2017年8月24日木曜日

学習環境

解析入門 原書第3版 (S.ラング(著)、松坂 和夫(翻訳)、片山 孝次(翻訳)、岩波書店)の第2部(微分と基本的な関数)、第6章(曲線をえがくこと)、4(極座標)、練習問題13、14、15.を取り組んでみる。


  1. x=rcosθ cosθ= x r x 2 + y 2 =a x x 2 + y 2 x 2 + y 2 =ax ( x 1 2 a ) 2 + y 2 = ( a 2 ) 2

    中心( a 2 , 0)、半径 a 2 の円。


  2. y=rsinθ sinθ= y r x 2 + y 2 =a y x 2 + y 2 x 2 + y 2 =ay x 2 + ( y a 2 ) 2 = ( a 2 ) 2

    中心(0, a 2 )、半径 a 2 の円。


  3. x=rcosθ cosθ= x r x 2 + y 2 =1+ x x 2 + y 2 x 2 + y 2 = x 2 + y 2 +x x 2 + y 2 x x 2 + y 2 =0

コード(Emacs)

Python 3

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from sympy import pprint, symbols, solve, sqrt, plot, Rational

x, y = symbols('x y', real=True)
a = symbols('a', positive=True)
eqs = [(x - Rational(1, 2) * a) ** 2 + y ** 2 - (a / 2) ** 2,
       x ** 2 + (y - a / 2) ** 2 - (a / 2) ** 2,
       x ** 2 + y ** 2 - x - sqrt(x ** 2 + y ** 2)]

for i, eq in enumerate(eqs, 13):
    eq = eq.subs({a: 2})
    print(f'{i}.')
    s = solve(eq, y)
    pprint(s)
    try:
        p = plot(*s, show=False, legend=True)
        p.save(f'sample{i}.svg')
    except Exception as err:
        print(type(err), err)
    print()

入出力結果(Terminal, IPython)

$ ./sample13.py
13.
⎡  ____________     ____________⎤
⎣╲╱ x⋅(-x + 2) , -╲╱ -x⋅(x - 2) ⎦

14.
⎡     __________         __________    ⎤
⎢    ╱    2             ╱    2         ⎥
⎣- ╲╱  - x  + 1  + 1, ╲╱  - x  + 1  + 1⎦

15.
⎡      ____________________________       ____________________________        
⎢     ╱              _________           ╱              _________            ╱
⎢    ╱     2       ╲╱ 4⋅x + 1    1      ╱     2       ╲╱ 4⋅x + 1    1       ╱ 
⎢-  ╱   - x  + x - ─────────── + ─ ,   ╱   - x  + x - ─────────── + ─ , -  ╱  
⎣ ╲╱                    2        2   ╲╱                    2        2    ╲╱   

____________________________       ____________________________⎤
              _________           ╱              _________     ⎥
    2       ╲╱ 4⋅x + 1    1      ╱     2       ╲╱ 4⋅x + 1    1 ⎥
 - x  + x + ─────────── + ─ ,   ╱   - x  + x + ─────────── + ─ ⎥
                 2        2   ╲╱                    2        2 ⎦

$

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="dΘ">dΘ = </label>
<input id="dΘ" type="number" min="0" step="0.0001" value="0.001">
<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="a0">a = </label>
<input id="a0" type="number" value="2">

<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="sample13.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_dΘ = document.querySelector('#dΘ'),
    input_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    input_y1 = document.querySelector('#y1'),
    input_y2 = document.querySelector('#y2'),
    input_a0 = document.querySelector('#a0'),    
    inputs = [input_r, input_dΘ, input_x1, input_x2, input_y1, input_y2,
              input_a0],
    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 f15 = (Θ) => 1 + Math.cos(Θ);
        
let draw = () => {
    pre0.textContent = '';

    let r = parseFloat(input_r.value),= parseFloat(input_dΘ.value),
        x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value),
        a0 = parseFloat(input_a0.value);

    if (r === 0 ||=== 0 || x1 > x2 || y1 > y2 || a0 <= 0) {
        return;
    }    

    let points = [],
        lines = [],
        f13 = (Θ) => a0 * Math.cos(Θ),
        f14 = (Θ) => a0 * Math.sin(Θ),
        fns = [[f13, 'red'],
               [f14, 'green'],
               [f15, 'blue']],
        fns1 = [],
        fns2 = [];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let Θ = 0; Θ <= 2 * Math.PI; Θ +=) {
                let r = f(Θ),
                    x = r * Math.cos(Θ),
                    y = r * Math.sin(Θ);

                points.push([x, y, 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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