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torusknot.html
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torusknot.html
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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<meta name="viewport" content="width=device-width, initial-scale=1">
<meta name="description" content="">
<meta name="author" content="">
<link rel="shortcut icon" href="assets/ico/favicon.ico">
<title>Torus Knot Fibration Visualization</title>
<!-- Bootstrap core CSS -->
<link href="assets/css/bootstrap.css" rel="stylesheet">
<!-- Custom styles for this template -->
<link href="assets/css/style.css" rel="stylesheet">
<link href="assets/css/font-awesome.min.css" rel="stylesheet">
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.0/jquery.min.js"></script>
<!-- <script src="assets/js/modernizr.js"></script>-->
<!-- mathbox -->
<script src="./lib/mathbox/build/mathbox-bundle.js"></script>
<link rel="stylesheet" href="./lib/mathbox/build/mathbox.css">
<meta name="viewport" content="initial-scale=1, maximum-scale=1">
<!--math-->
<script src="./lib/math.min.js"></script>
<!--jquery-->
<script src="assets/js/jquery.min.js"></script>
<!--bootstrap-->
<link rel="stylesheet" src="./lib/bootstrap-3.3.7-dist/css/bootstrap.min.css">
<link rel="stylesheet" src="./lib/bootstrap-3.3.7-dist/css/bootstrap-theme.min.css">
<script src="./lib/bootstrap-3.3.7-dist/js/bootstrap.min.js"></script>
<!--
bootstrap toggle
<link href="https://gitcdn.github.io/bootstrap-toggle/2.2.2/css/bootstrap-toggle.min.css" rel="stylesheet">
<script src="https://gitcdn.github.io/bootstrap-toggle/2.2.2/js/bootstrap-toggle.min.js"></script>
-->
<link href="./lib/switch/css/bootstrap3/bootstrap-switch.min.css" rel="stylesheet">
<script src="./lib/switch/js/bootstrap-switch.min.js"></script>
<!--bootstrap slider-->
<link href="./lib/slider/bootstrap-slider.min.css" rel="stylesheet">
<script src="./lib/slider/bootstrap-slider.min.js"></script>
<!-- my css -->
<link href="./lib/css/mystyle.css" rel="stylesheet">
<!-- mathjax-->
<script type="text/javascript" async src="assets/MathJax-2.6/MathJax.js?config=TeX-MML-AM_CHTML">
</script>
</head>
<body>
<!-- Fixed navbar -->
<div class="navbar navbar-default navbar-fixed-top" role="navigation">
<div class="container">
<div class="navbar-header">
<button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".navbar-collapse">
<span class="sr-only">Toggle navigation</span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<a class="navbar-brand" href="index.html">Torus Knot Fibration Visualization</a>
</div>
<div class="navbar-collapse collapse navbar-right">
<ul class="nav navbar-nav">
<li><a href="index.html">HOME</a></li>
<li class="active"><a href="torusknot.html">TORUS & KNOT</a></li>
<li><a href="boundary.html">BOUNDARY</a></li>
<li><a href="surface.html">SURFACE</a></li>
<li><a href="stereo.html">STEREOGRAPHIC</a></li>
</ul>
</div>
<!--/.nav-collapse -->
</div>
</div>
<!-- *****************************************************************************************************************
BLUE WRAP
***************************************************************************************************************** -->
<div id="blue">
<div class="container">
<div class="row">
<h3>Understanding the Torus and Trefoil Knot</h3>
</div>
<!-- /row -->
</div>
<!-- /container -->
</div>
<!-- /blue -->
<!-- *****************************************************************************************************************
AGENCY ABOUT
***************************************************************************************************************** -->
<div class="container-fluid mtb">
<div class="row">
<div id="torusvis" class="torusvis col-lg-7">
<script type="text/javascript" language="javascript">
console.log($('.torusvis'));
$('.torusvis').css('height', 0.8 * $(window).height() + 'px');
</script>
<script>
var mathbox = mathBox({
plugins: ['core', 'controls', 'cursor'],
element: document.getElementById("torusvis"),
controls: {
// Orbit controls, i.e. Euler angles, with gimbal lock
klass: THREE.OrbitControls,
parameters: {
noZoom: true
},
// Trackball controls, i.e. Free quaternion rotation
//klass: THREE.TrackballControls,
},
fill: {
block: true,
},
mathbox: {
warmup: 1
}
});
if (mathbox.fallback) throw "WebGL not supported"
var three = mathbox.three;
three.renderer.setClearColor(new THREE.Color('#f8f8f8'), 1.0);
// container.appendChild(three.renderer.domElement);
//
// console.log(container)
var time = three.Time.clock;
// Place camera
var camera =
mathbox
.camera({
proxy: true,
position: [0, 0, 3],
lookAt: [0, 0, 0]
});
// 2D cartesian
var view =
mathbox
.cartesian({
id: 'cartesian',
position: [0, 0, 0],
range: [
[-math.pi, math.pi],
[-math.pi, math.pi],
[-math.pi, math.pi]
],
scale: [1, 1, 1],
});
var alpha = 2;
//Quality of Life
var pi = math.pi;
var sin = math.sin;
var cos = math.cos;
function torus(x, y) {
R = 2;
r = 1;
var a = Number($('#slidera').val())
var b = Number($('#sliderb').val())
// return [(R-r*cos(x*b))*((1-a)*y+sin(a*y)),(R-r*cos(b*x))*-cos(a*y),(1-b)*x + r*sin(b*x)]
return [
(R - r * ((1 - a) + a * cos(x * b))) * ((1 - a) * y + sin(a * y)), (1 - b) * x + r * sin(b * x), (R - r * cos(b * x)) * -cos(a * y) + 1
]
}
view.area({
id: 'densearea',
width: 100,
height: 100,
// axes: [1, 3],
expr: function(emit, x, y, i, j, time) {
emit(...torus(x, y));
},
items: 1,
channels: 3,
});
view.surface({
id: 'torussurface',
shaded: false,
color: '#C5D8E0',
opacity: 1,
});
view.area({
id: 'sparsearea',
width: 64,
height: 10,
// axes: [1, 3],
expr: function(emit, x, y, i, j, time) {
emit(...torus(x, y));
emit(...torus(y, x));
},
items: 2,
channels: 3,
});
view.resample({
id: 'sparsesample',
source: mathbox.select('#densearea')[1],
width: 64,
height: 10
})
view.line({
id: 'torusgrid',
points: mathbox.select('#sparsesample')[1],
color: 'white',
zBias: 10,
opacity: 0.3
})
//Edges of Torus
view.interval({
id: 'redline',
width: 64,
expr: function(emit, x, i, time) {
emit(...torus(x, -pi));
emit(...torus(x, pi));
},
items: 2,
channels: 3,
})
view.line({
points: mathbox.select('#redline')[1],
color: '#FF4136',
zBias: 12
})
view.interval({
id: 'yellowline',
width: 64,
expr: function(emit, y, i, time) {
emit(...torus(-pi, y));
emit(...torus(pi, y));
},
items: 2,
channels: 3,
})
view.line({
points: mathbox.select('#yellowline')[1],
color: '#B10DC9',
zBias: 13
})
// Torus Knot Parameters
var p = 3;
var q = 2;
var m = p / q;
var m2 = q / p;
var eps = 0.01;
function mod2pi(x) {
return (x % (2 * pi) + 2 * pi) % (2 * pi)
}
// Knot Curve
function knot(x) {
return (m2 * x) % (2 * pi);
}
view.interval({
range: [0, 2 * p * pi],
expr: function(emit, x, i, t) {
//-(pi,pi,0) needed for new bounds
emit(...torus(x % (2 * pi) - pi, knot(x) - pi));
},
width: 400,
channels: 3,
});
view.line({
id: 'knotcurve',
color: '#2ECC40',
opacity: 1.0,
proximity: 2,
blending: 'no',
zBias: 14
})
// Knot Traveller
function sphere(r, th, ph, x, y, z) {
return [r * cos(th) * sin(ph) + x, r * sin(th) * sin(ph) + y, r * cos(ph) + z]
}
view.area({
id: 'spherearea',
width: 100,
height: 100,
// axes: [1, 3],
expr: function(emit, x, y, i, j, time) {
var t = time
var [X, Y, Z] = torus(t % (2 * pi) - pi, knot(t) - pi);
emit(...sphere(0.1, x, y, X, Y, Z));
},
items: 1,
channels: 3,
});
view.surface({
id: 'knottraveller',
shaded: false,
color: '#2ECC40',
opacity: 1.0,
zBias:10
});
mathbox.set('focus', 2);
</script>
</div>
<div class="col-lg-4">
<table class="panel" style="margin-left: auto;
margin-right: auto; width:80%">
<tr>
<th class="labels">
<label class="sliderlabel">Torus Surface</label>
</th>
<th>
<input id="torussurfacetoggle" type="checkbox" checked data-toggle="toggle" value="1.0" onchange="torustogfun(this);" data-onstyle="primary">
</th>
</tr>
<tr>
<th class="labels" style="color:#FF4136">
<label class="sliderlabel">Glue Latitudinal Edge</label>
</th>
<th>
<input id="slidera" data-slider-id='ex1Slider' type="text" data-slider-min="0" data-slider-max="1" data-slider-step="0.01" data-slider-value="0.0" data-slider-tooltip='hide' />
</th>
</tr>
<tr>
<th class="labels" style="color:#B10DC9">
<label class="sliderlabel">Glue Longitudinal Edge</label>
</th>
<th>
<input id="sliderb" data-slider-id='ex1Slider' type="text" data-slider-min="0" data-slider-max="1" data-slider-step="0.01" data-slider-value="0.0" data-slider-tooltip='hide' />
</th>
</tr>
<tr>
<th class="labels" style="color:#2ECC40">
<label class="sliderlabel">Trefoil</label>
</th>
<th>
<input id="knottoggle" type="checkbox" checked data-toggle="toggle" value="1.0" onchange="togfun(this);" data-onstyle="primary">
</th>
</tr>
<tr>
<th class="labels" style="color:#2ECC40">
<label class="sliderlabel">Trefoil Traveller</label>
</th>
<th>
<input id="knottravellertoggle" type="checkbox" checked data-toggle="toggle" value="1.0" onchange="travtogfun(this);" data-onstyle="primary">
</th>
</tr>
</table>
<script>
//Slider Script
$('#slidera').slider({
formatter: function(value) {
return 'Current value: ' + value;
}
});
$('#sliderb').slider({
formatter: function(value) {
return 'Current value: ' + value;
}
});
$('#slidera').on('slide', positionchanger)
function positionchanger() {
var Z = Number($('#slidera').val());
mathbox.select('#cartesian')[0].set('position', [0, 0, -0.33 * Z])
}
function slidefunction() {
var sliderval = Number($('#slider').val());
mathbox.select('surface').set('opacity', sliderval);
console.log(typeof sliderval)
}
//Button SCript
$('#knottravellertoggle').bootstrapSwitch({
size: 'small',
offColor: 'custom',
onColor: 'custom'
});
function travtogfun(button) {
var butprop = $('#knottravellertoggle').prop('checked');
var butval = $('#knottravellertoggle').val();
if (butprop == true) {
$('#knottravellertoggle').val(1.0);
console.log('true!')
mathbox.select('#knottraveller')[0].set('opacity', 1.0)
} else {
$('#knottravellertoggle').val();
mathbox.select('#knottraveller')[0].set('opacity', 0.0)
}
}
$('#torussurfacetoggle').bootstrapSwitch({
size: 'small',
offColor: 'custom',
onColor: 'custom'
});
function torustogfun(button) {
var butprop = $('#torussurfacetoggle').prop('checked');
var butval = $('#torussurfacetoggle').val();
if (butprop == true) {
$('#torussurfacetoggle').val(1.0);
console.log('true!')
mathbox.select('#torussurface')[0].set('opacity', 1.0)
mathbox.select('#torusgrid')[0].set('color', 'white')
} else {
$('#torussurfacetoggle').val();
mathbox.select('#torussurface')[0].set('opacity', 0.0)
mathbox.select('#torusgrid')[0].set('color', '#C5D8E0')
}
}
$('#knottoggle').bootstrapSwitch({
size: 'small',
offColor: 'custom',
onColor: 'custom'
});
function togfun(button) {
var butprop = $('#knottoggle').prop('checked');
var butval = $('#knottoggle').val();
if (butprop == true) {
$('#knottoggle').val(1.0);
mathbox.select('#knotcurve')[0].set('opacity', 1.0)
} else {
$('#knottoggle').val();
mathbox.select('#knotcurve')[0].set('opacity', 0.0)
}
}
</script>
<h3>Torus and Trefoil Knot</h3>
<p class="blurb">This preliminary visualization demonstrates how a
<torus>square</torus>
<!-- $$[-\pi,\pi]\times[-\pi,\pi]$$-->
can be glued along its oposite edges to produce a
<torus>torus</torus>.</p>
<p class="blurb">
Use the sliders in the parameters panel to manipulate the visualization. Gluing along either the <i style="color:#FF4136">latitudinal</i> or <i style="color:#B10DC9">longitudinal</i> will transform the square into a vertical or horizontal cylinder. Finally gluing the remaining circuluar edges of the cylinder produces a
<torus>torus</torus>.
<knot></knot>
</p>
<p class="blurb">
As defined previously, we consider a knot embedded in the 3-sphere, \(K \in S^3 \). A special kind of knots which are always fibred are called torus knots, which are embedded in the surface of a torus. In the following visualizations we consider the
<knot>trefoil knot</knot> or the (2,3)-torus knot. Note the (2,3)-torus knot is named because it touches the <i style="color:#FF4136">latitudinal</i> edge 2 times and the <i style="color:#B10DC9">longitudinal </i> edge 3 times. Finally,
<knot>trefoil traveller</knot> demonstrates how the knot connects along the edges.
</p>
<nav aria-label="...">
<ul class="pager">
<li><a href="index.html">Previous</a></li>
<li><a href="boundary.html">Next</a></li>
</ul>
</nav>
</div>
</div>
<! --/row -->
</div>
<! --/container -->
<!-- *****************************************************************************************************************
FOOTER
***************************************************************************************************************** -->
<div id="footerwrap">
<div class="container">
<div class="row">
<div class="col-lg-4">
<h4>About Project</h4>
<div class="hline-w"></div>
<p>This is my final deliverable for the summer research component of a Master's in Mathematics at the University of Toronto supervised by Dr. Dror Bar-Natan. </p>
</div>
<div class="col-lg-4">
<h4>Project Repository</h4>
<div class="hline-w"></div>
<p style="text-align:center;">
<a href="http://www.github.com/jessebett/TorusKnotFibration"><i class="fa fa-github" style="font-size: 80px;"></i></a>
</p>
</div>
<div class="col-lg-4">
<h4>About Me</h4>
<div class="hline-w"></div>
<p>
My name is Jesse Bettencourt and I've recently acquired an M.Sc. in Mathematics from U of T. I'm interested in vizualizations from the intersection of math and computer science.
</p>
</div>
</div>
<! --/row -->
</div>
<! --/container -->
</div>
<! --/footerwrap -->
<!-- Bootstrap core JavaScript
================================================== -->
<!-- Placed at the end of the document so the pages load faster -->
<!-- <script src="assets/js/bootstrap.min.js"></script>-->
<script src="assets/js/retina-1.1.0.js"></script>
<script src="assets/js/jquery.hoverdir.js"></script>
<script src="assets/js/jquery.hoverex.min.js"></script>
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</body>
</html>