2017年7月7日金曜日

学習環境

線型代数入門(松坂 和夫(著)、岩波書店)の第1章(2次元と3次元の簡単な幾何学)、7(直線の方程式(2))、問1、2、3、4、5、6.を取り組んでみる。


  1. 平行であるための必要十分条件。

    ( x,y )=( x 0 , y 0 )+t( α,β ) ( α,β )·( a,b )=0 aα+bβ=0

    直交するための必要十分条件。

    aβαb=0

  2. ( 3,2 )·( 4,2 )= 9+4 · 16+4 cos( ±θ ) ±cosθ= 124 13 20 =± 8 13 20 =± 4 13 5 =± 4 65

  3. ( ab )·x=( ab )·t( a+b ) t( ab )·( a+b ) =t( | a | 2 | b | 2 ) =0 ( ab )·x=0

  4. | a | a | |=| b | b | |=10 ab a | a | b | b | ( a | a | b | b | )·x=0

  5. BA =( 12,5 ) BC =( 3,4 ) ( ( 12,5 ) 144+25 ( 3,4 ) 9+16 )·( x( 1,2 ) )=0 ( ( 12,5 ) 13 ( 3,4 ) 5 )·( x( 1,2 ) )=0 ( 60+39,2552 )·( x( 1,2 ) )=0 ( 99,27 )·( x( 1,2 ) )=0 ( 11,3 )·( x1,y2 )=0 11x3y5=0

  6. ( a,b )·( x,y )=c a·x=c q=p a·p+c | a | 2 a ( x',y' )=( x 0 , y 0 ) a x 0 +b y 0 +c a 2 + b 2 ( a,b ) x'= x 0 a x 0 +b y 0 +c a 2 + b 2 a y'= y 0 a x 0 +b y 0 +c a 2 + b 2 b d= | a·p+c | | a | = | a x 0 +b y 0 +c | a 2 + b 2

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, solve, cos, Matrix, sqrt

print('2.')
Θ = symbols('Θ')
eq = Matrix([3, -2]).dot(Matrix([4, 2])) - sqrt(9 + 4) * sqrt(16 + 4) * cos(Θ)
pprint(solve(eq, cos(Θ)))

入出力結果(Terminal, IPython)

$ ./sample1.py
2.
⎡4⋅√65⎤
⎢─────⎥
⎣  65 ⎦
$

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="0.0001" 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">

<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="sample1.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',
    range = (start, end, step=1) => {
        let res = [];
        for (let i = start; i < end; i += step) {
            res.push(i);
        }
        return res;
    };

let A = [13, 7],
    B = [1, 2],
    C = [-2, 6],
    f = (x) => 11 / 3 * x - 5 / 3;

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 = [[A[0], A[1], B[0], B[1], 'green'],
                 [C[0], C[1], B[0], B[1], 'green']],
        fns = [[f, 'orange']],
        fns1 = [],
        fns2 = [];

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

                if (Math.abs(y) < Infinity) {
                    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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