mirror of
https://github.com/MartinThoma/LaTeX-examples.git
synced 2025-04-24 22:08:04 +02:00
fixed some errors
This commit is contained in:
Martin Thoma
parent
76ec8c9cab
commit
7fdc530b6c
5 changed files with 283 additions and 12 deletions
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<!DOCTYPE html>
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<!-- saved from url=(0112)file:///home/moose/Downloads/LaTeX-examples/documents/math-minimal-distance-to-cubic-function/quadratic-vis.html -->
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<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
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<meta charset="UTF-8">
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<title>Quadratic visualization</title>
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<style type="text/css">
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span.hint {
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border-bottom:1px dotted #666;
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}
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</style>
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</head>
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<body>
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<table>
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<tbody><tr>
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<td><span class="hint" title="constat">c</span></td>
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<td><input type="number" step="1" value="1" id="c" onchange="updateBoard()"></td>
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</tr>
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<tr>
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<td><span class="hint" title="">STRETCH_X</span></td>
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<td><input type="number" step="1" value="3" id="STRETCH_X" min="1" onchange="updateBoard()"></td>
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<td><span class="hint" title="">STRETCH_Y</span></td>
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<td><input type="number" step="0.25" value="0.5" id="STRETCH_Y" min="0.1" onchange="updateBoard()"></td>
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<td><span class="hint" title="">X_OFFSET</span></td>
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<td><input type="number" step="16" value="-128" id="X_OFFSET" onchange="updateBoard()"></td>
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<td><span class="hint" title="">Y_OFFSET</span></td>
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<td><input type="number" step="16" value="256" id="Y_OFFSET" onchange="updateBoard()"></td>
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</tr>
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<tr>
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<td><span class="hint" title="visualize distance">pDistance</span></td>
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<td><input type="checkbox" id="pDistance" onchange="updateBoard()"></td>
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<td><span class="hint" title="How much will points be spread for voronoi? USE 1 WITH CAUTION! The bigger the value, the quicker the computation.">spread</span></td>
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<td><input type="number" step="1" value="1" id="density" min="1" onchange="updateBoard()"></td>
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<td><a href="constant-vis.html">clear board</a></td>
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</tr>
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</tbody></table>
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<canvas id="myCanvas" width="1316" height="535" style="border: 1px solid rgb(0, 0, 0); cursor: crosshair;"> </canvas>
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<script type="text/javascript" src="constant-vis.js">
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</script>
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</body></html>
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@ -0,0 +1,233 @@
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'use strict';
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/* global variables */
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var STRETCH_X = 3;
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var STRETCH_Y = 0.5;
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var X_MIN = -10;
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var X_MAX = +10;
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var Y_MIN = -10;
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var Y_MAX = +10;
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var X_OFFSET = -128;
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var Y_OFFSET = 256;
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var INITIAL_RADIUS = 20;
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var POINT_RADIUS = 5;
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/*******************************************************************/
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/* Graphics */
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/*******************************************************************/
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/**
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* Calculates coordinates from worldspace to screenspace
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* @param {Number} x the coordinate you want to transform
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* @param {bool} isX true iff x is a x-coordinate, otherwise false
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* @return {Number} transformed coordinate
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*/
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function c(x, isX) {
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if (isX) {
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return STRETCH_X * (x - X_OFFSET);
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}
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return STRETCH_Y * (-x + Y_OFFSET);
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}
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/**
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* Calculates coordinates from screenspace to worldspace
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* @param {Number} x the coordinate you want to transform
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* @param {bool} isX true iff x is a x-coordinate, otherwise false
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* @return {Number} transformed coordinate
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*/
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function r(x, isX) {
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if (isX) {
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return (x / STRETCH_X) + X_OFFSET;
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}
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return (-x / STRETCH_Y) + Y_OFFSET;
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}
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function setCursorByID(id,cursorStyle) {
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var elem;
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if (document.getElementById &&
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(elem=document.getElementById(id)) ) {
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if (elem.style) elem.style.cursor=cursorStyle;
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}
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}
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function drawEllipse(centerX, centerY, width, height) {
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context.beginPath() ;
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var x = centerX;
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var y = centerY;
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var rx = width;
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var ry = height;
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var rotation = 0; // The rotation of the ellipse (in radians)
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var start = 0; // The start angle (in radians)
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var end = 2 * Math.PI; // The end angle (in radians)
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var anticlockwise = false;
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context.ellipse(x, y, rx, ry, rotation, start, end, anticlockwise);
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context.fillStyle = "rgba(255, 0, 0, 0.5)";
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context.fill();
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}
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function getColor(i, transparency) {
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//var t = (i+1)*(360/k);
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//var color = 'hsla('+t+', 100%, 50%, '+transparency+')';
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var x = i / 256;
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if (x > 1) {x = 1.0;}
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x = parseInt(x*255);
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var color = 'rgba('+x+','+x+','+x+','+transparency+')';
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return color;
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}
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function drawFunction(canvas) {
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var add = parseInt(document.getElementById("density").value);
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context.beginPath();
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context.fillStyle = 'red';
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context.moveTo(0, c(getValue(r(0))), false);
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for (var xS=0; xS < canvas.width; xS+=add) {
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var x = r(xS);
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var y = getValue(x);
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context.lineTo(c(x, true), c(y, false));
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}
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context.closePath();
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context.stroke();
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}
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/*******************************************************************/
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/* Math */
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/*******************************************************************/
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function euklideanDist(p1, p2) {
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return Math.sqrt(
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Math.pow(p1["x"]-p2["x"], 2)
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+ Math.pow(p1["y"]-p2["y"], 2));
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}
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/**
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* Calculates the value of a constant function at x
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* @param {Number} x
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* @return {Number} f(x)
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*/
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function getValue(x) {
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var c1 = parseFloat(document.getElementById("c").value);
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return c1;
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}
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/**
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* Calculates the drivate f'(x)
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* @param {Number} x
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* @return {Number} f'(x)
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*/
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function getDValue(x) {
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var a = parseFloat(document.getElementById("a").value);
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var b = parseFloat(document.getElementById("b").value);
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return 2*a*x+b;
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}
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/**
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* Calculates the drivate f''(x)
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* @param {Number} x
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* @return {Number} f''(x)
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*/
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function getDDValue(x) {
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var a = parseFloat(document.getElementById("a").value);
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return 2*a;
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}
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/**
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* Calculates (f(x)^2)' = ((a*x*x+b*x+c)^2)' = 2 (b + 2 a x) (c + x (b + a x))
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* @param {Number} x
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* @return {Number} (f(x)^2)'
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*/
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function gedSquaredValueD(x) {
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var a = parseFloat(document.getElementById("a").value);
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var b = parseFloat(document.getElementById("b").value);
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var c1 = parseFloat(document.getElementById("c").value);
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return 2*(2*a*x+b)*(x*(a*x+b)+c1);
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}
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/**
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* Calculates (f(x)^2)'' = ((a*x*x+b*x+c)^2)'' = 2 (b^2 + 6 a b x + 2 a (c + 3 a x^2))
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* @param {Number} x
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* @return {Number} (f(x)^2)''
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*/
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function gedSquaredValueDD(x) {
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var a = parseFloat(document.getElementById("a").value);
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var b = parseFloat(document.getElementById("b").value);
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var c1 = parseFloat(document.getElementById("c").value);
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return 2*(b*b+6*a*b*x+2*a*(c1+3*a*x*x));
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}
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function findMin(p) {
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return p.x;
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}
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/*******************************************************************/
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/* Start / Update */
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/*******************************************************************/
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function drawBoard(canvas, mouseCoords, radius) {
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var context = canvas.getContext('2d');
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context.canvas.width = window.innerWidth - 50;
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context.canvas.height = window.innerHeight - 120;
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context.clearRect(0, 0, canvas.width, canvas.height);
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drawFunction(canvas);
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if (document.getElementById("pDistance").checked) {
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var add = parseInt(document.getElementById("density").value)+10;
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for (var x=0; x < canvas.width; x+=add) {
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for (var y=0; y < canvas.height; y+=add) {
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var dist = getDist({"x":r(x,true), "y":r(y,false)}, findMin({"x":r(x,true), "y":r(y,false)}));
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context.fillStyle = getColor(dist,0.5);
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context.fillRect(x, y, add/2, add/2);
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}
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}
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}
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}
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function updateBoard(){
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var canvas = document.getElementById("myCanvas");
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STRETCH_X = parseFloat(document.getElementById("STRETCH_X").value);
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STRETCH_Y = parseFloat(document.getElementById("STRETCH_Y").value);
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X_OFFSET = parseFloat(document.getElementById("X_OFFSET").value);
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Y_OFFSET = parseFloat(document.getElementById("Y_OFFSET").value);
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drawBoard(canvas, {"x":0,"y":0}, INITIAL_RADIUS);
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}
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var canvas = document.getElementById("myCanvas");
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var context = canvas.getContext("2d");
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drawBoard(canvas, {"x":0,"y":0}, INITIAL_RADIUS);
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setCursorByID("myCanvas", "crosshair");
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/** get the current position of the mouse */
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function getMouseCoords(canvas, evt) {
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var rect = canvas.getBoundingClientRect();
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return {
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"x": evt.clientX - rect.left,
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"y": evt.clientY - rect.top
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};
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}
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/** event listeners */
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canvas.addEventListener('mousemove',
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function (evt) {
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var mouseCoords = getMouseCoords(canvas, evt);
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drawBoard(canvas, mouseCoords, 10);
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// draw coordinates next to mouse
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context.fillStyle = "blue";
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context.font = "bold 16px Arial";
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var x = r(mouseCoords.x, true).toFixed(3);
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var y = r(mouseCoords.y, false).toFixed(3);
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context.fillText("(" + x + ", " + y + ")", mouseCoords.x + 5, mouseCoords.y - 5);
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var minX = findMin({"x": r(mouseCoords.x,true), "y": r(mouseCoords.y,false)});
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var minY = getValue(minX);
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context.beginPath();
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context.moveTo(c(minX,true), c(minY, false));
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context.lineTo(mouseCoords.x, mouseCoords.y, false);
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context.stroke();
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var minRadius = euklideanDist({"x": r(mouseCoords.x,true), "y": r(mouseCoords.y,false)}, {"x":minX,"y":minY});
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/* Draw circle */
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drawEllipse(mouseCoords.x, mouseCoords.y, minRadius*STRETCH_X, minRadius*STRETCH_Y);
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}, false);
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Binary file not shown.
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@ -1,18 +1,15 @@
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\chapter{Description of the Problem}
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Let $f: D \rightarrow \mdr$ with $D \subseteq \mdr$ be a polynomial function and $P \in \mdr^2$
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be a point. Let $d_{P,f}: \mdr \rightarrow \mdr_0^+$
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be the Euklidean distance of a point $P$ and a point $\left (x, f(x) \right )$
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be the Euklidean distance of a point $P$ to a point $\left (x, f(x) \right )$
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on the graph of $f$:
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\[d_{P,f} (x) := \sqrt{(x_P - x)^2 + (y_P - f(x))^2}\]
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Now there is finite set $M = \Set{x_1, \dots, x_n} \subseteq D$ of minima for given $f$ and $P$:
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\[M = \Set{x \in D | d_{P,f}(x) = \min_{\overline{x} \in D} d_{P,f}(\overline{x})}\]
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But minimizing $d_{P,f}$ is the same as minimizing $d_{P,f}^2$:
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\begin{align}
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d_{P,f}(x)^2 &= \sqrt{(x_P - x)^2 + (y_P - f(x))^2}^2\\
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&= x_p^2 - 2x_p x + x^2 + y_p^2 - 2y_p f(x) + f(x)^2
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\end{align}
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But minimizing $d_{P,f}$ is the same as minimizing
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$d_{P,f}^2 = x_p^2 - 2x_p x + x^2 + y_p^2 - 2y_p f(x) + f(x)^2$.
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\begin{theorem}[Fermat's theorem about stationary points]\label{thm:required-extremum-property}
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Let $x_0$ be a local extremum of a differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$.
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@ -78,7 +78,7 @@ function getColor(i, transparency) {
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return color;
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}
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function drawQuadraticFunction(canvas) {
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function drawFunction(canvas) {
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var add = parseInt(document.getElementById("density").value);
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context.beginPath();
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@ -88,7 +88,6 @@ function drawQuadraticFunction(canvas) {
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for (var xS=0; xS < canvas.width; xS+=add) {
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var x = r(xS);
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var y = getValue(x);
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//context.fillRect(c(x), c(y, false), add/2, add/2);
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context.lineTo(c(x, true), c(y, false));
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}
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@ -215,7 +214,7 @@ function drawBoard(canvas, mouseCoords, radius) {
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context.canvas.height = window.innerHeight - 120;
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context.clearRect(0, 0, canvas.width, canvas.height);
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drawQuadraticFunction(canvas);
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drawFunction(canvas);
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if (document.getElementById("pDistance").checked) {
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var add = parseInt(document.getElementById("density").value)+10;
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for (var x=0; x < canvas.width; x+=add) {
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@ -262,13 +261,13 @@ canvas.addEventListener('mousemove',
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var x = r(mouseCoords.x, true).toFixed(3);
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var y = r(mouseCoords.y, false).toFixed(3);
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context.fillText("(" + x + ", " + y + ")", mouseCoords.x + 5, mouseCoords.y - 5);
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var minX = findMin({"x": mouseCoords.x, "y": mouseCoords.y});
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var minX = findMin({"x": r(mouseCoords.x,true), "y": r(mouseCoords.y,false)});
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var minY = getValue(minX);
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context.beginPath();
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context.moveTo(c(minX, true), c(minY, false), false);
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context.moveTo(c(minX,true), c(minY, false));
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context.lineTo(mouseCoords.x, mouseCoords.y, false);
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context.stroke();
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var minRadius = getDist({"x":x, "y":y}, minX);
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var minRadius = euklideanDist({"x": r(mouseCoords.x,true), "y": r(mouseCoords.y,false)}, {"x":minX,"y":minY});
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/* Draw circle */
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drawEllipse(mouseCoords.x, mouseCoords.y, minRadius*STRETCH_X, minRadius*STRETCH_Y);
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