数学 - Python - JavaScript - 線型代数 - 複素数、複素ベクトル空間 - 極形式(2点と商、作図方法、絶対値と偏角)

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

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)、3(極形式)、問題3.を取り組んでみる。

2点の複素数の極形式をそれぞれ、

$\begin{array}{l} α_{1} = r_{1} (\cos θ_{1} + i \sin θ_{2}) \\ α_{2} = r_{2} (\cos θ_{2} - 1 i \sin θ_{2}) \end{array}$

とする。
このとき、商と絶対値と偏角ついて、

$\begin{array}{l} \frac{α_{1}}{α_{2}} \\ = \frac{r_{1} (\cos θ_{1} + i \sin θ_{1})}{r_{2} (\cos θ_{2} - - l i \sin θ_{2})} \\ = \frac{r_{1}}{r_{2}} (\cos (θ_{1} - θ_{2}) + i \sin (θ_{1} - θ_{2})) \\ | \begin{matrix} α_{1} \\ \bar{α_{2}} \end{matrix} | \end{array} = \frac{r_{1}}{r_{2}}, \arg (\frac{α_{1}}{α_{2}}) = θ_{1} - θ_{2}$

となる。
このことから作図すればいい。

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, sqrt, sin, cos, solve, I

θ1, θ2 = symbols('θ1, θ2', real=True)
r1, r2 = symbols('r1, r2', positive=True)

z1 = r1 * (cos(θ1) + I * sin(θ1))
z2 = r2 * (cos(θ2) + I * sin(θ2))
z3 = z1 / z2
for z in [z1, z2, z3]:
    for s in [z, abs(z)]:
        pprint(s)
        print()
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample3.py
r₁⋅(ⅈ⋅sin(θ1) + cos(θ1))

      _____________________
     ╱    2          2     
r₁⋅╲╱  sin (θ1) + cos (θ1) 


r₂⋅(ⅈ⋅sin(θ2) + cos(θ2))

      _____________________
     ╱    2          2     
r₂⋅╲╱  sin (θ2) + cos (θ2) 


r₁⋅(ⅈ⋅sin(θ1) + cos(θ1))
────────────────────────
r₂⋅(ⅈ⋅sin(θ2) + cos(θ2))

      _____________________
     ╱    2          2     
r₁⋅╲╱  sin (θ1) + cos (θ1) 
───────────────────────────
      _____________________
     ╱    2          2     
r₂⋅╲╱  sin (θ2) + cos (θ2) 


$

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>
<label for="r1">r1 = </label>
<input id="r1" type="number" min="0" value="8">
<label for="θ1">θ1 = </label>
<input id="θ1" type="number" min="0" value="1.5">
<br>
<label for="r2">r2 = </label>
<input id="r2" type="number" min="0" value="4">
<label for="θ2">θ2 = </label>
<input id="θ2" type="number" min="0" value="1">

<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="sample3.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_r1 = document.querySelector('#r1'),
    input_θ1 = document.querySelector('#θ1'),
    input_r2 = document.querySelector('#r2'),
    input_θ2 = document.querySelector('#θ2'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_r1, input_θ1, input_r2, input_θ2],
    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 fx = (a, x, y) => x * Math.cos(a) - y * Math.sin(a),
    fy = (a, x, y) => x * Math.sin(a) + y * Math.cos(a);

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),
        r1 = parseFloat(input_r1.value),
        θ1 = parseFloat(input_θ1.value),
        r2 = parseFloat(input_r2.value),
        θ2 = parseFloat(input_θ2.value);
        

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

    let points = [],
        r3 = r1 / r2,
        θ3 = θ1 - θ2,
        real1 = r1 * Math.cos(θ1),
        imag1 = r1 * Math.sin(θ1),
        real2 = r2 * Math.cos(θ2),
        imag2 = r2 * Math.sin(θ2),
        real3 = r3 * Math.cos(θ3),
        imag3 = r3 * Math.sin(θ3),
        lines = [[0, 0, real1, imag1, 'red'],
                 [0, 0, real2, imag2, 'green'],
                 [0, 0, real3, imag3, '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 =
r1 = θ1 =
r2 = θ2 =

Kamimura's blog

2017年12月7日木曜日

数学 - Python - JavaScript - 線型代数 - 複素数、複素ベクトル空間 - 極形式(2点と商、作図方法、絶対値と偏角)

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