## 2018年1月31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 - 領域の像(3つの直線、三角形の内部)

1. $\frac{1}{-5+6}\left(\begin{array}{cc}5& 2\\ -3& -1\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}5x+2y\\ -3x-y\end{array}\right)$
$\begin{array}{}25x+10y-3x-y+2=0\\ 22x+9y+2=0\end{array}$
$\begin{array}{}-3x-y=5x+2y-2\\ 8x+3y-2=0\end{array}$
$\begin{array}{}5x+2y-6x-2y-5=0\\ x=-5\end{array}$
$\begin{array}{}22x+9y+2=0\\ 24x+9y-6=0\\ 2x-8=0\\ x=4\\ y=-10\end{array}$
$\begin{array}{}x=-5\\ -40+3y-2=0\\ y=14\end{array}$
$\begin{array}{}x=-5\\ -110+9y+2=0\\ 9y=108\\ y=12\end{array}$

よって、求める像は、

$\left(4,-10\right),\left(-5,14\right),\left(-5.12\right)$

の3点を頂点とする三角形の内部。

コード(Emacs)

Python 3

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

x, y = symbols('x, y')
fs = [5 * x + y + 2,
x - y - 2,
x + 2 * y - 5]

for i in range(3):
pprint(solve((fs[i], fs[i + 1 if i < 2 else 0]), dict=True))
print()

p = plot(*[solve(f, y)[0] for f in fs], (x, -5, 5), show=False, legend=True)
colors = ['red', 'green', 'blue']
for i, color in enumerate(colors):
p[i].line_color = color
p.save('sample18.svg')


$./sample18.py [{x: 0, y: -2}] [{x: 3, y: 1}] [{x: -1, y: 3}]$


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="-15">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="15">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-15">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="15">

<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="sample18.js"></script>


JavaScript

let div0 = document.querySelector('#graph0'),
pre0 = document.querySelector('#output0'),
width = 600,
height = 600,
padding = 50,
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 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 = [[0, -2, 3, 1, 'red'],
[3, 1, -1, 3, 'green'],
[-1, 3, 0, -2, 'blue'],
[4, -10, -5, 14, 'orange'],
[-5, 14, -5, 12, 'brown'],
[-5, 12, 4, -10, 'purple']],
fns = [],
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])
.range([padding, width - padding]);
let yscale = d3.scaleLinear()
.domain([y1, y2])
.range([height - padding, padding]);

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();