## 2018年2月9日金曜日

### 数学 - Python - JavaScript - 図形の変換の方法 - 線形写像・1次変換 – 1次変換による色々な図形の像 - 1次変換で2次曲線は2次曲線に移る(単位円、楕円の長軸、短軸上の頂点)

1. 長軸上の頂点。

$\frac{1}{4-1}\left(\begin{array}{cc}2& -1\\ -1& 2\end{array}\right)\left(\begin{array}{c}\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\end{array}\right)=\left(\begin{array}{c}\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\end{array}\right)$
$\frac{1}{3}\left(\begin{array}{cc}2& -1\\ -1& 2\end{array}\right)\left(\begin{array}{c}-\frac{3}{\sqrt{2}}\\ -\frac{3}{\sqrt{2}}\end{array}\right)=\left(\begin{array}{c}-\frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}}\end{array}\right)$

短軸上の頂点。

$\frac{1}{3}\left(\begin{array}{cc}2& -1\\ -1& 2\end{array}\right)\left(\begin{array}{c}-\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\end{array}\right)=\frac{1}{3}\left(\begin{array}{c}\frac{-2-1}{\sqrt{2}}\\ \frac{1+2}{\sqrt{2}}\end{array}\right)=\left(\begin{array}{c}-\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\end{array}\right)$
$\frac{1}{3}\left(\begin{array}{cc}2& -1\\ -1& 2\end{array}\right)\left(\begin{array}{c}\frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}}\end{array}\right)=\frac{1}{3}\left(\begin{array}{c}\frac{2+1}{\sqrt{2}}\\ \frac{-1-2}{\sqrt{2}}\end{array}\right)=\left(\begin{array}{c}\frac{1}{\sqrt{2}}\\ \frac{-1}{\sqrt{2}}\end{array}\right)$

コード(Emacs)

Python 3

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

points = [(3 / sqrt(2), 3 / sqrt(2)),
(-3 / sqrt(2), -3 / sqrt(2)),
(-1 / sqrt(2), 1 / sqrt(2)),
(1 / sqrt(2), -1 / sqrt(2))]
A = Matrix([[2, 1],
[1, 2]])
A1 = A ** -1
for p in points:
X = Matrix(p).reshape(2, 1)
for t in [X, A1 * X]:
pprint(t.T)
print()
print()

x, y = symbols('x, y')
eqs = [x ** 2 + y ** 2 - 1,
5 * x ** 2 - 8 * x * y + 5 * y ** 2 - 9]

for eq in eqs:
pprint(solve(eq, y))


$./sample21.py ⎡3⋅√2 3⋅√2⎤ ⎢──── ────⎥ ⎣ 2 2 ⎦ ⎡√2 √2⎤ ⎢── ──⎥ ⎣2 2 ⎦ ⎡-3⋅√2 -3⋅√2 ⎤ ⎢────── ──────⎥ ⎣ 2 2 ⎦ ⎡-√2 -√2 ⎤ ⎢──── ────⎥ ⎣ 2 2 ⎦ ⎡-√2 √2⎤ ⎢──── ──⎥ ⎣ 2 2 ⎦ ⎡-√2 √2⎤ ⎢──── ──⎥ ⎣ 2 2 ⎦ ⎡√2 -√2 ⎤ ⎢── ────⎥ ⎣2 2 ⎦ ⎡√2 -√2 ⎤ ⎢── ────⎥ ⎣2 2 ⎦ ⎡ __________ __________⎤ ⎢ ╱ 2 ╱ 2 ⎥ ⎣-╲╱ - x + 1 , ╲╱ - x + 1 ⎦ ⎡ __________ __________⎤ ⎢ ╱ 2 ╱ 2 ⎥ ⎢4⋅x 3⋅╲╱ - x + 5 4⋅x 3⋅╲╱ - x + 5 ⎥ ⎢─── - ───────────────, ─── + ───────────────⎥ ⎣ 5 5 5 5 ⎦$


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="sample21.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) => - Math.sqrt(- (x ** 2) + 1),
f2 = (x) => -f1(x),
fns = [[f1, 'red'],
[f2, 'green'],
[(x) => 4 * x / 5 - 3 * Math.sqrt(-(x ** 2) + 5) / 5, 'blue'],
[(x) => 4 * x / 5 + 3 * Math.sqrt(-(x ** 2) + 5) / 5, 'orange']];

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 = [],
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();