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 - アフィン変換(スカラー倍の和)

学習環境

Surface 3 (4G LTE)、Surface 3 タイプカバー、Surface ペン(端末)
Windows 10 Pro (OS)
Nebo(Windows アプリ)
iPad Pro + Apple Pencil
MyScript Nebo(iPad アプリ)
参考書籍

数学読本〈6〉線形写像・1次変換/数論へのプレリュード/集合論へのプレリュード/εとδ/落ち穂拾いなど(松坂和夫(著)、岩波書店)の第22章(図形の変換の方法 - 線形写像・1次変換)、22.2(平面の1次変換)、アフィン変換、問12.を取り組んでみる。

アフィン変換 f を

$f (\vec{p}) = g (\vec{p}) + \vec{k}$

とおく。(g は1次変換。）

$\begin{array}{l} f (r {\vec{p}}_{1} + s {\vec{p}}_{2}) \\ = g (r {\vec{p}}_{1} + s {\vec{p}}_{2}) + \vec{k} \\ = r g ({\vec{p}}_{1}) + s g ({\vec{p}}_{2}) + \vec{k} \\ = r (g ({\vec{p}}_{1}) + \vec{k}) + s (g ({\vec{p}}_{2}) + \vec{k}) - (r + s) \vec{k} + \vec{k} \\ = r f ({\vec{p}}_{1}) + s f ({\vec{p}}_{2}) - \vec{k} + \vec{k} \\ = r f ({\vec{p}}_{1}) + s f ({\vec{p}}_{2}) \end{array}$

コード(Emacs)

Python 3

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

x = symbols('x')

g = 2 * x
f = 2 * x + 1

p = plot(f, show=False, legend=True)
p.save('sample12.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample12.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="-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="k1">k1 = </label>
<input id="k1" type="number" value="1">
<label for="r1">r1 = </label>
<input id="r1" type="number" step="0.1" value="0.1">
<label for="s1">s1 = </label>
<input id="s1" type="number" step="0.1" value="0.9">
<label for="p1">p1 = </label>
<input id="p1" type="number" value="2">
<label for="p2">p2 = </label>
<input id="p2" type="number" value="3">

<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="sample12.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'),
    input_k1 = document.querySelector('#k1'),
    input_r1 = document.querySelector('#r1'),
    input_s1 = document.querySelector('#s1'),
    input_p1 = document.querySelector('#p1'),
    input_p2 = document.querySelector('#p2'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_k1, input_r1, input_s1, input_p1, input_p2],
    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 g = (x) => 2 * x;

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),
        k1 = parseFloat(input_k1.value),
        r1 = parseFloat(input_r1.value),
        s1 = parseFloat(input_s1.value),
        p1 = parseFloat(input_p1.value),
        p2 = parseFloat(input_p2.value);

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

    let points = [],
        lines = [],
        f = (x) => g(x) + k1,
        fns = [[g, 'red'],
               [f, '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])
        .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')));
    p(f(r1 * p1 + s1 * p2));
    p(r1 * f(p1) + s1 * f(p2));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

r = dx =
x1 = x2 =
y1 = y2 =
k1 = r1 = s1 = p1 = p2 =

Kamimura's blog

2018年1月20日土曜日

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 - アフィン変換(スカラー倍の和)

0 コメント:

コメントを投稿