## 2018年1月18日木曜日

### 数学 - Python - JavaScript - 図形の変換の方法 - 線形写像・1次変換 – 平面の1次変換 - 原点のまわりの回転(回転を表す行列、逆変換、直線、楕円、曲線)

1. 問題の直線上の点(u, v)を原点のまわりに60度だけ回転した点を(x, y)とすれば、

$\left(\begin{array}{cc}\mathrm{cos}\left(-\frac{\pi }{3}\right)& -\mathrm{sin}\left(-\frac{\pi }{3}\right)\\ \mathrm{sin}\left(-\frac{\pi }{3}\right)& \mathrm{cos}\left(-\frac{\pi }{3}\right)\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}u\\ v\end{array}\right)$
$\left(\begin{array}{cc}\frac{1}{2}& \frac{\sqrt{3}}{2}\\ -\frac{\sqrt{3}}{2}& \frac{1}{2}\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}u\\ v\end{array}\right)$
$\begin{array}{}u=\frac{1}{2}x+\frac{\sqrt{3}}{2}y\\ v=-\frac{\sqrt{3}}{2}x+\frac{1}{2}y\end{array}$

また、

$u+v=1$

が成り立つので、

$\begin{array}{}\frac{1}{2}x+\frac{\sqrt{3}}{2}y-\frac{\sqrt{3}}{2}x+\frac{1}{2}y=1\\ \left(1-\sqrt{3}\right)x+\left(\sqrt{3}+1\right)y=2\end{array}$

2. $\left(\begin{array}{cc}\mathrm{cos}\left(-\frac{\pi }{6}\right)& -\mathrm{sin}\left(-\frac{\pi }{6}\right)\\ \mathrm{sin}\left(-\frac{\pi }{6}\right)& \mathrm{cos}\left(-\frac{\pi }{6}\right)\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}u\\ v\end{array}\right)$
$\left(\begin{array}{cc}\frac{\sqrt{3}}{2}& \frac{1}{2}\\ -\frac{1}{2}& \frac{\sqrt{3}}{2}\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}u\\ v\end{array}\right)$
$\begin{array}{}\frac{\sqrt{3}}{2}x+\frac{1}{2}y=u\\ -\frac{1}{2}x+\frac{\sqrt{3}}{2}y=v\end{array}$
$\begin{array}{}\frac{{u}^{2}}{4}+{v}^{2}=1\\ \frac{1}{4}\left(\frac{3}{4}{x}^{2}+\frac{1}{4}{y}^{2}+\frac{\sqrt{3}}{2}xy\right)+\left(\frac{1}{4}{x}^{2}+\frac{3}{4}{y}^{2}-\frac{\sqrt{3}}{2}xy\right)=1\\ 3{x}^{2}+{y}^{2}+2\sqrt{3}xy+4{x}^{2}+12{y}^{2}-8\sqrt{3}xy=16\\ 7{x}^{2}+13{y}^{2}-6\sqrt{3}xy-16=0\end{array}$

3. $\left(\begin{array}{cc}\mathrm{cos}\left(-\frac{\pi }{4}\right)& -\mathrm{sin}\left(-\frac{\pi }{4}\right)\\ \mathrm{sin}\left(-\frac{x}{4}\right)& los\left(-\frac{\pi }{4}\right)\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}u\\ v\end{array}\right)$
$\left(\begin{array}{cc}\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}u\\ v\end{array}\right)$
$\begin{array}{}u=\frac{1}{\sqrt{2}}\left(x+y\right)\\ v=\frac{1}{\sqrt{2}}\left(-x+y\right)\end{array}$
$\begin{array}{}\sqrt{\frac{1}{\sqrt{2}}\left(x+y\right)}+\sqrt{\frac{1}{\sqrt{2}}\left(-x+y\right)}=1\\ \sqrt{x+y}+\sqrt{-x+y}=\sqrt{2}\end{array}$

コード(Emacs)

Python 3

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

x, y = symbols('x, y')
a = [
(
x + y - 1,
(1 - sqrt(3)) * x + (sqrt(3) + 1) * y - 2
),

(
x ** 2 / 4 + y ** 2 - 1,
7 * x ** 2 + 13 * y ** 2 - 6 * sqrt(3) * x * y - 16
),
(
sqrt(x) + sqrt(y) - 1,
sqrt(x + y) + sqrt(-x + y) - sqrt(2)
)
]

fs = []
for i, (f, g) in enumerate(a, 1):
print(f'({i})')
for h in [f, g]:
fs += solve(h, y)
pprint(solve(h, y))
print()
print()

p = plot(*fs, show=False, legend=True)
p.save('sample11.svg')


$./sample11.py (1) [-x + 1] [-√3⋅x + 2⋅x - 1 + √3] (2) ⎡ __________ __________⎤ ⎢ ╱ 2 ╱ 2 ⎥ ⎢-╲╱ - x + 4 ╲╱ - x + 4 ⎥ ⎢───────────────, ─────────────⎥ ⎣ 2 2 ⎦ ⎡ _____________ _____________⎤ ⎢ ╱ 2 ╱ 2 ⎥ ⎢3⋅√3⋅x 4⋅╲╱ - 4⋅x + 13 3⋅√3⋅x 4⋅╲╱ - 4⋅x + 13 ⎥ ⎢────── - ──────────────────, ────── + ──────────────────⎥ ⎣ 13 13 13 13 ⎦ (3) ⎡ 2⎤ ⎣(√x - 1) ⎦ ⎡ 2 ⎤ ⎢x 1⎥ ⎢── + ─⎥ ⎣2 2⎦$


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="0.001" value="0.005">
<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="sample11.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'),
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 f1 = (x) => -x + 1,
f2 = (x) => -Math.sqrt(3) * x + 2 * x - 1 + Math.sqrt(3),
g11 = (x) => -Math.sqrt(- (x ** 2) + 4) / 2,
g12 = (x) => -g11(x),
g21 = (x) => (3 * Math.sqrt(3) * x - 4 * Math.sqrt(-4 * x ** 2 + 13)) / 13,
g22 = (x) => (3 * Math.sqrt(3) * x + 4 * Math.sqrt(-4 * x ** 2 + 13)) / 13,
h1 = (x) => (Math.sqrt(x) - 1) ** 2,
h2 = (x) => 1 / 2 * (x ** 2 + 1);

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 = [],
fns = [[f1, 'red'],
[f2, 'blue'],
[g11, 'green'],
[g12, 'green'],
[g21, 'orange'],
[g22, 'orange'],
[h1, 'brown'],
[h2, 'purple']],
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]);
}
});

fns1
.forEach((o) => {
let [f, color] = o;

lines.push([x1, f(x1), x2, f(x2), 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();