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stereo.html
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stereo.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><a href="surface.html">SURFACE</a></li>
<li class="active"><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 in Stereographic Projection</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, 2, 0],
lookAt: [0, 0, 0]
});
// 2D cartesian
var view =
mathbox
// .cartesian({
// position: [0, 0, -0.33],
// range: [
// [-math.pi, math.pi],
// [-math.pi, math.pi],
// [-math.pi, math.pi]
// ],
// scale: [1, 1, 1],
// });
.cartesian({
position: [0, 0, 0],
range: [
[-math.pi, math.pi],
[-math.pi, math.pi],
[-1, 1]
],
scale: [1, 1, 0.5],
});
function mod2pi(x) {
return (x % (2 * pi) + 2 * pi) % (2 * pi)
}
function linecut(x) {
var X = x
var thresh = 0.002;
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)
}
var alpha = 2;
//Quality of Life
var pi = math.pi;
var sin = math.sin;
var cos = math.cos;
// Torus Knot Parameters
var p = 2;
var q = 3;
var m = p / q;
var m2 = q / p;
var eps = 0.01;
// Global Animation Time
var tau_last= 0
function SurfaceCut(x) {
if (x <= 0) {
x = NaN;
} else if (x >= 2 * pi) {
x = NaN;
}
return x
}
function ZW(gamma, alpha, beta) {
var z = polarComplex(sin(gamma), alpha);
var w = polarComplex(cos(gamma), beta);
return [z, w];
};
function ReIm(z, w) {
return [z.re, z.im, w.re, w.im];
};
function polarComplex(r, phi) {
return math.complex({
r: r,
phi: phi
});
}
function sigma(z, w) {
return [w, z];
}
function SterProj(x, y, u, v) {
return [x / (1 - v), y / (1 - v), u / (1 - v)];
};
//First Piecewise Curve
function piecewise(x, y, t, n) {
x = fakemodpi(x);
var period = (pi) / q;
var amplitude = pi / (2 * p);
var wave = Math.pow(math.abs(4 / (pi) * y), 4) * ((2 * amplitude) / pi) * math.asin(sin((x + t / q) * pi / period)) + 2 * pi * n / p - (1 / p) * t
return wave
}
function SurfaceCut2(x) {
var cuteps = 0.2;
if ((x < 0) && (x > (0 - cuteps))) {
x = 0;
} else if ((x > 2 * pi) && (x < (2 * pi + cuteps))) {
x = 2 * pi;
} else if (x < (0 - cuteps)) {
x = NaN;
} else if (x > (2 * pi + cuteps)) {
x = NaN;
}
return x
}
console.log(piecewise(0, pi / 2, 0, 0))
view.area({
id: 'outsidearea',
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($('#slidertime').val());
var tau = tau_last;
}
// var xdom = SurfaceCut(x);
var xdom = SurfaceCut2(x);
// emit(...ReIm(...ZW(y,xdom,piecewise(x,y,tau,-1))));
emit(...SterProj(...ReIm(...ZW(y, xdom, piecewise(xdom, y, tau, 0)))));
emit(...SterProj(...ReIm(...ZW(y, xdom, piecewise(xdom, y, tau, 1)))));
// emit(...ReIm(...ZW(y,xdom,piecewise(x,y,tau,2))));
},
rangeX: [0, 3 * pi],
rangeY: [0, pi / 4],
width: 123,
height: 30,
channels: 3,
items: 2
});
view.surface({
id: 'outsidesurface',
color: '#fab125',
proximity: 10000000000,
shaded: false,
crossed: false,
lineX: false,
visible: true,
zBias: 10
});
view.line({
id: 'outsidemesh',
color: '#FF4136',
zBias: 10,
opacity: 0.4,
visible: true,
})
// view.resample({
// id: 'outsidesample',
// source: mathbox.select('#outsidearea')[0],
// width: 80,
// height: 20,
// })
//
// view.surface({
// id: 'outsidemesh',
// points: mathbox.select('#outsidesample')[0],
// color: '#FF4136',
// zBias: 10,
// opacity: 0.2,
// visible: false,
// lineX: true,
// lineY: true,
// fill: false
// })
//Second Stereographic Surface
//Second Piecewise Curve
function piecewise2(x, y, t, n) {
x = x % (2 * pi);
var period = (pi) / p;
var amplitude = pi / (2 * q);
var wave = Math.pow(math.abs(4 / pi * y), 4) * ((2 * amplitude) / pi) * math.asin(sin((x - t) * pi / period)) - 2 * pi * n / q
return -(wave - pi / q + (1 / q) * t)
}
view.area({
id: 'insidearea',
expr: function(emit, x, y, i, j, t) {
if ($('#timetoggle').prop('checked') == true) {
var tau = t % (2 * pi);
} else {
// var tau = Number($('#slidertime').val());
var tau = tau_last;
}
// var xdom = SurfaceCut(x);
var xdom = -(x - (1 / p) * tau) - pi / p;
emit(...SterProj(...ReIm(...ZW(-y + pi / 4 + pi / 4, piecewise2(x, y, tau, 0), xdom))));
emit(...SterProj(...ReIm(...ZW(-y + pi / 4 + pi / 4, piecewise2(x, y, tau, 1), xdom))));
emit(...SterProj(...ReIm(...ZW(-y + pi / 4 + pi / 4, piecewise2(x, y, tau, 2), xdom))));
// emit(...ReIm(...ZW(y,xdom,piecewise(x,y,tau,-1))));
// emit(...SterProj(...ReIm(...sigma(...ZW(y,xdom,piecewise2(xdom,y,tau,0))))));
// emit(...SterProj(...ReIm(...sigma(...ZW(y,xdom,piecewise2(xdom,y,tau,1))))));
// emit(...SterProj(...ReIm(...sigma(...ZW(y,xdom,piecewise2(xdom,y,tau,2))))));
// emit(...ReIm(...ZW(y,xdom,piecewise(x,y,tau,2))));
},
rangeX: [0, 3 * pi],
rangeY: [0, pi / 4],
width: 200,
height: 20,
channels: 3,
items: 3,
});
view.surface({
id: 'insidesurface',
color: '#49a7e9',
proximity: 10000000000,
shaded: false,
visible: true,
zBias: 10
});
view.line({
id: 'insidemesh',
color: '#001f3f',
zBias: 10,
opacity: 0.2,
visible: true,
})
// view.resample({
// id: 'insidesample',
// source: mathbox.select('#insidearea')[0],
// width: 64,
// height: 10,
// })
//
// view.surface({
// id: 'insidemesh',
// points: mathbox.select('#insidesample')[0],
// color: '#001f3f',
// zBias: 11,
// opacity: 0.1,
// visible: false,
// lineX: true,
// lineY: true,
// fill: false
// })
// Knot Curve
function knot(x) {
return (m2 * x) % (2 * pi);
}
view.interval({
range: [0, 2 * p * pi + 0.5],
expr: function(emit, x, i, t) {
emit(...SterProj(...ReIm(...ZW(π / 4, x % (2 * pi), knot(x)))));
},
width: 100,
channels: 4,
live: false
});
view.line({
id: 'knotcurve',
color: '#2ECC40',
proximity: 10,
blending: 'no',
zBias: 20
})
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" style="color:#0fbfcf">
<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" style="color:#0fbfcf">
<label class="sliderlabel">
<mark style="color:#AAAAAA">Glue Latitudinal Edge</mark>
</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:#0fbfcf">
<label class="sliderlabel">
<mark style="color:#AAAAAA">Glue Longitudinal Edge</mark>
</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-->
<!--
<tr>
<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' data-slider-enabled="false"/>
</th>
</tr>
-->
</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;
}
});
// $('#slider').on('slide', slidefunction)
// function slidefunction() {
//
// var sliderval = Number($('#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('#insidesurface')[0].set('visible', true)
mathbox.select('#insidemesh')[0].set('visible', true)
} else {
$('#bluetoggle').val();
mathbox.select('#insidesurface')[0].set('visible', false)
mathbox.select('#insidemesh')[0].set('visible', false)
}
}
//RedToggle
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});
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var butprop = $('#redtoggle').prop('checked');
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if (butprop == true) {
$('#redtoggle').val(1.0);
mathbox.select('#outsidesurface')[0].set('visible', true)
mathbox.select('#outsidemesh')[0].set('visible', true)
} else {
$('#redtoggle').val();
mathbox.select('#outsidesurface')[0].set('visible', false)
mathbox.select('#outsidemesh')[0].set('visible', false)
}
}
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</script>
<div class="info">
<h3> Stereographic Projection from \( S^3\)</h3>
<p class="blurb">
The previous visualization is flawed because circular boundaries appear as the non-boundary edge (away from the torus) of the <mark style="color:#FF851B">outside</mark> subsurface 'tears apart' as we animate through \(\theta\). Since we require that the only boundary for our fibration surface must be exactly the <knot>knot </knot>, we identify points along this tear.
</p>
<p class="blurb">
To acheive this, we define our surfaces as embedded in \(S^3 \in \mathbb{R}^4 \). Then we use stereographic projection \( \rho : S^3 \rightarrow\mathbb{R}^3 \) to visualize our surfaces by projecting from the point \(\{0,0,0,1\}\):
$$\rho(x,y,u,v)=\{ \frac{x}{1-v}, \frac{y}{1-v}, \frac{u}{1-v}\}$$
</p>
<p class="blurb">
By construction, this identifies all points on the non-boundary edge (away from the torus) of our subsurfaces. Now our visualization only 'tears' when the surface passes through the stereographic projection point, which is at infinity.
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<h4>About Project</h4>
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<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>
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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.
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