## 2017年12月1日金曜日

### 数学 - Python - JavaScript - 線型代数 - 複素数、複素ベクトル空間 - 複素平面(絶対値、円)

• 等号の場合は、αを中心とする半径rの円の円周上の点の集合。

• αを中心とする半径rの円の円周上の点と内部の点の集合。

• 直交座標で確認。

$\begin{array}{}{\left|z-\alpha \right|}^{2}={r}^{2}\\ \left(z-\alpha \right)\stackrel{-}{\left(z-\alpha \right)}={r}^{2}\\ \left(z-\alpha \right)\left(\stackrel{-}{z}-\stackrel{-}{\alpha }\right)={r}^{2}\\ {\left|z\right|}^{2}+{\left|\alpha \right|}^{2}-\alpha \stackrel{-}{z}-\stackrel{-}{\alpha }z={r}^{2}\\ {\left|z\right|}^{2}+{\left|\alpha \right|}^{2}-2\mathrm{Re}\left(\alpha \stackrel{-}{z}\right)={r}^{2}\\ \alpha =a+bi\\ z=x+yi\\ \mathrm{Re}\left(\alpha \stackrel{-}{z}\right)\\ =\mathrm{Re}\left(\left(a+bi\right)\left(x+yi\right)\right)\\ =ax+by\\ {x}^{2}+{y}^{2}+{a}^{2}+{b}^{2}-2ax-2by={r}^{2}\\ {\left(x-a\right)}^{2}+{\left(y-b\right)}^{2}={r}^{2}\end{array}$
$y=±\sqrt{{r}^{2}-{\left(x-a\right)}^{2}}+b$

コード(Emacs)

Python 3

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

a, b, x, y = symbols('a, b, x, y', real=True)
r = symbols('r', positive=True)
α = a + b * I
z = x + y * I
eq = abs(z - α) - r

ys = solve(eq, y)
for t in ys:
for s in [t, t.expand(), t.factor()]:
pprint(s)
print()
print()

p = plot(
*map(lambda y0: y0.subs({a: 1, b: 2, r: 3}), ys), show=False, legend=True)
p.save('sample7.svg')


$./sample7.py __________________________ b - ╲╱ (-a + r + x)⋅(a + r - x) ________________________ ╱ 2 2 2 b - ╲╱ - a + 2⋅a⋅x + r - x ________________________ ╱ 2 2 2 b - ╲╱ - a + 2⋅a⋅x + r - x __________________________ b + ╲╱ (-a + r + x)⋅(a + r - x) ________________________ ╱ 2 2 2 b + ╲╱ - a + 2⋅a⋅x + r - x ________________________ ╱ 2 2 2 b + ╲╱ - a + 2⋅a⋅x + r - x$


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="0.005">
<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="2">
<label for="r1">r1 = </label>
<input id="r1" type="number" step="1" min="0" 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="sample7.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_r1 = document.querySelector('#r1'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_a0, input_b0, input_r1],
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 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),
r1 = parseFloat(input_r1.value);

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

let points = [],
lines = [[a0, y1, a0, y2, 'red'],
[x1, b0, x2, b0, 'red']],
f1 = (x) => Math.sqrt(r1 ** 2 - (x - a0) ** 2) + b0,
f2 = (x) => -Math.sqrt(r1 ** 2 - (x - a0) ** 2) + b0,
fns = [[f1, 'green'],
[f2, 'blue']],
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();