2017年10月20日金曜日

学習環境

オイラーの贈物―人類の至宝eiπ=-1を学ぶ (吉田 武(著)、東海大学出版会)の第III部(オイラーの公式とその応用(Euler's Formula & Its Applications))、第8章(オイラーの公式(Euler's Formula))、8.2(オイラーの公式の応用)、8.2.2(代数方程式への応用: 1のn乗根)、問題3.を取り組んでみる。


  1. sin( 2π 5 + 2π 5 )=2sin 2π 5 cos 2π 5 sin 4π 5 =2· 1 4 10+2 5 · 5 1 4 = ( 10+2 5 )( 62 5 ) 2·4 = 2 ( 5+ 5 )( 3 5 ) 2·4 = 102 5 4 sin π 5 =sin( π π 5 ) =sin 4π 5 = 102 5 4 cos( 2 5 π+ 2 5 π )= cos 2 2π 5 sin 2 2π 5 cos 4π 5 = ( 5 1 4 ) 2 ( 10+2 5 4 ) 2 = ( 62 5 )( 10+2 5 ) 4 2 = 44 5 4 2 = 1+ 5 4 cos π 5 =cos( π 4π 5 ) =cos 4π 5 = 1+ 5 4

コード(Emacs)

Python 3

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

print('問題3.')
fs = [(sin, sqrt(10 - 2 * sqrt(5)) / 4),
      (cos, (1 + sqrt(5)) / 4)]

for (f, t) in fs:
    for s in [f(pi / 5), t, f(pi / 5).factor() == t.factor()]:
        pprint(s)
        print()
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample3.py
問題3.
    __________
   ╱   √5   5 
  ╱  - ── + ─ 
╲╱     8    8 

  ____________
╲╱ -2⋅√5 + 10 
──────────────
      4       

True


1   √5
─ + ──
4   4 

1   √5
─ + ──
4   4 

True


$

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="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'),
    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) => Math.sin(x),
    g = (x) => Math.cos(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 = [[Math.PI / 5, y1, Math.PI / 5, y2, 'red'],
                 [x1, Math.sqrt(10 - 2 * Math.sqrt(5)) / 4,
                  x2, Math.sqrt(10 - 2 * Math.sqrt(5)) / 4,
                  'red'],
                 [x1, (1 + Math.sqrt(5)) / 4,
                  x2, (1 + Math.sqrt(5)) / 4,
                  'red']],
        fns = [[f, 'green'],
               [g, 'blue']],
        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();







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