## 2018年1月17日水曜日

### 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、像、単位ベクトル、三角関数(正弦、余弦))

1. $\left(\begin{array}{cc}\mathrm{cos}\frac{\pi }{3}& -\mathrm{sin}\frac{\pi }{3}\\ \mathrm{sin}\frac{\pi }{3}& \mathrm{cos}\frac{\pi }{3}\end{array}\right)=\left(\begin{array}{cc}\frac{1}{2}& -\frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2}& \frac{1}{2}\end{array}\right)$
$\left(\begin{array}{cc}\mathrm{cos}\frac{\pi }{2}& -\mathrm{sin}\frac{\pi }{2}\\ \mathrm{sin}\frac{\pi }{2}& \mathrm{cos}\frac{-n}{2}\end{array}\right)=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$
$\left(\begin{array}{cc}\mathrm{cos}\frac{2}{3}\pi & -\mathrm{sin}\frac{2}{3}\pi \\ \mathrm{sin}\frac{2}{3}\pi & \mathrm{cos}\frac{2}{3}\pi \end{array}\right)=\left(\begin{array}{cc}-\frac{1}{2}& -\frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2}& -\frac{1}{2}\end{array}\right)$
$\left(\begin{array}{cc}\mathrm{cos}\frac{3}{4}\pi & -\mathrm{sin}\frac{3}{4}\pi \\ \mathrm{sin}\frac{3}{4}\pi & \mathrm{cos}\frac{3}{4}\pi \end{array}\right)=\left(\begin{array}{cc}-\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\end{array}\right)$
$\left(\begin{array}{cc}\mathrm{cos}\pi & -\mathrm{sin}\pi \\ \mathrm{sin}\pi & \mathrm{cos}\pi \end{array}\right)=\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)$
$\left(\begin{array}{cc}\mathrm{cos}\left(-\frac{\pi }{3}\right)& -\mathrm{sin}\left(-\frac{\pi }{3}\right)\\ \mathrm{sin}\left(-\frac{\pi }{3}\right)& \mathrm{cos}\left(-\frac{\pi }{3}\right)\end{array}\right)=\left(\begin{array}{cc}\frac{1}{2}& \frac{\sqrt{3}}{2}\\ -\frac{\sqrt{3}}{2}& \frac{1}{2}\end{array}\right)$
$\left(\begin{array}{cc}\mathrm{cos}\left(-\frac{3}{4}\pi \right)& -\mathrm{sin}\left(-\frac{3}{4}\pi \right)\\ \mathrm{sin}\left(-\frac{3}{4}\pi \right)& \mathrm{cos}\left(-\frac{3}{4}\pi \right)\end{array}\right)=\left(\begin{array}{cc}-\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\end{array}\right)$

これらの回転による点（2，1）の像。

$\left(1-\frac{\sqrt{3}}{2},\sqrt{3}+\frac{1}{2}\right)$
$\left(-1,2\right)$
$\left(-1-\frac{\sqrt{3}}{2},\sqrt{3}-\frac{1}{2}\right)$
$\left(-\frac{2+1}{\sqrt{2}},\frac{2-1}{\sqrt{2}}\right)=\left(-\frac{3}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)$
$\left(-2,-1\right)$
$\left(1+\frac{\sqrt{3}}{2},-\sqrt{3}+\frac{1}{2}\right)$
$\left(\frac{-2+1}{\sqrt{2}},-\frac{2+1}{\sqrt{2}}\right)=\left(-\frac{1}{\sqrt{2}},-\frac{3}{\sqrt{2}}\right)$

コード(Emacs)

Python 3

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

θ = symbols('θ')
A = Matrix([[cos(θ), -sin(θ)],
[sin(θ), cos(θ)]])

X = Matrix([[2],
[1]])

for θ0 in [pi / 3, pi / 2, 2 * pi / 3, 3 * pi / 4, pi, -pi / 3, -3 * pi / 4]:
B = A.subs({θ: θ0})
for t in [θ0, B, (B * X).T]:
pprint(t)
print()
print()


$./sample10.py π ─ 3 ⎡ -√3 ⎤ ⎢1/2 ────⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢√3 ⎥ ⎢── 1/2 ⎥ ⎣2 ⎦ ⎡ √3 ⎤ ⎢- ── + 1 1/2 + √3⎥ ⎣ 2 ⎦ π ─ 2 ⎡0 -1⎤ ⎢ ⎥ ⎣1 0 ⎦ [-1 2] 2⋅π ─── 3 ⎡ -√3 ⎤ ⎢-1/2 ────⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ √3 ⎥ ⎢ ── -1/2⎥ ⎣ 2 ⎦ ⎡ √3 ⎤ ⎢-1 - ── -1/2 + √3⎥ ⎣ 2 ⎦ 3⋅π ─── 4 ⎡-√2 -√2 ⎤ ⎢──── ────⎥ ⎢ 2 2 ⎥ ⎢ ⎥ ⎢ √2 -√2 ⎥ ⎢ ── ────⎥ ⎣ 2 2 ⎦ ⎡-3⋅√2 √2⎤ ⎢────── ──⎥ ⎣ 2 2 ⎦ π ⎡-1 0 ⎤ ⎢ ⎥ ⎣0 -1⎦ [-2 -1] -π ─── 3 ⎡ √3 ⎤ ⎢1/2 ── ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢-√3 ⎥ ⎢──── 1/2⎥ ⎣ 2 ⎦ ⎡√3 ⎤ ⎢── + 1 -√3 + 1/2⎥ ⎣2 ⎦ -3⋅π ───── 4 ⎡-√2 √2 ⎤ ⎢──── ── ⎥ ⎢ 2 2 ⎥ ⎢ ⎥ ⎢-√2 -√2 ⎥ ⎢──── ────⎥ ⎣ 2 2 ⎦ ⎡-√2 -3⋅√2 ⎤ ⎢──── ──────⎥ ⎣ 2 2 ⎦$


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">
<br>
<label for="θ0">θ0 = </label>
<input id="m0" type="number" step="0.01" value="3.14">

<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="sample10.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_θ0 = document.querySelector('#m0'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_θ0],
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),
θ0 = parseFloat(input_θ0.value);

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

let points = [],
lines = [[0, 0, 2, 1, 'red'],
[0, 0,
2 * Math.cos(θ0) - Math.sin(θ0),
2 * Math.sin(θ0) + Math.cos(θ0),
'green']],
fns = [[(x) => m0 * x, 'green']]
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();