2017年9月1日金曜日

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40、41、42、43、44.を取り組んでみる。


  1. x=rcosθ cosθ= x r = x x 2 + y 2 x 2 + y 2 = 1 1 x x 2 + y 2 x 2 + y 2 = x 2 + y 2 x 2 + y 2 x 1= 1 x 2 + y 2 x x 2 + y 2 x1=0

  2. x=rcosθ cosθ= x r = x x 2 + y 2 x 2 + y 2 = 2 2 x x 2 + y 2 x 2 + y 2 = 2 x 2 + y 2 2 x 2 + y 2 x 1= 2 2 x 2 + y 2 x 2 x 2 + y 2 x2=0

  3. y=0( x0 )

    半直線。


  4. y=xtan3( x0 )

    半直線。


  5. y=xtan 25 10 =xtan 5 2 ( x0 )

    半直線。

コード(Emacs)

Python 3

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

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

x, y = symbols('x y', real=True)
eqs = [sqrt(x ** 2 + y ** 2) - x - 1,
       2 * sqrt(x ** 2 + y ** 2) - x - 2,
       y - 0,
       y - x * tan(3),
       y - x * tan(Rational(5, 2))]


for i, eq in enumerate(eqs, 40):
    try:
        print(f'{i}.')
        s = solve(eq, y)
        pprint(s)
        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)

$ ./sample40.py
40.
⎡   _________    _________⎤
⎣-╲╱ 2⋅x + 1 , ╲╱ 2⋅x + 1 ⎦

41.
⎡   ____________________     ____________________⎤
⎢-╲╱ -(x - 2)⋅(3⋅x + 2)    ╲╱ -(x - 2)⋅(3⋅x + 2) ⎥
⎢────────────────────────, ──────────────────────⎥
⎣           2                        2           ⎦

42.
[0]
<class 'TypeError'> __init__() missing 1 required positional argument: 'var_start_end'

43.
[x⋅tan(3)]

44.
[x⋅tan(5/2)]

$

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">

<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="sample40.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'),
    inputs = [input_r, input_dΘ, 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 f40 = (Θ) => 1 / (1 - Math.cos(Θ)),
    f41 = (Θ) => 2 / (2 - 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);

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

    let points = [],
        lines = [],
        fns = [[f40, 'red'],
               [f41, 'green']],
        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);

                if (r >= 0) {
                    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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