2017年7月11日火曜日

学習環境

数学読本〈5〉微分法の応用/積分法/積分法の応用/行列と行列式(松坂 和夫(著)、岩波書店)の第18章(曲線の性質、最大・最小 - 微分法の応用)、18.4(媒介変数で表される曲線)、平面上の点の運動、速度・加速度、問50、51.を取り組んでみる。


    1. r= 2 cosθ= 1 2 sinθ= 1 2 θ= π 4 ( 2 , π 4 )

    2. r= 3+1 =2 cosθ= 3 2 sinθ= 1 2 θ= π 6 ( 2, π 6 )

    3. r= 4 =2 cosθ= 2 2 =1 sinθ=0 θ=π ( 2,π )

    4. r= 4+4 =2 2 cosθ= 2 2 2 = 1 2 sinθ= 1 2 θ= 3 4 π ( 2 2 , 3 4 π )

    1. x=2cos π 4 = 2 2 = 2 y=2sin π 4 = 2

    2. x=2 3 cos π 3 =2 3 · 3 2 = 3 y=2 3 sin π 3 =2 3 · 1 2 =3

    3. x=cos π 2 =0 y=sin π 2 =1

    4. x=4cosπ=4 y=4sinπ=0

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, sqrt, solve, cos, sin, pi

print('50.')

Θ = symbols('Θ')


def f(x, y):
    r = sqrt(x ** 2 + y ** 2)
    s = solve((r * cos(Θ) - x, r * sin(Θ) - y), Θ, dict=True)
    return (r, s[0][Θ])

for i, (x, y) in enumerate([(1, 1), (sqrt(3), -1), (-2, 0), (-2, -2)], 1):
    print(f'({i})')
    pprint(f(x, y))
    print()

print('51.')


def g(r, Θ):
    return (r * cos(Θ), r * sin(Θ))

points = [(2, pi / 4), (2 * sqrt(3), pi / 3), (1, -pi / 2), (4, pi)]
for i, (r, Θ) in enumerate(points, 1):
    print(f'({i})')
    pprint(g(r, Θ))
    print()

入出力結果(Terminal, IPython)

$ ./sample50.py
50.
(1)
⎛    π⎞
⎜√2, ─⎟
⎝    4⎠

(2)
⎛   -π ⎞
⎜2, ───⎟
⎝    6 ⎠

(3)
(2, π)

(4)
⎛      5⋅π⎞
⎜2⋅√2, ───⎟
⎝       4 ⎠

51.
(1)
(√2, √2)

(2)
(√3, 3)

(3)
(0, -1)

(4)
(-4, 0)

$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="2">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.0001" value="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-10">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-10">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">
<br>
<label for="r1">r = </label>
<input id="r1" type="number" min="0" value="1">
<label for="Θ0">Θ = </label>
<input id="Θ0" type="number" step="0.1" value="1">

<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="sample50.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_r1 = document.querySelector('#r1'),
    input_Θ0 = document.querySelector('#Θ0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_r1, input_Θ0],
    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 f = (r, Θ) => [r * Math.cos(Θ), r * Math.sin(Θ), 'green'];

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),
        r1 = parseFloat(input_r1.value),
        Θ0 = parseFloat(input_Θ0.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }
    
    let points = [f(r1, Θ0)],
        lines = [],
        fns = [],
        fns1 = [],
        fns2 = [];

    fns.forEach((o) => {
        let [fn, color] = o;
        for (let x = x1; x <= x2; x += dx) {
            let y = fn(x);

            if (Math.abs(y) < Infinity) {
                points.push([x, y, color]);
            }
        }
    });
    fns1.forEach((o) => {
        let [fn, color] = o;
        
        lines.push([x1, fn(x1), x2, fn(x2), color]);
    });
    fns2.forEach((o) => {
        let [fn, color] = o;

        for (let x = x1; x <= x2; x += dx0) {
            let g = fn(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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