/
surface.html
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surface.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="assets/js/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="https://ajax.googleapis.com/ajax/libs/jquery/1.12.4/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><a href="torusknot.html">TORUS & KNOT</a></li>
<li><a href="boundary.html">BOUNDARY</a></li>
<li class="active"><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>Fibration Surface</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 * $(window).height() + 'px');
</script>
<script>
var mathbox = mathBox({
plugins: ['core', 'controls', 'cursor'],
controls: {
// Orbit controls, i.e. Euler angles, with gimbal lock
klass: THREE.OrbitControls,
parameters: {
noZoom: true
},
// Trackball controls, i.e. Free quaternion rotation
//klass: THREE.TrackballControls,
},
element: document.getElementById("torusvis"),
mathbox: {
warmup: 1
}
});
if (mathbox.fallback) throw "WebGL not supported"
var three = mathbox.three;
three.renderer.setClearColor(new THREE.Color('#f8f8f8'), 1.0);
var time = three.Time.clock;
// Place camera
var camera =
mathbox
.camera({
proxy: true,
position: [0, 0, 3],
lookAt: [0, 0, 0]
});
// 2D cartesian
var view =
mathbox
.cartesian({
id:'cartesian',
position: [0, 0, 0.33],
range: [
[-math.pi, math.pi],
[-math.pi, math.pi],
[-math.pi, math.pi]
],
scale: [1, 1, 1],
});
var alpha = 2;
//Quality of Life
var pi = math.pi;
var sin = math.sin;
var cos = math.cos;
function torus(x, y) {
R = 2;
r = pi / 4;
var a = Number($('#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)
]
}
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({
shaded: false,
color: '#0FBFCF',
opacity: 0.0,
});
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('#sparsearea')[0],
width: 64,
height: 10
})
view.line({
id: 'torusmesh',
points: mathbox.select('#sparsesample')[0],
color: '#6a9fb5',
zBias: 1,
opacity: 0.4,
visible: true,
})
// //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: '#FFDC00',
// zBias: 13
// })
// Torus Knot Parameters
var p = 3;
var q = 2;
var m = p / q;
var m2 = q / p;
var eps = 0.01;
//Global Animation Time
var tau_last = 0;
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: 200,
channels: 3,
});
view.line({
id: 'knotcurve',
color: '#2ECC40',
visible: true,
proximity: 1,
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: 'green'
//
// });
//Piecewise Stuff
// //First Piecewise Curve
// function piecewise(x, n) {
// //Triangle Wave Function to match knot + rungs!
// x = x % (2 * pi);
// var period = (pi) / q;
// var amplitude = pi / (2 * p);
// var wave = ((2 * amplitude) / pi) * math.asin(sin(x * pi / period)) + 2 * pi * n / p
//
// return mod2pi(wave)
// }
//
function modpi(x) {
//Function for Modular Arithmetic on [-pi,pi]
return ((x + 2 * pi) % (2 * pi) - pi)
}
function linecut(x) {
var X = x
var thresh = 0.02;
if (Math.abs(pi) - X < thresh) {
X = NaN
}
return X
}
function fakemodpi(x) {
var tpi = 2 * pi;
var X = x;
//To stop edge errors:
while (Math.abs(X) > pi) {
if (X <= -pi) {
X = X + 2 * pi
} else if (X >= -pi + tpi) {
X = X - tpi
}
}
return linecut(X)
}
function truemodpi(x) {
return (x + pi) % (2 * pi) - pi
}
console.log(fakemodpi(0))
console.log(fakemodpi(pi + 0.1))
//((2*amplitude)/pi)*math.asin(sin((x+t/q)*pi/period))+2*pi*n/p-(1/p)*t
//First Piecewise Curve
function piecewise(x, n) {
//Triangle Wave Function to match knot + rungs!
x = x % (2 * pi);
var t = Number($('#slidertime').val());
var period = (pi) / q;
var amplitude = pi / (2 * p);
var wave = ((2 * amplitude) / pi) * math.asin(sin((x + t / q) * pi / period)) + 2 * pi * n / p - (t / p)
return fakemodpi(wave - pi / p)
}
view.interval({
range: [0, 4 * pi],
expr: function(emit, x, i, t) {
//(0,-pi,0) shift to match new domains
emit(...torus(fakemodpi(x) + eps, linecut(piecewise(x, 0))));
emit(...torus(fakemodpi(x) + eps, linecut(piecewise(x, 1))));
emit(...torus(fakemodpi(x) + eps, linecut(piecewise(x, 2))));
},
width: 1000,
channels: 3,
items: 3
});
view.line({
id: 'blueboundary',
color: '#49a7e9',
visible: false,
zBias: 15,
proximity: 0.5
});
//Second Piecewise Curve
function piecewise2(x, n) {
x = x % (2 * pi);
var t = Number($('#slidertime').val());
var period = (pi) / p;
var amplitude = pi / (2 * q);
var wave = ((2 * amplitude) / pi) * math.asin(sin((x - t / p) * pi / period)) - 2 * pi * n / q
// var wave = ((2 * amplitude) / pi) * math.asin(sin((x+ t / q) * pi / period)) + 2 * pi * n / p - (t / p)
return fakemodpi(-(wave - pi / q + (1 / q) * t))
}
view.interval({
expr: function(emit, x, i, t) {
emit(...torus(linecut(piecewise2(x, 0) - eps), -fakemodpi(x)));
emit(...torus(linecut(piecewise2(x, 1) - eps), -fakemodpi(x)));
},
width: 600,
channels: 3,
items: 2
});
view.line({
id: 'redboundary',
color: '#fab125',
proximity: 1,
visible: false,
zBias: 15
});
function SurfaceCut(x) {
var thresh = 0.15;
if (x <= -pi - thresh) {
x = -NaN;
} else if (x >= pi + thresh) {
x = NaN;
}
return x
}
function threetorus(x, y, z) {
R = 2;
r = pi / 4;
var a = Number($('#slidera').val())
var b = Number($('#sliderb').val())
// // return [
// (R - r * ((1 - a) + a * cos(x * b))) * ((1 - a) * y + sin(a * y)),
//(1 - b) * x + r * sin(b * x), (R - r * cos(b * x)) * -cos(a * y) + 1
// ]
return [
(R - (r - a * z) * ((1 - a) + a * cos(x * b))) * ((1 - a) * y + sin(a * y)), (1 - b) * x + (r - z) * sin(b * x), (R - (r - z) * cos(b * x)) * -cos(a * y)
]
}
//First Piecewise Surface
function piecewise(x, y, t, n) {
x = x % (2 * pi);
var period = (pi) / q;
var amplitude = pi / (2 * p);
var wave = Math.pow(math.abs(4 / pi * (pi / 4 - y)), 4) * ((2 * amplitude) / pi) * math.asin(sin((x + t / q) * pi / period)) + 2 * pi * n / p - (1 / p) * t
return SurfaceCut(wave - pi / p)
}
var edgethresh = 0.5;
view.area({
expr: function(emit, x, y, i, j, t) {
if ($('#timetoggle').prop('checked') == true) {
var tau = t % (2 * pi);
tau_last = tau
} else {
// var tau = Number($('#slidertime').val());
var tau = tau_last;
}
var xdom = SurfaceCut(x);
emit(...threetorus(xdom, piecewise(x, y, tau, -1), y));
emit(...threetorus(xdom, piecewise(x, y, tau, 0), y));
emit(...threetorus(xdom, piecewise(x, y, tau, 1), y));
emit(...threetorus(xdom, piecewise(x, y, tau, 2), y));
emit(...threetorus(xdom, piecewise(x, y, tau, 3), y));
// emit(xdom, piecewise(x, tau, 0), -y);
// emit(xdom, piecewise(x, tau, 1), -y);
// emit(xdom, piecewise(x, tau, 2), -y);
},
rangeX: [-pi - edgethresh, pi + edgethresh],
rangeY: [0, pi / 4],
width: 80,
height: 20,
channels: 3,
items: 5,
centeredX: false,
centeredY: false
});
view.surface({
id: 'bluesurface',
color: '#49a7e9',
proximity: 0.001,
shaded: false,
crossed: false,
zBias: 9,
visible: true,
});
view.line({
id: 'bluemesh',
color: '#001f3f',
zBias: 10,
opacity: 0.2,
visible: true,
})
// var wave = ((2 * amplitude) / pi) * math.asin(sin((x - t / p) * pi / period)) - 2 * pi * n / q
// // var wave = ((2 * amplitude) / pi) * math.asin(sin((x+ t / q) * pi / period)) + 2 * pi * n / p - (t / p)
//
// return fakemodpi(-(wave - pi / q + (1 / q) * t))
//Second Piecewise Surface
function piecewise2(x, y, t, n) {
x = x % (2 * pi);
var period = (pi) / p;
var amplitude = pi / (2 * q);
var wave = Math.pow(math.abs(4 / pi * (pi / 4 - y)), 4) * ((2 * amplitude) / pi) * math.asin(sin((x + t / p - pi / p) * pi / period)) - 2 * pi * n / q
return SurfaceCut(-(wave - pi / q + (1 / q) * t))
}
var edgethresh = 0.5;
view.area({
expr: function(emit, x, y, i, j, t) {
if ($('#timetoggle').prop('checked') == true) {
var tau = t % (2 * pi);
} else {
// var tau = Number($('#slidertime').val());
var tau = tau_last;
}
var xdom = SurfaceCut(x);
// emit(...threetorus(xdom, piecewise2(x, y, tau, -1), y));
emit(...threetorus(piecewise2(x, y, tau, -1), xdom, -y));
emit(...threetorus(piecewise2(x, y, tau, 0), xdom, -y));
emit(...threetorus(piecewise2(x, y, tau, 1), xdom, -y));
// emit(...threetorus(piecewise2(x, y, tau, ),xdom, -y));
// emit(...threetorus(xdom, piecewise2(x, y, tau, 1), y));
// emit(...threetorus(xdom, piecewise2(x, y, tau, 2), y));
// emit(...threetorus(xdom, piecewise2(x, y, tau, 3), y));
// // emit(xdom, piecewise(x, tau, 0), -y);
// emit(xdom, piecewise(x, tau, 1), -y);
// emit(xdom, piecewise(x, tau, 2), -y);
},
rangeX: [-pi - edgethresh, pi + edgethresh],
rangeY: [0, pi / 4 + 0.4],
width: 80,
height: 20,
channels: 3,
items: 3,
centeredX: false,
centeredY: false
});
view.surface({
id: 'redsurface',
color: '#fab125',
proximity: 0.001,
shaded: false,
crossed: false,
zBias: 10,
visible: true,
});
view.line({
id: 'redmesh',
color: '#FF4136',
zBias: 10,
opacity: 0.4,
visible: true,
})
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 Mesh</label>
</th>
<th>
<input id="meshtoggle" type="checkbox" checked data-toggle="toggle" value="1.0" onchange="meshtogfun(this);" data-onstyle="primary">
</th>
</tr>
<tr>
<th class="labels">
<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="1.0" data-slider-tooltip='hide' />
</th>
</tr>
<tr>
<th class="labels">
<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>
<!--Knot Curve Toggle-->
<tr>
<th class="labels" style="color:#0fbfcf">
<label class="sliderlabel">
<mark style="color:#2ECC40">Knot Curve</mark>
</label>
</th>
<th>
<input id="knottoggle" type="checkbox" checked data-toggle="toggle" value="1.0" onchange="togfun(this);" data-onstyle="primary">
</th>
</tr>
<!--Blue Piecewise Toggle-->
<tr>
<th class="labels" style="color:#0fbfcf">
<label class="sliderlabel">
<mark style="color:#49a7e9">Inside Surface</mark>
</label>
</th>
<th>
<input id="bluetoggle" type="checkbox" checked data-toggle="toggle" value="1.0" onchange="bluetogfun(this);" data-onstyle="primary">
</th>
</tr>
<!--Red Piecewise Toggle-->
<tr>
<th class="labels" style="color:#0fbfcf">
<label class="sliderlabel">
<mark style="color:#fab125">Outside Surface</mark>
</label>
</th>
<th>
<input id="redtoggle" type="checkbox" checked data-toggle="toggle" value="1.0" onchange="redtogfun(this);" data-onstyle="primary">
</th>
</tr>
<!--Time Animate Toggle-->
<tr>
<th class="labels">
<label class="sliderlabel">
Animate \(\theta\)
</label>
</th>
<th>
<input id="timetoggle" type="checkbox" checked data-toggle="toggle" value="1.0" onchange="timetogfun(this);" data-onstyle="primary">
</th>
</tr>
<!--
Time Slider
<th class="labels">
<label class="sliderlabel">
\(\theta\)
</label>
</th>
<th>
<input id="slidertime" data-slider-id='ex1Slider' type="text" data-slider-min="0" data-slider-max="6.28" data-slider-step="0.01" data-slider-value="3.14" data-slider-tooltip='hide' />
</th>
-->
</table>
<script>
//Slider Script
$('#slidera').slider({
formatter: function(value) {
return 'Current value: ' + value;
}
});
$('#sliderb').slider({
formatter: function(value) {
return 'Current value: ' + value;
}
});
$('#slidertime').slider({
formatter: function(value) {
return 'Current value: ' + value;
}
});
$('#slidera').on('slide', positionchanger)
function positionchanger() {
var Z = Number($('#slidera').val());
// console.log(Z)
mathbox.select('#cartesian')[0].set('position', [0, 0, 0.33 -0.33 * Z])
// console.log(mathbox.select('#cartesian')[0].get('position'))
}
// function slidefunction() {
//
// var sliderval = Number($('#slider').val());
//
// mathbox.select('surface').set('opacity', sliderval);
// console.log(typeof sliderval)
// }
//Button SCript
//MeshToggle
$('#meshtoggle').bootstrapSwitch({
size: 'small',
offColor: 'custom', onColor:'custom'
});
function meshtogfun(button) {
var butprop = $('#meshtoggle').prop('checked');
var butval = $('#meshtoggle').val();
if (butprop == true) {
$('#meshtoggle').val(1.0);
mathbox.select('#torusmesh')[0].set('visible', true)
} else {
$('#meshtoggle').val();
mathbox.select('#torusmesh')[0].set('visible', false)
}
}
//BlueToggle
$('#bluetoggle').bootstrapSwitch({
size: 'small',
offColor: 'custom', onColor:'custom'
});
function bluetogfun(button) {
var butprop = $('#bluetoggle').prop('checked');
var butval = $('#bluetoggle').val();
if (butprop == true) {
$('#bluetoggle').val(1.0);
mathbox.select('#bluesurface')[0].set('visible', true)
mathbox.select('#bluemesh')[0].set('visible', true)
} else {
$('#bluetoggle').val();
mathbox.select('#bluesurface')[0].set('visible', false)
mathbox.select('#bluemesh')[0].set('visible', false)
}
}
//RedToggle
$('#redtoggle').bootstrapSwitch({
size: 'small',
offColor: 'custom', onColor:'custom'
});
function redtogfun(button) {
var butprop = $('#redtoggle').prop('checked');
var butval = $('#redtoggle').val();
if (butprop == true) {
$('#redtoggle').val(1.0);
mathbox.select('#redsurface')[0].set('visible', true)
mathbox.select('#redmesh')[0].set('visible', true)
} else {
$('#redtoggle').val();
mathbox.select('#redsurface')[0].set('visible', false)
mathbox.select('#redmesh')[0].set('visible', false)
}
}
//timeToggle
$('#timetoggle').bootstrapSwitch({
size: 'small',
offColor: 'custom', onColor:'custom'
});
function timetogfun(button) {
var butprop = $('#timetoggle').prop('checked');
var butval = $('#timetoggle').val();
if (butprop == true) {
$('#timetoggle').val(1.0);
$('#slidertime').slider("disable")
} else {
$('#timetoggle').val();
$('#slidertime').slider("enable")
}
}
//KnotToggle
$('#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('visible', true)
} else {
$('#knottoggle').val();
mathbox.select('#knotcurve')[0].set('visible', false)
}
}
</script>
<h3>Constructing the Fibration Surface</h3>
<p class="blurb">
Now we see our <mark style="color:#0074D9">inside</mark> and <mark style="color:#FF851B">outside</mark> surfaces whose boundaries on the torus are exactly the boundary curves found previously. </p>
<p class="blurb"> When the torus is unwrapped as a square these surfaces look like curtains <mark style="color:#0074D9">below</mark> and <mark style="color:#FF851B">above</mark> the plane. By transforming the square to torus with the gluing operation as before, we see begin to see our fibration subsurfaces.
</p>
<p class="blurb"> Note that the <torus>rungs</torus> from the previous visualization correspond exactly to where <mark style="color:#0074D9">inside</mark> and <mark style="color:#FF851B">outside</mark> subsurfaces meet, and thus is not a boundary of their union.
</p>
<p class="blurb">
This visualization gives good intuition for how the complete fibration surface is constructed from the subsurfaces. However, it is flawed because there appears to be a circular boundary on the <mark style="color:#FF851B">outside</mark> subsurfaces as they 'tear apart'. These circular boundaries must also be glued closed so that the only boundary for our complete fibration surface is the <knot>trefoil knot</knot>.
</p>
<!-- <p>As naming suggests, the boundary curves we previously defined are actually boundaries to surfaces above and below our plane.</p>-->
<!--
<p>We can wrap these surfaces <mark style="color:#0074D9">inside</mark> and <mark style="color:#FF851B">outside</mark> the torus by our gluing transformations as before.</p>
<p>Notice that our rungs occur where the two surfaces join, and therefore are not actual boundaries to our fibration surface, which must be exactly the trefoil.</p>
<p>However, there appears to be a circular boundary on the outside surface away from the torus. To remove this as a boundary we identify these points, which we can visualize with stereographic projection.</p>
-->
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<li><a href="boundary.html">Previous</a></li>
<li><a href="stereo.html">Next</a></li>
</ul>
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<!-- *****************************************************************************************************************
FOOTER
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<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>
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