数学 - Python - JavaScript - 曲線の性質、最大・最小 - 微分法の応用 – 媒介変数で表される曲線 - 平面上の点の運動、速度・加速度(等速円運動、位置ベクトル、速度ベクトル、加速度ベクトル)

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)
数式入力ソフト(TeX, MathML): MathType
MathML対応ブラウザ: Firefox、Safari
MathML非対応ブラウザ(Internet Explorer, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
参考書籍

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48、49.を取り組んでみる。

1. $\begin{array}{l} (a \cos ω t, a \sin ω t) \cdot (- a ω \sin ω t, a ω \cos ω t) \\ = - a^{2} ω (\sin ω t) (\cos ω t) + a^{2} ω (\sin ω t) (\cos ω t) \\ = 0 \end{array}$
2. $\begin{array}{l} \vec{α} (t) = (- a ω^{2} (\cos ω t), - a ω^{2} (\sin ω t)) \\ = - ω^{2} (a \cos ω t, a \sin ω t) \\ = - ω^{2} \vec{O P} \end{array}$
$\begin{array}{l} x = f (t) \\ y = g (t) \\ \frac{d x}{d t} = f' (t) \\ \frac{d y}{d t} = g' (t) \\ \sqrt{{(f' (t))}^{2} + {(g' (t))}^{2}} = c \\ \frac{d^{2} x}{d t^{2}} = f'' (t) \\ \frac{d^{2} y}{d t^{2}} = g'' (t) \\ (f' (t), g' (t)) \cdot (f'' (t), g'' (t)) \\ = f' (t) f'' (t) + g' (t) g'' (t) \\ {(f' (t))}^{2} + {(g' (t))}^{2} = c^{2} \\ 2 (f' (t)) f'' (t) + 2 (g' (t)) g'' (t) = 0 \\ f' (t) f'' (t) + g' (t) g'' (t) = 0 \\ (f' (t), g' (t)) \cdot (f'' (t), g'' (t)) = 0 \end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, sin, cos, Derivative, Matrix

print('48.')

a, ω, t = symbols('a ω t')

x = a * cos(ω * t)
y = a * sin(ω * t)

pprint(x)
pprint(y)

dx1 = Derivative(x, t, 1)
dy1 = Derivative(y, t, 1)
x1 = dx1.doit()
y1 = dy1.doit()

pprint(dx1)
pprint(dy1)

pprint(x1)
pprint(y1)

dx2 = Derivative(x, t, 2)
dy2 = Derivative(y, t, 2)
x2 = dx2.doit()
y2 = dy2.doit()

pprint(dx2)
pprint(dy2)

pprint(x2)
pprint(y2)

pprint(Matrix([x, y]).dot(Matrix([x1, y1])))

pprint(Matrix([x2, y2]) + ω ** 2 * Matrix([x, y]))

入出力結果(Terminal, IPython)

$ ./sample48.py
48.
a⋅cos(t⋅ω)
a⋅sin(t⋅ω)
∂             
──(a⋅cos(t⋅ω))
∂t            
∂             
──(a⋅sin(t⋅ω))
∂t            
-a⋅ω⋅sin(t⋅ω)
a⋅ω⋅cos(t⋅ω)
  2            
 ∂             
───(a⋅cos(t⋅ω))
  2            
∂t             
  2            
 ∂             
───(a⋅sin(t⋅ω))
  2            
∂t             
    2         
-a⋅ω ⋅cos(t⋅ω)
    2         
-a⋅ω ⋅sin(t⋅ω)
0
⎡0⎤
⎢ ⎥
⎣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="a0">a = </label>
<input id="a0" type="number" min="0" value="5">
<label for="ω0">ω = </label>
<input id="ω0" type="number" 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="sample48.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_a0 = document.querySelector('#a0'),
    input_ω0 = document.querySelector('#ω0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_a0, input_ω0],
    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 draw = (t) => {
    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),
        a0 = parseFloat(input_a0.value),
        ω0 = parseFloat(input_ω0.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }
    
    let points = [[a0 * Math.cos(ω0 * t), a0 * Math.sin(ω0 * t), 'green']],
        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')));
};

let t = 0;
setInterval(() => {
    draw(t);
    t += 0.01;
}, 10);
// inputs.forEach((input) => input.onchange = draw);
// btn0.onclick = draw;
// btn1.onclick = () => pre0.textContent = '';
// draw();

r = dx =
x1 = x2 =
y1 = y2 =
a = ω =

Kamimura's blog

2017年7月10日月曜日

数学 - Python - JavaScript - 曲線の性質、最大・最小 - 微分法の応用 – 媒介変数で表される曲線 - 平面上の点の運動、速度・加速度(等速円運動、位置ベクトル、速度ベクトル、加速度ベクトル)

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