## 2018年1月24日水曜日

### 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 - 直線の像の求め方(x軸に平行な直線、直線を通る2点、方向ベクトル、媒介変数、逆変換、逆像)

1. $\begin{array}{}y=0,x=3\\ x=0,y=\frac{3}{2}\end{array}$
$\left(\begin{array}{cc}1& 3\\ 2& 4\end{array}\right)\left(\begin{array}{c}3\\ 0\end{array}\right)=\left(\begin{array}{c}3\\ 6\end{array}\right)$
$\left(\begin{array}{cc}1& 3\\ 2& 4\end{array}\right)\left(\begin{array}{c}0\\ \frac{3}{2}\end{array}\right)=\left(\begin{array}{c}\frac{9}{2}\\ 6\end{array}\right)$
$y=6$

2. 直線は

$\left(0,\frac{3}{2}\right)$

を通る。
方向ベクトルは、

$\left(-2,1\right)$
よって、 t を媒介変数として、
$\begin{array}{}x=-2t\\ y=t+\frac{3}{2}\end{array}$
$\left(\begin{array}{cc}1& 3\\ 2& 4\end{array}\right)\left(\begin{array}{c}-2t\\ t+\frac{3}{2}\end{array}\right)=\left(\begin{array}{c}-2t+3t+\frac{9}{2}\\ -4t+4t+6\end{array}\right)=\left(\begin{array}{c}t+\frac{9}{2}\\ 6\end{array}\right)$
$\begin{array}{}x=t+\frac{9}{2}\\ y=6\end{array}$

よって求める直線は、

$y=6$

3. $\left(\begin{array}{c}x\text{'}\\ y\text{'}\end{array}\right)=\left(\begin{array}{cc}1& 3\\ 2& 4\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$
$\frac{1}{4-6}\left(\begin{array}{cc}4& -3\\ -2& 1\end{array}\right)\left(\begin{array}{c}x\text{'}\\ y\text{'}\end{array}\right)=\left(\begin{array}{c}x\\ y\end{array}\right)$
$\begin{array}{}x=-\frac{1}{2}\left(4x\text{'}-3y\text{'}\right)\\ y=-\frac{1}{2}\left(-2x\text{'}+y\text{'}\right)\end{array}$
$\begin{array}{}-\frac{1}{2}\left(4x-3y\right)+2\left(-\frac{1}{2}\left(-2x+y\right)\right)=3\\ -2x+\frac{3}{2}y+2x-y=3\\ \frac{1}{2}y=3\\ y=6\end{array}$

コード(Emacs)

Python 3

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

x = symbols('x')
f = - x / 2 + Rational(3, 2)
g = 6
p = plot(f, 6, show=False, legend=True)
for i, color in enumerate(['red', 'green']):
p[i].line_color = color
p.save('sample16_1.svg')


$./sample16.py$


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="-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">

<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="sample16.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 f = (x) => -1 / 2 * x + 3 / 2,
gx = (x, y) => x + 3 * y,
gy = (x, y) => 2 * x + 4 * y;

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 = [[f, 'red']],
fns1 = [],
fns2 = [];

for (let x = 5 * x1; x <= 5 * x2; x += dx) {
let y = f(x);

points.push([gx(x, y), gy(x, y), 'green']);
}
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();