2018年2月9日金曜日

学習環境

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)、22.3(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、問21.を取り組んでみる。


  1. 長軸上の頂点。

    1 4 - 1 ( 2 - 1 - 1 2 ) ( 1 2 1 2 ) = ( 1 2 1 2 )
    1 3 ( 2 - 1 - 1 2 ) ( - 3 2 - 3 2 ) = ( - 1 2 - 1 2 )

    短軸上の頂点。

    1 3 ( 2 - 1 - 1 2 ) ( - 1 2 1 2 ) = 1 3 ( - 2 - 1 2 1 + 2 2 ) = ( - 1 2 1 2 )
    1 3 ( 2 - 1 - 1 2 ) ( 1 2 - 1 2 ) = 1 3 ( 2 + 1 2 - 1 - 2 2 ) = ( 1 2 - 1 2 )

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))

入出力結果(Terminal, Jupyter(IPython))

$ ./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,
    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 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])
        .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();







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