{a}{\left|a\right|}\ne \frac{b}{\left|b\right|}\\ \left(\frac{a}{\left|a\right|}-\frac{b}{\left|b\right|}\right)·x=0\end{array}$
5. 
   $\begin{array}{l}\overrightarrow{BA}=\left(12,5\right)\\ \overrightarrow{BC}=\left(-3,4\right)\\ \left(\frac{\left(12,5\right)}{\sqrt{144+25}}-\frac{\left(-3,4\right)}{\sqrt{9+16}}\right)·\left(x-\left(1,2\right)\right)=0\\ \left(\frac{\left(12,5\right)}{13}-\frac{\left(-3,4\right)}{5}\right)·\left(x-\left(1,2\right)\right)=0\\ \left(60+39,25-52\right)·\left(x-\left(1,2\right)\right)=0\\ \left(99,-27\right)·\left(x-\left(1,2\right)\right)=0\\ \left(11,-3\right)·\left(x-1,y-2\right)=0\\ 11x-3y-5=0\end{array}$
6. 
   $\begin{array}{l}\left(a,b\right)·\left(x,y\right)=-c\\ a·x=-c\\ q=p-\frac{a·p+c}{{\left|a\right|}^2}a\\ \left(x\text{'},y\text{'}\right)=\left(x_0,y_0\right)-\frac{a{x}_0+b{y}_0+c}{\sqrt{a^2+b^2}}\left(a,b\right)\\ x\text{'}=x_0-\frac{a{x}_0+b{y}_0+c}{\sqrt{a^2+b^2}}a\\ y\text{'}=y_0-\frac{a{x}_0+b{y}_0+c}{\sqrt{a^2+b^2}}b\\ \\ d=\frac{\left|a·p+c\right|}{\left|a\right|}\\ =\frac{\left|a{x}_0+b{y}_0+c\right|}{\sqrt{a^2+b^2}}\end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, solve, cos, Matrix, sqrt

print('2.')
Θ = symbols('Θ')
eq = Matrix([3, -2]).dot(Matrix([4, 2])) - sqrt(9 + 4) * sqrt(16 + 4) * cos(Θ)
pprint(solve(eq, cos(Θ)))

入出力結果(Terminal, IPython)

$ ./sample1.py
2.
⎡4⋅√65⎤
⎢─────⎥
⎣  65 ⎦
$

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">

<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'),
    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 A = [13, 7],
    B = [1, 2],
    C = [-2, 6],
    f = (x) => 11 / 3 * x - 5 / 3;

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 = [[A[0], A[1], B[0], B[1], 'green'],
                 [C[0], C[1], B[0], B[1], 'green']],
        fns = [[f, 'orange']],
        fns1 = [],
        fns2 = [];

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

                if (Math.abs(y) < Infinity) {
                    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 =