2018年11月19日月曜日

学習環境

ラング線形代数学(上)(S.ラング (著)、芹沢 正三 (翻訳)、ちくま学芸文庫)の1章(R^n におけるベクトル)、5(直線と平面)、練習問題11.を取り組んでみる。



    1. L = 1 , - 1 , 3 , 1 + t 1 , - 3 , 2 , 1 = 1 + t , - 1 - 3 t , 3 + 2 t , 1 + t

      距離。

      1 + t - 1 2 + - 1 - 3 t - 1 2 + 3 + 2 t + 1 2 + 1 + t - 2 2 = t 2 + 3 t + 2 2 + 2 t + 4 2 + t - 1 2 = 1 + 9 + 4 + 1 t 2 + 12 + 16 - 2 t + 4 + 16 + 1 = 15 t 2 + 26 t + 21

    2. 15 t 2 + 26 t + 21 = 15 t 2 + 26 15 t + 21 = 15 t + 13 15 2 + 21 - 169 15 = 15 t + 13 15 2 + 146 15

      よって、距離が最小値となるような点がただ1つ存在して、 その点の t の値、最小距離は、

      t = - 13 15 146 15

    3. X 0 - Q = 1 - 13 15 , - 1 + 39 15 , 3 - 26 15 , 1 - 13 15 - 1 , 1 , - 1 , 2 = - 13 15 , - 2 + 39 15 , 4 - 26 15 , - 1 - 13 15 = 1 15 - 13 , 9 , 34 , - 28 X 0 - Q · A = 1 15 - 13 - 27 + 68 - 28 = 0

      よって、垂直。

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Matrix, solve, Rational

P = Matrix([1, -1, 3, 1])
Q = Matrix([1, 1, -1, 2])
A = Matrix([1, -3, 2, 1])
t = symbols('t', real=True)
L = P + t * A

XQ = (L - Q).norm().simplify()
pprint(XQ)
X0 = L.subs({t: - Rational(13, 15)})
pprint(X0)
pprint((X0 - Q).dot(A))

入出力結果(Terminal, Jupyter(IPython))

$ ./sample11.py
   ___________________
  ╱     2             
╲╱  15⋅t  + 26⋅t + 21 
⎡2/15⎤
⎢    ⎥
⎢8/5 ⎥
⎢    ⎥
⎢ 19 ⎥
⎢ ── ⎥
⎢ 15 ⎥
⎢    ⎥
⎣2/15⎦
0
$

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="0.001">
<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">

<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="sample11.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';

let fns = [[x => Math.sqrt(15 * x ** 2 + 26 * x + 21), 'red']];

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 = [[-13 / 15, y1, -13 / 15, y2, 'green'],
                 [x1, Math.sqrt(146 / 15), x2, Math.sqrt(146 / 15), 'blue']];

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

                points.push([x, y, 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].forEach((fs) => p(fs.join('\n')));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();







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