## 2017年11月28日火曜日

### 数学 - Python - JavaScript - 線型代数 - 複素数、複素ベクトル空間 - 複素平面(幾何学的解釈、平行四辺形、対角線と辺の長さ)

1. $\begin{array}{}{\left|\alpha +\beta \right|}^{2}+{\left|\alpha -\beta \right|}^{2}\\ =\left(\alpha +\beta \right)\stackrel{-}{\left(\alpha -1\beta \right)}+\left(\alpha -\beta \right)\left(\stackrel{-}{\alpha -\beta }\right)\\ =\left(\alpha +\beta \right)\left(\stackrel{-}{\alpha }-1\stackrel{-}{\beta }\right)+\left(\alpha -\beta \right)\left(\stackrel{-}{\alpha }-\stackrel{-}{\beta }\right)\\ =\alpha \stackrel{-}{\alpha }+\beta \stackrel{-}{\beta }+\stackrel{-}{\alpha }\beta +\stackrel{-}{\alpha }\beta +\alpha \stackrel{-}{\alpha }+\beta \stackrel{-}{\beta }-\alpha \stackrel{-}{\beta }-\stackrel{-}{\alpha }\beta \\ ={\left|\alpha \right|}^{2}+{\left|\beta \right|}^{2}+{\left|\alpha \right|}^{2}+{\left|\beta \right|}^{2}\\ =2\left({\left|\alpha \right|}^{2}+{\left|\beta \right|}^{2}\right)\end{array}$

幾何学的解釈。

平行四辺形の2つの対角線の2乗の和は、2つの辺の2乗の和の2倍と等しい。

コード(Emacs)

Python 3

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

a, b = symbols('a, b', imag=True)

l = abs(a + b) ** 2 + abs(a - b) ** 2
r = 2 * (abs(a) ** 2 + abs(b) ** 2)

for t in [l, r, l.factor() == r.factor()]:
pprint(t)
print()

a, b, c, d = symbols('a, b, c, d', real=True)
za = a + b * I
zb = c + d * I

l = abs(za + zb) ** 2 + abs(za - zb) ** 2
r = 2 * (abs(za) ** 2 + abs(zb) ** 2)

for t in [l, r, l == r]:
pprint(t)
print()


$./sample3.py 2 2 │a - b│ + │a + b│ 2 2 2⋅│a│ + 2⋅│b│ False 2 2 2 2 2⋅a + 2⋅b + 2⋅c + 2⋅d 2 2 2 2 2⋅a + 2⋅b + 2⋅c + 2⋅d True$


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="-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="a0">a0 = </label>
<input id="a0" type="number" step="1" value="1">
<label for="b0">b0 = </label>
<input id="b0" type="number" step="1" value="4">
<br>
<label for="c0">c0 = </label>
<input id="c0" type="number" step="1" value="2">
<label for="d0">d0 = </label>
<input id="d0" type="number" step="1" value="3">

<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,
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_a0 = document.querySelector('#a0'),
input_b0 = document.querySelector('#b0'),
input_c0 = document.querySelector('#c0'),
input_d0 = document.querySelector('#d0'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_a0, input_b0, input_c0, input_d0],
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),
a0 = parseFloat(input_a0.value),
b0 = parseFloat(input_b0.value),
c0 = parseFloat(input_c0.value),
d0 = parseFloat(input_d0.value);

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

let points = [],
lines = [[0, 0, a0, b0, 'red'],
[0, 0, c0, d0, 'green'],
[0, 0, a0 + c0, b0 + d0, '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])
let yscale = d3.scaleLinear()
.domain([y1, y2])

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();