07Manifold_FunPullb_Curves.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 7. Smooth functions and pullbacks\n",
    "\n",
    "This notebook is part of the [Introduction to manifolds in SageMath](https://sagemanifolds.obspm.fr/intro_to_manifolds.html) by Andrzej Chrzeszczyk (Jan Kochanowski University of Kielce, Poland)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'SageMath version 9.6, Release Date: 2022-05-15'"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "version()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Recall that if $M$ is a smooth manifold, the set of all  smooth functions from $M$ to $R$ is denoted by $C^∞(M)$.\n",
    "This set is a **ring** with the operations given by<br>\n",
    "\\begin{equation}\n",
    "\\begin{matrix}\n",
    "( f + g)( p) = f ( p) + g( p),\\\\\n",
    "( f g)( p) = f ( p)g( p),\n",
    "\\end{matrix}\n",
    "\\label{}\\tag{7.1}\n",
    "\\end{equation}<br>\n",
    "for $f, g ∈ C^\\infty(M),\\ \\  a ∈ R$  and $p ∈ M.$\n",
    "\n",
    "One can also define an additional operation\n",
    "$$(a f )( p) = a f ( p),$$\n",
    "\n",
    "for $f∈ C^\\infty(M),\\ \\  a ∈ R$  and $p ∈ M.$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If $ψ$ is a smooth map from $M$ to a smooth manifold $N$ and $f ∈\n",
    "C^∞ (N )$, the **pullback $ \\ \\ ψ ^* f\\ \\ $ of $f$ under** $\\ \\ ψ\\ \\ $ is defined by\n",
    "\n",
    "\\begin{equation}\n",
    "ψ^∗ f = f ◦ ψ.\n",
    "\\label{}\\tag{7.2}\n",
    "\\end{equation}\n",
    "\n",
    "If $\\phi\\ $ and $\\ \\chi\\ $ are charts on $M$ and $N$, respectively, then\n",
    "from  $\\ \\ (ψ^∗ f ) ◦ φ^{−1} = ( f ◦ χ^{−1} ) ◦ (χ ◦ ψ ◦ φ^{−1} )\\ \\ $ it follows that $ψ^∗ f ∈\n",
    "C^∞ (M).$  \n",
    "\n",
    "The operation $ψ^∗$ is applied to functions defined on $N$\n",
    "to produce functions defined on $M$, hence the name pullback for $ψ^∗$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "**Example 7.1**\n",
    "\n",
    "If we denote by $\\psi$ the coordinate change defined by\n",
    "$x=r\\cos(\\phi),\\; y=r\\sin(\\phi)$  on the open set $U=\\{(r,\\phi): r>0, 0<\\phi <2\\pi \\},$ then for a scalar function $f$ of variables $x,y,$ the pullback\n",
    "$\\psi^*f$ is the function $(r,\\phi)\\to f(r\\cos(\\phi),r\\sin(\\phi))$, defined on $U$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\begin{array}{llcl} {\\psi}^*f:& U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(r, {\\phi}\\right) & \\longmapsto & f\\left(r \\cos\\left({\\phi}\\right), r \\sin\\left({\\phi}\\right)\\right) \\end{array}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\begin{array}{llcl} {\\psi}^*f:& U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(r, {\\phi}\\right) & \\longmapsto & f\\left(r \\cos\\left({\\phi}\\right), r \\sin\\left({\\phi}\\right)\\right) \\end{array}$"
      ],
      "text/plain": [
       "\\psi^*(f): U → ℝ\n",
       "   (r, phi) ↦ f(r*cos(phi), r*sin(phi))"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%display latex\n",
    "M = Manifold(2, 'R^2p')   # manifold R^2\n",
    "U = M.open_subset('U')    # open subset of R2, \n",
    "                          # polar coord. on U: \n",
    "c_rphi.<r,phi>=U.chart(r'r:(0,oo) phi:(0,2*pi):\\phi')\n",
    "N = Manifold(2, 'R^2c')   # second copy of R^2                         \n",
    "c_xy.<x,y> = N.chart()    # Cartesian coordinates:\n",
    "                          # psi: M --> N : polar to Cart.:\n",
    "psi = U.diff_map(N, (r*cos(phi), r*sin(phi)),name=r'\\psi')\n",
    "                          # f -scalar function on N \n",
    "f = N.scalar_field(function('f')(x,y), name='f') \n",
    "\n",
    "fplb=psi.pullback(f)      # psi^*f -scalar func. on M\n",
    "fplb.disp()               # show pullback"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "If $ψ$ is a smooth map  $M\\to N$ then\n",
    "\\begin{equation}\n",
    "ψ^∗ (a f + bg) = aψ^∗ f + bψ^∗ g,\n",
    "\\tag{7.3}\n",
    "\\end{equation}<br>\n",
    "for $f, g ∈ C^∞ (N )$ and $a, b ∈ R.$\n",
    "\n",
    "In fact\n",
    "$$ψ^∗ (a f + bg) ( p) = (a f + bg)(ψ( p)) = a f (ψ( p)) + bg(ψ( p))\\\\\n",
    "= a(ψ^∗ f )( p) + b(ψ^∗ g)( p) = (aψ^∗ f + bψ^∗ g)( p).\n",
    "$$\n",
    "\n",
    "If $ψ$ is a smooth map  $M\\to N$ then\n",
    "\\begin{equation}\n",
    "ψ^∗ ( f g) = (ψ^∗ f )(ψ^∗ g),\n",
    "\\tag{7.4}\n",
    "\\end{equation}<br>\n",
    "for $f, g ∈ C^∞ (N )$.\n",
    "\n",
    "\n",
    "The last relation follows from\n",
    "\n",
    "$$ψ^∗ ( f g) ( p) = ( f g)(ψ( p)) = f (ψ( p))g(ψ( p)) = (ψ^∗ f )( p)(ψ^∗ g)( p)\n",
    "= (ψ^∗ f )(ψ^∗ g) ( p).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If $M_1,M_2,M_3$ are smooth manifolds and $ψ_1 : M_1 → M_2$ and $ψ_2 : M_2 → M_3$ are smooth\n",
    "maps, then \n",
    "\n",
    "\\begin{equation}\n",
    "(ψ_2 ◦ ψ_1 )^∗ = ψ_1^∗ ◦ ψ_2^∗ .\n",
    "\\tag{7.5}\n",
    "\\end{equation}\n",
    "\n",
    "This is consequence of"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$(ψ_2 ◦ ψ_1 )^∗ f = f ◦ (ψ_2 ◦ ψ_1 ) = ( f ◦ ψ_2 ) ◦ ψ_1 = (ψ_2^∗ f ) ◦ ψ_1 = ψ_1^∗ (ψ_2^∗ f )= (ψ_1^∗ ◦ ψ_2^∗ ) f.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "<br>\n",
    "\n",
    "## Curves in a manifold\n",
    "\n",
    "<br> \n",
    "\n",
    "\n",
    "Let $M$ be a smooth manifold. A **smooth curve** $\\gamma$  in\n",
    "$M$, is a smooth mapping $\\gamma : I → M$ where $I\\subset R$ is an open interval.<br>\n",
    "Equivalently:\n",
    "$\\gamma : I → M$ is a smooth curve in $M$ if $I$ is an open interval of $R$ and\n",
    "$φ ◦ \\gamma: I\\to R^n$ is a smooth map   for every chart $(U, φ)$ of the atlas of $M$.\n",
    "\n",
    "In local coordinates $\\ \\ (x^1,\\ldots,x^n)\\ $ the curve is defined if we define $\\ n\\ $  real functions \n",
    "\n",
    "$$\\gamma^1(t)=x^1(\\gamma(t)),\\ldots, \\gamma^n(t)=x^n(\\gamma(t)),\\quad t\\in I. $$\n",
    "\n",
    "<br>\n",
    "\n",
    "**Example 7.2**\n",
    "\n",
    "Assume, that  in an open subset $U\\subset R^2$ we have a curve defined in polar coordinates: $\\ r=1,\\phi=t, \\ t\\in(0,2\\pi)$.<br>  Since in this example the transition maps are defined, the representation of the curve in Cartesian coordinates is determined by the system."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Check of the inverse coordinate transformation:\n",
      "  x == x  *passed*\n",
      "  y == y  *passed*\n",
      "  r == r  *passed*\n",
      "  ph == arctan2(r*sin(ph), r*cos(ph))  **failed**\n",
      "NB: a failed report can reflect a mere lack of simplification.\n",
      "\n",
      "Representations in polar and Cartesian coordinates:\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(1, t\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(1, t\\right)$"
      ],
      "text/plain": [
       "(1, t)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\cos\\left(t\\right), \\sin\\left(t\\right)\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\cos\\left(t\\right), \\sin\\left(t\\right)\\right)$"
      ],
      "text/plain": [
       "(cos(t), sin(t))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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\n",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "M = Manifold(2, 'R^2', r'\\RR^2')  # the Euclidean plane R^2\n",
    "c_xy.<x,y> = M.chart()            # Cartesian coordinates on R^2\n",
    "U = M.open_subset('U', coord_def={c_xy: (y!=0, x<0)}) \n",
    "                   # the complement of the segment y=0 and x>0\n",
    "c_cart = c_xy.restrict(U)         # Cartesian coordinates on U\n",
    "                                  # spherical coordinates on U:\n",
    "c_spher.<r,ph> = U.chart(r'r:(0,+oo) ph:(0,2*pi):\\phi') \n",
    "                                  # transition spher. to Cartesian\n",
    "ch_cart_spher=c_cart.transition_map(c_spher, [sqrt(x*x+y*y),\n",
    "                                              atan2(y,x)])\n",
    "                                  # transition Cartesian to spher.\n",
    "ch_cart_spher.set_inverse(r*cos(ph), r*sin(ph))\n",
    "R.<t> = manifolds.RealLine()      # manifold R\n",
    "c = U.curve({c_spher: (1,t)}, (t, 0, 2*pi), name='c') # curve c\n",
    "print()\n",
    "print(\"Representations in polar and Cartesian coordinates:\")\n",
    "show(c.coord_expr(c_spher))       # curve c in spherical coord.\n",
    "show(c.coord_expr(c_cart))        # curve c in Cartesian coord.\n",
    "cp=c.plot(color='black',aspect_ratio=1)   # curve plot\n",
    "cp.show(figsize=[3,3])            # show plot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "**Example 7.3**\n",
    "\n",
    "Consider the curve in $R^3$  defined in Cartesian coordinates by \n",
    "$\\ x=\\cos 3t,\\ y=\\sin 3t,\\ z=2t$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<iframe srcdoc=\"<!DOCTYPE html>\n",
       "<html>\n",
       "<head>\n",
       "<title></title>\n",
       "<meta charset=&quot;utf-8&quot;>\n",
       "<meta name=viewport content=&quot;width=device-width, user-scalable=no, minimum-scale=1.0, maximum-scale=1.0&quot;>\n",
       "<style>\n",
       "\n",
       "    body { margin: 0px; overflow: hidden; }\n",
       "\n",
       "    #menu-container { position: absolute; bottom: 30px; right: 40px; cursor: default; }\n",
       "\n",
       "    #menu-message { position: absolute; bottom: 0px; right: 0px; white-space: nowrap;\n",
       "                    display: none; background-color: #F5F5F5; padding: 10px; }\n",
       "\n",
       "    #menu-content { position: absolute; bottom: 0px; right: 0px;\n",
       "                    display: none; background-color: #F5F5F5; border-bottom: 1px solid black;\n",
       "                    border-right: 1px solid black; border-left: 1px solid black; }\n",
       "\n",
       "    #menu-content div { border-top: 1px solid black; padding: 10px; white-space: nowrap; }\n",
       "\n",
       "    #menu-content div:hover { background-color: #FEFEFE; }\n",
       "\n",
       "    .dark-theme #menu-container { color: white; }\n",
       "\n",
       "    .dark-theme #menu-message { background-color: #181818; }\n",
       "\n",
       "    .dark-theme #menu-content { background-color: #181818; border-color: white; }\n",
       "\n",
       "    .dark-theme #menu-content div { border-color: white; }\n",
       "\n",
       "    .dark-theme #menu-content div:hover { background-color: #303030; }\n",
       "\n",
       "</style>\n",
       "\n",
       "</head>\n",
       "\n",
       "<body>\n",
       "\n",
       "<script src=&quot;/nbextensions/threejs-sage/r122/three.min.js&quot;></script>\n",
       "<script>\n",
       "  if ( !window.THREE ) document.write(' \\\n",
       "<script src=&quot;https://cdn.jsdelivr.net/gh/sagemath/threejs-sage@r122/build/three.min.js&quot;><\\/script> \\\n",
       "            ');\n",
       "</script>\n",
       "        <script>// fat_lines.js\n",
       "\n",
       "var _fatLines = [];\n",
       "\n",
       "function _createFatLine( lineStrip, geometry, materialOptions ) {\n",
       "\n",
       "    var vertexCount = geometry.vertices.length;\n",
       "    var positions = [];\n",
       "    for ( var i=0 ; i < vertexCount ; i++ ) {\n",
       "        var v = geometry.vertices[i];\n",
       "        positions.push( v.x, v.y, v.z );\n",
       "        // For a line strip, duplicate all but the first and last vertices.\n",
       "        if ( lineStrip && i > 0 && i < vertexCount - 1 ) {\n",
       "            positions.push( v.x, v.y, v.z );\n",
       "        }\n",
       "    }\n",
       "\n",
       "    geometry = new THREE.LineSegmentsGeometry();\n",
       "    geometry.setPositions( positions );\n",
       "\n",
       "    var material = new THREE.LineMaterial( materialOptions );\n",
       "    material.resolution = new THREE.Vector2( window.innerWidth, window.innerHeight );\n",
       "\n",
       "    var line = new THREE.LineSegments2( geometry, material );\n",
       "    line.computeLineDistances();\n",
       "    line.scale.set( 1, 1, 1 );\n",
       "\n",
       "    _fatLines.push( line );\n",
       "\n",
       "    return line;\n",
       "\n",
       "}\n",
       "\n",
       "function createFatLineStrip( geometry, materialOptions ) {\n",
       "    return _createFatLine( true, geometry, materialOptions );\n",
       "}\n",
       "\n",
       "function createFatLineSegments( geometry, materialOptions ) {\n",
       "    return _createFatLine( false, geometry, materialOptions );\n",
       "}\n",
       "\n",
       "function rescaleFatLines() {\n",
       "    var res = new THREE.Vector2( window.innerWidth, window.innerHeight );\n",
       "    var n = _fatLines.length;\n",
       "    for ( var i=0 ; i < n ; i++ ) {\n",
       "        _fatLines[i].material.resolution = res;\n",
       "    }\n",
       "}\n",
       "</script>\n",
       "<script>\n",
       "\n",
       "    var options = {&quot;animate&quot;: false, &quot;animationControls&quot;: true, &quot;aspectRatio&quot;: [1.0, 1.0, 0.3], &quot;autoScaling&quot;: [false, false, false], &quot;autoPlay&quot;: true, &quot;axes&quot;: false, &quot;axesLabels&quot;: [&quot;x&quot;, &quot;y&quot;, &quot;z&quot;], &quot;axesLabelsStyle&quot;: null, &quot;decimals&quot;: 2, &quot;delay&quot;: 20, &quot;frame&quot;: true, &quot;loop&quot;: true, &quot;projection&quot;: &quot;perspective&quot;, &quot;theme&quot;: &quot;light&quot;, &quot;viewpoint&quot;: false};\n",
       "    var animate = options.animate;\n",
       "\n",
       "    if ( options.theme === 'dark' )\n",
       "        document.body.className = 'dark-theme';\n",
       "\n",
       "    var scene = new THREE.Scene();\n",
       "\n",
       "    var renderer = new THREE.WebGLRenderer( { antialias: true, preserveDrawingBuffer: true } );\n",
       "    renderer.setPixelRatio( window.devicePixelRatio );\n",
       "    renderer.setSize( window.innerWidth, window.innerHeight );\n",
       "    renderer.setClearColor( options.theme === 'dark' ? 0 : 0xffffff, 1 );\n",
       "    document.body.appendChild( renderer.domElement );\n",
       "\n",
       "    var b = [{&quot;x&quot;:-0.9999688468941564, &quot;y&quot;:-1.0, &quot;z&quot;:-3.141592653589793}, {&quot;x&quot;:0.9997196336934782, &quot;y&quot;:1.0, &quot;z&quot;:3.1415926535897833}]; // bounds\n",
       "\n",
       "    if ( b[0].x === b[1].x ) {\n",
       "        b[0].x -= 1;\n",
       "        b[1].x += 1;\n",
       "    }\n",
       "    if ( b[0].y === b[1].y ) {\n",
       "        b[0].y -= 1;\n",
       "        b[1].y += 1;\n",
       "    }\n",
       "    if ( b[0].z === b[1].z ) {\n",
       "        b[0].z -= 1;\n",
       "        b[1].z += 1;\n",
       "    }\n",
       "\n",
       "    var rxRange = Math.sqrt( Math.pow( b[1].z - b[0].z, 2 )\n",
       "                            + Math.pow( b[1].y - b[0].y, 2 ) );\n",
       "    var ryRange = Math.sqrt( Math.pow( b[1].x - b[0].x, 2 )\n",
       "                            + Math.pow( b[1].z - b[0].z, 2 ) );\n",
       "    var rzRange = Math.sqrt( Math.pow( b[1].x - b[0].x, 2 )\n",
       "                            + Math.pow( b[1].y - b[0].y, 2 ) );\n",
       "    var xRange = b[1].x - b[0].x;\n",
       "    var yRange = b[1].y - b[0].y;\n",
       "    var zRange = b[1].z - b[0].z;\n",
       "\n",
       "    var ar = options.aspectRatio;\n",
       "    var a = [ ar[0], ar[1], ar[2] ]; // aspect multipliers\n",
       "\n",
       "    var autoScaling = options.autoScaling;\n",
       "    var autoAspect = 2.5;\n",
       "    if ( xRange > autoAspect * rxRange && autoScaling[0] ) a[0] = autoAspect * rxRange / xRange;\n",
       "    if ( yRange > autoAspect * ryRange && autoScaling[1] ) a[1] = autoAspect * ryRange / yRange;\n",
       "    if ( zRange > autoAspect * rzRange && autoScaling[2] ) a[2] = autoAspect * rzRange / zRange;\n",
       "\n",
       "    // Distance from (xMid,yMid,zMid) to any corner of the bounding box, after applying aspectRatio\n",
       "    var midToCorner = Math.sqrt( a[0]*a[0]*xRange*xRange + a[1]*a[1]*yRange*yRange + a[2]*a[2]*zRange*zRange ) / 2;\n",
       "\n",
       "    var xMid = ( b[0].x + b[1].x ) / 2;\n",
       "    var yMid = ( b[0].y + b[1].y ) / 2;\n",
       "    var zMid = ( b[0].z + b[1].z ) / 2;\n",
       "\n",
       "    var box = new THREE.Geometry();\n",
       "    box.vertices.push( new THREE.Vector3( a[0]*b[0].x, a[1]*b[0].y, a[2]*b[0].z ) );\n",
       "    box.vertices.push( new THREE.Vector3( a[0]*b[1].x, a[1]*b[1].y, a[2]*b[1].z ) );\n",
       "    var boxMesh = new THREE.Line( box );\n",
       "    var boxColor = options.theme === 'dark' ? 'white' : 'black';\n",
       "    if ( options.frame ) scene.add( new THREE.BoxHelper( boxMesh, boxColor ) );\n",
       "\n",
       "    if ( options.axesLabels ) {\n",
       "\n",
       "        var d = options.decimals; // decimals\n",
       "        var offsetRatio = 0.1;\n",
       "        var al = options.axesLabels;\n",
       "        var als = options.axesLabelsStyle || [{}, {}, {}];\n",
       "\n",
       "        var offset = offsetRatio * a[1]*( b[1].y - b[0].y );\n",
       "        var xm = xMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(xm) ) xm = xm.substr(1);\n",
       "        addLabel( al[0] + '=' + xm, a[0]*xMid, a[1]*b[1].y+offset, a[2]*b[0].z, als[0] );\n",
       "        addLabel( ( b[0].x ).toFixed(d), a[0]*b[0].x, a[1]*b[1].y+offset, a[2]*b[0].z, als[0] );\n",
       "        addLabel( ( b[1].x ).toFixed(d), a[0]*b[1].x, a[1]*b[1].y+offset, a[2]*b[0].z, als[0] );\n",
       "\n",
       "        var offset = offsetRatio * a[0]*( b[1].x - b[0].x );\n",
       "        var ym = yMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(ym) ) ym = ym.substr(1);\n",
       "        addLabel( al[1] + '=' + ym, a[0]*b[1].x+offset, a[1]*yMid, a[2]*b[0].z, als[1] );\n",
       "        addLabel( ( b[0].y ).toFixed(d), a[0]*b[1].x+offset, a[1]*b[0].y, a[2]*b[0].z, als[1] );\n",
       "        addLabel( ( b[1].y ).toFixed(d), a[0]*b[1].x+offset, a[1]*b[1].y, a[2]*b[0].z, als[1] );\n",
       "\n",
       "        var offset = offsetRatio * a[1]*( b[1].y - b[0].y );\n",
       "        var zm = zMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(zm) ) zm = zm.substr(1);\n",
       "        addLabel( al[2] + '=' + zm, a[0]*b[1].x, a[1]*b[0].y-offset, a[2]*zMid, als[2] );\n",
       "        addLabel( ( b[0].z ).toFixed(d), a[0]*b[1].x, a[1]*b[0].y-offset, a[2]*b[0].z, als[2] );\n",
       "        addLabel( ( b[1].z ).toFixed(d), a[0]*b[1].x, a[1]*b[0].y-offset, a[2]*b[1].z, als[2] );\n",
       "\n",
       "    }\n",
       "\n",
       "    function addLabel( text, x, y, z, style ) {\n",
       "\n",
       "        var color = style.color || 'black';\n",
       "        var fontSize = style.fontSize || 14;\n",
       "        var fontFamily = style.fontFamily || 'monospace';\n",
       "        var fontStyle = style.fontStyle || 'normal';\n",
       "        var fontWeight = style.fontWeight || 'normal';\n",
       "        var opacity = style.opacity || 1;\n",
       "\n",
       "        if ( options.theme === 'dark' )\n",
       "            if ( color === 'black' || color === '#000000' )\n",
       "                color = 'white';\n",
       "\n",
       "        if ( Array.isArray( fontStyle ) ) {\n",
       "            fontFamily = fontFamily.map( function( f ) {\n",
       "                // Need to put quotes around fonts that have whitespace in their names.\n",
       "                return /\\s/.test( f ) ? '&quot;' + f + '&quot;' : f;\n",
       "            }).join(', ');\n",
       "        }\n",
       "\n",
       "        var canvas = document.createElement( 'canvas' );\n",
       "        var context = canvas.getContext( '2d' );\n",
       "        var pixelRatio = Math.round( window.devicePixelRatio );\n",
       "\n",
       "        // For example: italic bold 20px &quot;Times New Roman&quot;, Georgia, serif\n",
       "        var font = [fontStyle, fontWeight, fontSize + 'px', fontFamily].join(' ');\n",
       "\n",
       "        context.font = font;\n",
       "        var width = context.measureText( text ).width;\n",
       "        var height = fontSize;\n",
       "\n",
       "        // The dimensions of the canvas's underlying image data need to be powers\n",
       "        // of two in order for the resulting texture to support mipmapping.\n",
       "        canvas.width = THREE.MathUtils.ceilPowerOfTwo( width * pixelRatio );\n",
       "        canvas.height = THREE.MathUtils.ceilPowerOfTwo( height * pixelRatio );\n",
       "\n",
       "        // Re-compute the unscaled dimensions after the power of two conversion.\n",
       "        width = canvas.width / pixelRatio;\n",
       "        height = canvas.height / pixelRatio;\n",
       "\n",
       "        canvas.style.width = width + 'px';\n",
       "        canvas.style.height = height + 'px';\n",
       "\n",
       "        context.scale( pixelRatio, pixelRatio );\n",
       "        context.fillStyle = color;\n",
       "        context.font = font; // Must be set again after measureText.\n",
       "        context.textAlign = 'center';\n",
       "        context.textBaseline = 'middle';\n",
       "        context.fillText( text, width/2, height/2 );\n",
       "\n",
       "        var texture = new THREE.Texture( canvas );\n",
       "        texture.needsUpdate = true;\n",
       "\n",
       "        var materialOptions = { map: texture, sizeAttenuation: false, depthWrite: false };\n",
       "        if ( opacity < 1 ) {\n",
       "            // Setting opacity=1 would cause the texture's alpha component to be\n",
       "            // discarded, giving the text a black background instead of the\n",
       "            // background being transparent.\n",
       "            materialOptions.opacity = opacity;\n",
       "        }\n",
       "        var sprite = new THREE.Sprite( new THREE.SpriteMaterial( materialOptions ) );\n",
       "        sprite.position.set( x, y, z );\n",
       "\n",
       "        // Scaling factor, chosen somewhat arbitrarily so that the size of the text\n",
       "        // is consistent with previously generated plots.\n",
       "        var scale = 1/625;\n",
       "        if ( options.projection === 'orthographic' ) {\n",
       "            scale = midToCorner/256; // Needs to scale along with the plot itself.\n",
       "        }\n",
       "        sprite.scale.set( scale * width, scale * height, 1 );\n",
       "\n",
       "        scene.add( sprite );\n",
       "\n",
       "        return sprite;\n",
       "\n",
       "    }\n",
       "\n",
       "    if ( options.axes ) scene.add( new THREE.AxesHelper( Math.min( a[0]*b[1].x, a[1]*b[1].y, a[2]*b[1].z ) ) );\n",
       "\n",
       "    var camera = createCamera();\n",
       "    camera.up.set( 0, 0, 1 );\n",
       "    camera.position.set( a[0]*xMid, a[1]*yMid, a[2]*zMid );\n",
       "\n",
       "    var offset = new THREE.Vector3( a[0]*xRange, a[1]*yRange, a[2]*zRange );\n",
       "\n",
       "    if ( options.viewpoint ) {\n",
       "\n",
       "        var aa = options.viewpoint;\n",
       "        var axis = new THREE.Vector3( aa[0][0], aa[0][1], aa[0][2] ).normalize();\n",
       "        var angle = aa[1] * Math.PI / 180;\n",
       "        var q = new THREE.Quaternion().setFromAxisAngle( axis, angle ).inverse();\n",
       "\n",
       "        offset.set( 0, 0, offset.length() );\n",
       "        offset.applyQuaternion( q );\n",
       "\n",
       "    }\n",
       "\n",
       "    camera.position.add( offset );\n",
       "\n",
       "    function createCamera() {\n",
       "\n",
       "        var aspect = window.innerWidth / window.innerHeight;\n",
       "\n",
       "        // Scale the near and far clipping planes along with the overall plot size.\n",
       "        var nearClip = 0.01 * midToCorner;\n",
       "        var farClip = 100 * midToCorner;\n",
       "\n",
       "        if ( options.projection === 'orthographic' ) {\n",
       "            var camera = new THREE.OrthographicCamera( -1, 1, 1, -1, -farClip, farClip );\n",
       "            updateCameraAspect( camera, aspect );\n",
       "            return camera;\n",
       "        }\n",
       "\n",
       "        return new THREE.PerspectiveCamera( 45, aspect, nearClip, farClip );\n",
       "\n",
       "    }\n",
       "\n",
       "    function updateCameraAspect( camera, aspect ) {\n",
       "\n",
       "        if ( camera.isPerspectiveCamera ) {\n",
       "            camera.aspect = aspect;\n",
       "        } else if ( camera.isOrthographicCamera ) {\n",
       "            // Fit the camera frustum to the bounding box's diagonal so that the entire plot fits\n",
       "            // within at the default zoom level and camera position.\n",
       "            if ( aspect > 1 ) { // Wide window\n",
       "                camera.top = midToCorner;\n",
       "                camera.right = midToCorner * aspect;\n",
       "            } else { // Tall or square window\n",
       "                camera.top = midToCorner / aspect;\n",
       "                camera.right = midToCorner;\n",
       "            }\n",
       "            camera.bottom = -camera.top;\n",
       "            camera.left = -camera.right;\n",
       "        }\n",
       "\n",
       "        camera.updateProjectionMatrix();\n",
       "\n",
       "    }\n",
       "\n",
       "    var lights = [{&quot;x&quot;:-5, &quot;y&quot;:3, &quot;z&quot;:0, &quot;color&quot;:&quot;#7f7f7f&quot;, &quot;parent&quot;:&quot;camera&quot;}];\n",
       "    for ( var i=0 ; i < lights.length ; i++ ) {\n",
       "        var light = new THREE.DirectionalLight( lights[i].color, 1 );\n",
       "        light.position.set( a[0]*lights[i].x, a[1]*lights[i].y, a[2]*lights[i].z );\n",
       "        if ( lights[i].parent === 'camera' ) {\n",
       "            light.target.position.set( a[0]*xMid, a[1]*yMid, a[2]*zMid );\n",
       "            scene.add( light.target );\n",
       "            camera.add( light );\n",
       "        } else scene.add( light );\n",
       "    }\n",
       "    scene.add( camera );\n",
       "\n",
       "    var ambient = {&quot;color&quot;:&quot;#7f7f7f&quot;};\n",
       "    scene.add( new THREE.AmbientLight( ambient.color, 1 ) );\n",
       "\n",
       "    var controls = new THREE.OrbitControls( camera, renderer.domElement );\n",
       "    controls.target.set( a[0]*xMid, a[1]*yMid, a[2]*zMid );\n",
       "    controls.addEventListener( 'change', function() { if ( !animate ) render(); } );\n",
       "\n",
       "    window.addEventListener( 'resize', function() {\n",
       "\n",
       "        renderer.setSize( window.innerWidth, window.innerHeight );\n",
       "        updateCameraAspect( camera, window.innerWidth / window.innerHeight );\n",
       "        if ( window.rescaleFatLines ) rescaleFatLines();\n",
       "        if ( !animate ) render();\n",
       "\n",
       "    } );\n",
       "\n",
       "    var texts = [];\n",
       "    for ( var i=0 ; i < texts.length ; i++ ) addText( texts[i] );\n",
       "\n",
       "    function addText( json ) {\n",
       "        var sprite = addLabel( json.text, a[0]*json.x, a[1]*json.y, a[2]*json.z, json );\n",
       "        sprite.userData = json;\n",
       "    }\n",
       "\n",
       "    var points = [];\n",
       "    for ( var i=0 ; i < points.length ; i++ ) addPoint( points[i] );\n",
       "\n",
       "    function addPoint( json ) {\n",
       "\n",
       "        var geometry = new THREE.Geometry();\n",
       "        var v = json.point;\n",
       "        geometry.vertices.push( new THREE.Vector3( a[0]*v[0], a[1]*v[1], a[2]*v[2] ) );\n",
       "\n",
       "        var canvas = document.createElement( 'canvas' );\n",
       "        canvas.width = 128;\n",
       "        canvas.height = 128;\n",
       "\n",
       "        var context = canvas.getContext( '2d' );\n",
       "        context.arc( 64, 64, 64, 0, 2 * Math.PI );\n",
       "        context.fillStyle = json.color;\n",
       "        context.fill();\n",
       "\n",
       "        var texture = new THREE.Texture( canvas );\n",
       "        texture.needsUpdate = true;\n",
       "\n",
       "        var transparent = json.opacity < 1 ? true : false;\n",
       "        var size = camera.isOrthographicCamera ? json.size : json.size/100;\n",
       "        var material = new THREE.PointsMaterial( { size: size, map: texture,\n",
       "                                                   transparent: transparent, opacity: json.opacity,\n",
       "                                                   alphaTest: .1 } );\n",
       "\n",
       "        var c = new THREE.Vector3();\n",
       "        geometry.computeBoundingBox();\n",
       "        geometry.boundingBox.getCenter( c );\n",
       "        geometry.translate( -c.x, -c.y, -c.z );\n",
       "\n",
       "        var mesh = new THREE.Points( geometry, material );\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        mesh.userData = json;\n",
       "        scene.add( mesh );\n",
       "\n",
       "    }\n",
       "\n",
       "    var lines = [{&quot;points&quot;: [[-1.8369701987210297e-16, 1.0, -3.141592653589793], [-0.047342989971558266, 0.9988786919844436, -3.110018858076328], [-0.09457980779484532, 0.9955172826011058, -3.0784450625628628], [-0.1416045194249512, 0.9899233102005572, -3.0468712670493976], [-0.18831166648971756, 0.9821093199149804, -3.0152974715359324], [-0.2345965027923685, 0.9720928355242571, -2.9837236760224672], [-0.28035522921701345, 0.9598963201568573, -2.952149880509002], [-0.32548522651021083, 0.9455471259136674, -2.920576084995537], [-0.3698852854165461, 0.9290774325277308, -2.8890022894820717], [-0.4134558336521334, 0.9105241751974625, -2.8574284939686065], [-0.4560991592070154, 0.8899289617551805, -2.8258546984551414], [-0.4977196294756819, 0.8673379793567153, -2.794280902941676], [-0.5382239057242875, 0.8428018909013514, -2.762707107428211], [-0.5775211524135876, 0.8163757214143998, -2.731133311914746], [-0.6155232409081776, 0.7881187346471936, -2.6995595164012807], [-0.6521449471151851, 0.7580943001712468, -2.6679857208878155], [-0.6873041426091829, 0.7263697512646407, -2.6364119253743503], [-0.7209219788147152, 0.693016233909333, -2.604838129860885], [-0.7529230638333757, 0.6581085472380392, -2.57326433434742], [-0.7832356315188889, 0.6217249757884971, -2.541690538833955], [-0.8117917024210194, 0.5839471139413082, -2.5101167433204896], [-0.8385272362373761, 0.5448596829350724, -2.4785429478070244], [-0.863382275431222, 0.5045503408691797, -2.4469691522935593], [-0.8863010796932073, 0.4631094861203504, -2.415395356780094], [-0.9072322509454802, 0.42063005461378683, -2.383821561266629], [-0.9261288486078402, 0.37720731140357855, -2.3522477657531637], [-0.942948494867436, 0.332938637029764, -2.3206739702396986], [-0.9576534697159286, 0.28792330913116976, -2.2891001747262334], [-0.9702107955409854, 0.24226227980378642, -2.257526379212768], [-0.9805923110824034, 0.1960579492039812, -2.225952583699303], [-0.988774734587002, 0.1494139359042645, 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[-0.2954734496346631, 0.955350951515197, 1.2471649227818729], [-0.3403713034041059, 0.940291112272678, 1.278738718295338], [-0.38450583503200386, 0.9231225611078636, 1.3103125138088032], [-0.4277780677102029, 0.9038838004888267, 1.3418863093222684], [-0.47009095843599613, 0.8826179755685506, 1.3734601048357336], [-0.5113496156423196, 0.8593727774269161, 1.4050339003491987], [-0.5514615120031008, 0.834200336117922, 1.436607695862664], [-0.5903366919365219, 0.80715710376199, 1.468181491376129], [-0.6278879733408519, 0.7783037279455352, 1.4997552868895943], [-0.6640311431104248, 0.7477049157117148, 1.5313290824030594], [-0.6986851459933003, 0.7154292884473773, 1.5629028779165246], [-0.731772266367071, 0.68154922799164, 1.5944766734299898], [-0.7632183025251628, 0.6461407143112164, 1.626050468943455], [-0.792952733082773, 0.609283155106524, 1.6576242644569201], [-0.8209088751292574, 0.5710592077307022, 1.6891980599703853], [-0.8470240337722941, 0.531554593820909, 1.7207718554838505], [-0.871239642738455, 0.4908579070576018, 1.7523456509973157], [-0.8935013957148698, 0.44906041448292877, 1.7839194465107808], [-0.9137593681374329, 0.40625585182379775, 1.815493242024246], [-0.9319681291524314, 0.3625402132786333, 1.8470670375377112], [-0.9480868435005064, 0.31801153623924755, 1.8786408330511764], [-0.9620793630944601, 0.27276968143061286, 1.9102146285646415], [-0.9739143080855355, 0.22691610896159994, 1.9417884240781067], [-0.9835651372363681, 0.180553650788912, 1.9733622195915719], [-0.9910102074427908, 0.13378628010448934, 2.004936015105037], [-0.9962328222710057, 0.0867188781635614, 2.0365098106185022], [-0.9992212694012752, 0.03945699907626331, 2.0680836061319674], [-0.9999688468941564, -0.007893366909702814, 2.0996574016454326], [-0.9984738782203795, -0.055226031104497336, 2.1312311971588977], [-0.9947397160206581, -0.10243484451660219, 2.162804992672363], [-0.9887747345870043, -0.14941393590425, 2.194378788185828], [-0.9805923110824063, -0.19605794920396727, 2.2259525836992933], [-0.970210795540989, -0.2422622798037722, 2.2575263792127584], [-0.9576534697159329, -0.28792330913115527, 2.2891001747262236], [-0.9429484948674409, -0.3329386370297502, 2.320673970239689], [-0.9261288486078455, -0.3772073114035654, 2.352247765753154], [-0.9072322509454863, -0.4206300546137735, 2.383821561266619], [-0.8863010796932143, -0.46310948612033703, 2.4153953567800843], [-0.8633822754312295, -0.5045503408691671, 2.4469691522935495], [-0.8385272362373839, -0.5448596829350604, 2.4785429478070147], [-0.8117917024210279, -0.5839471139412962, 2.51011674332048], [-0.7832356315188983, -0.6217249757884853, 2.541690538833945], [-0.7529230638333854, -0.6581085472380281, 2.57326433434741], [-0.7209219788147251, -0.6930162339093228, 2.6048381298608754], [-0.6873041426091936, -0.7263697512646305, 2.6364119253743405], [-0.6521449471151965, -0.7580943001712369, 2.6679857208878057], [-0.6155232409081896, -0.7881187346471843, 2.699559516401271], [-0.5775211524135992, -0.8163757214143916, 2.731133311914736], [-0.5382239057243002, -0.8428018909013432, 2.7627071074282012], [-0.497719629475695, -0.8673379793567079, 2.7942809029416664], [-0.45609915920702804, -0.8899289617551741, 2.8258546984551316], [-0.4134558336521464, -0.9105241751974567, 2.8574284939685968], [-0.3698852854165593, -0.9290774325277256, 2.889002289482062], [-0.3254852265102251, -0.9455471259136624, 2.920576084995527], [-0.28035522921702793, -0.959896320156853, 2.9521498805089923], [-0.2345965027923823, -0.9720928355242537, 2.9837236760224575], [-0.18831166648973238, -0.9821093199149776, 3.0152974715359226], [-0.14160451942496616, -0.989923310200555, 3.046871267049388], [-0.09457980779485947, -0.9955172826011045, 3.078445062562853], [-0.04734298997157246, -0.9988786919844429, 3.110018858076318], [-1.4394551735074106e-14, -1.0, 3.1415926535897833]], &quot;color&quot;: &quot;#808080&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 4.0}];\n",
       "    for ( var i=0 ; i < lines.length ; i++ ) addLine( lines[i] );\n",
       "\n",
       "    function addLine( json ) {\n",
       "\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i < json.points.length ; i++ ) {\n",
       "            var v = json.points[i];\n",
       "            geometry.vertices.push( new THREE.Vector3( a[0]*v[0], a[1]*v[1], a[2]*v[2] ) );\n",
       "        }\n",
       "\n",
       "        var c = new THREE.Vector3();\n",
       "        geometry.computeBoundingBox();\n",
       "        geometry.boundingBox.getCenter( c );\n",
       "        geometry.translate( -c.x, -c.y, -c.z );\n",
       "\n",
       "        var transparent = json.opacity < 1 ? true : false;\n",
       "        var materialOptions = { color: json.color, linewidth: json.linewidth,\n",
       "                                transparent: transparent, opacity: json.opacity };\n",
       "\n",
       "        var mesh;\n",
       "        if ( json.linewidth > 1 && window.createFatLineStrip ) {\n",
       "            mesh = createFatLineStrip( geometry, materialOptions );\n",
       "        } else {\n",
       "            var material = new THREE.LineBasicMaterial( materialOptions );\n",
       "            mesh = new THREE.Line( geometry, material );\n",
       "        }\n",
       "\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        mesh.userData = json;\n",
       "        scene.add( mesh );\n",
       "\n",
       "    }\n",
       "\n",
       "    var surfaces = [];\n",
       "    for ( var i=0 ; i < surfaces.length ; i++ ) addSurface( surfaces[i] );\n",
       "\n",
       "    function addSurface( json ) {\n",
       "\n",
       "        var useFaceColors = 'faceColors' in json ? true : false;\n",
       "\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i < json.vertices.length ; i++ ) {\n",
       "            var v = json.vertices[i];\n",
       "            geometry.vertices.push( new THREE.Vector3( a[0]*v.x, a[1]*v.y, a[2]*v.z ) );\n",
       "        }\n",
       "        for ( var i=0 ; i < json.faces.length ; i++ ) {\n",
       "            var f = json.faces[i];\n",
       "            for ( var j=0 ; j < f.length - 2 ; j++ ) {\n",
       "                var face = new THREE.Face3( f[0], f[j+1], f[j+2] );\n",
       "                if ( useFaceColors ) face.color.set( json.faceColors[i] );\n",
       "                geometry.faces.push( face );\n",
       "            }\n",
       "        }\n",
       "        geometry.computeVertexNormals();\n",
       "\n",
       "        var side = json.singleSide ? THREE.FrontSide : THREE.DoubleSide;\n",
       "        var transparent = json.opacity < 1 ? true : false;\n",
       "        var depthWrite = 'depthWrite' in json ? json.depthWrite : !transparent;\n",
       "        var flatShading = json.useFlatShading ? json.useFlatShading : false;\n",
       "\n",
       "        var material = new THREE.MeshPhongMaterial( { side: side,\n",
       "                                     color: useFaceColors ? 'white' : json.color,\n",
       "                                     vertexColors: useFaceColors ? THREE.FaceColors : THREE.NoColors,\n",
       "                                     transparent: transparent, opacity: json.opacity,\n",
       "                                     shininess: 20, flatShading: flatShading,\n",
       "                                     depthWrite: depthWrite } );\n",
       "\n",
       "        var c = new THREE.Vector3();\n",
       "        geometry.computeBoundingBox();\n",
       "        geometry.boundingBox.getCenter( c );\n",
       "        geometry.translate( -c.x, -c.y, -c.z );\n",
       "\n",
       "        var mesh = new THREE.Mesh( geometry, material );\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        if ( transparent && json.renderOrder ) mesh.renderOrder = json.renderOrder;\n",
       "        mesh.userData = json;\n",
       "        scene.add( mesh );\n",
       "\n",
       "        if ( json.showMeshGrid ) addSurfaceMeshGrid( json );\n",
       "\n",
       "    }\n",
       "\n",
       "    function addSurfaceMeshGrid( json ) {\n",
       "\n",
       "        var geometry = new THREE.Geometry();\n",
       "\n",
       "        for ( var i=0 ; i < json.faces.length ; i++ ) {\n",
       "            var f = json.faces[i];\n",
       "            for ( var j=0 ; j < f.length ; j++ ) {\n",
       "                var k = j === f.length-1 ? 0 : j+1;\n",
       "                var v1 = json.vertices[f[j]];\n",
       "                var v2 = json.vertices[f[k]];\n",
       "                // vertices in opposite directions on neighboring faces\n",
       "                var nudge = f[j] < f[k] ? .0005*zRange : -.0005*zRange;\n",
       "                geometry.vertices.push( new THREE.Vector3( a[0]*v1.x, a[1]*v1.y, a[2]*(v1.z+nudge) ) );\n",
       "                geometry.vertices.push( new THREE.Vector3( a[0]*v2.x, a[1]*v2.y, a[2]*(v2.z+nudge) ) );\n",
       "            }\n",
       "        }\n",
       "\n",
       "        var c = new THREE.Vector3();\n",
       "        geometry.computeBoundingBox();\n",
       "        geometry.boundingBox.getCenter( c );\n",
       "        geometry.translate( -c.x, -c.y, -c.z );\n",
       "\n",
       "        var gridColor = options.theme === 'dark' ? 'white' : 'black';\n",
       "        var linewidth = json.linewidth || 1;\n",
       "        var materialOptions = { color: gridColor, linewidth: linewidth };\n",
       "\n",
       "        var mesh;\n",
       "        if ( linewidth > 1 && window.createFatLineSegments ) {\n",
       "            mesh = createFatLineSegments( geometry, materialOptions );\n",
       "        } else {\n",
       "            var material = new THREE.LineBasicMaterial( materialOptions );\n",
       "            mesh = new THREE.LineSegments( geometry, material );\n",
       "        }\n",
       "\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        mesh.userData = json;\n",
       "        scene.add( mesh );\n",
       "\n",
       "    }\n",
       "\n",
       "    function render() {\n",
       "\n",
       "        if ( window.updateAnimation ) animate = updateAnimation();\n",
       "        if ( animate ) requestAnimationFrame( render );\n",
       "\n",
       "        renderer.render( scene, camera );\n",
       "\n",
       "    }\n",
       "\n",
       "    render();\n",
       "    controls.update();\n",
       "    if ( !animate ) render();\n",
       "\n",
       "\n",
       "    // menu functions\n",
       "\n",
       "    function toggleMenu() {\n",
       "\n",
       "        var m = document.getElementById( 'menu-content' );\n",
       "        if ( m.style.display === 'block' ) m.style.display = 'none'\n",
       "        else m.style.display = 'block';\n",
       "\n",
       "    }\n",
       "\n",
       "\n",
       "    function saveAsPNG() {\n",
       "\n",
       "        var a = document.body.appendChild( document.createElement( 'a' ) );\n",
       "        a.href = renderer.domElement.toDataURL( 'image/png' );\n",
       "        a.download = 'screenshot';\n",
       "        a.click();\n",
       "\n",
       "    }\n",
       "\n",
       "    function saveAsHTML() {\n",
       "\n",
       "        toggleMenu(); // otherwise visible in output\n",
       "        event.stopPropagation();\n",
       "\n",
       "        var blob = new Blob( [ '<!DOCTYPE html>\\n' + document.documentElement.outerHTML ] );\n",
       "        var a = document.body.appendChild( document.createElement( 'a' ) );\n",
       "        a.href = window.URL.createObjectURL( blob );\n",
       "        a.download = suggestFilename();\n",
       "        a.click();\n",
       "\n",
       "        function suggestFilename() {\n",
       "            if ( !document.title ) {\n",
       "                return 'graphic.html';\n",
       "            } else if ( /\\.html?$/i.test( document.title ) ) {\n",
       "                return document.title; // already ends in .htm or .html\n",
       "            } else {\n",
       "                return document.title + '.html';\n",
       "            }\n",
       "        }\n",
       "\n",
       "    }\n",
       "\n",
       "    function getViewpoint() {\n",
       "\n",
       "        function roundTo( x, n ) { return +x.toFixed(n); }\n",
       "\n",
       "        var v = camera.quaternion.inverse();\n",
       "        var r = Math.sqrt( v.x*v.x + v.y*v.y + v.z*v.z );\n",
       "        var axis = [ roundTo( v.x / r, 4 ), roundTo( v.y / r, 4 ), roundTo( v.z / r, 4 ) ];\n",
       "        var angle = roundTo( 2 * Math.atan2( r, v.w ) * 180 / Math.PI, 2 );\n",
       "\n",
       "        var textArea = document.createElement( 'textarea' );\n",
       "        textArea.textContent = JSON.stringify( axis ) + ',' + angle;\n",
       "        textArea.style.csstext = 'position: absolute; top: -100%';\n",
       "        document.body.append( textArea );\n",
       "        textArea.select();\n",
       "        document.execCommand( 'copy' );\n",
       "\n",
       "        var m = document.getElementById( 'menu-message' );\n",
       "        m.innerHTML = 'Viewpoint copied to clipboard';\n",
       "        m.style.display = 'block';\n",
       "        setTimeout( function() { m.style.display = 'none'; }, 2000 );\n",
       "\n",
       "    }\n",
       "\n",
       "    function getCamera() {\n",
       "\n",
       "        function roundTo( x, n ) { return +x.toFixed(n); }\n",
       "\n",
       "        var pos = camera.position;\n",
       "        var pos_r = [ roundTo( pos.x, 4 ), roundTo( pos.y, 4 ), roundTo( pos.z, 4 ) ];\n",
       "   //     var up = camera.up; // up is always (0,0,1)\n",
       "        var textArea = document.createElement('textarea');\n",
       "        var cam_position = JSON.stringify(pos_r);\n",
       "        textArea.textContent = ',camera_position=' + cam_position;\n",
       "        textArea.style.csstext = 'position: absolute; top: -100%';\n",
       "        document.body.append( textArea );\n",
       "        textArea.select();\n",
       "        document.execCommand( 'copy' );\n",
       "\n",
       "        var m = document.getElementById( 'menu-message' );\n",
       "        m.innerHTML = 'Camera position '+ cam_position+' copied to clipboard';\n",
       "        m.style.display = 'block';\n",
       "        setTimeout( function() { m.style.display = 'none'; }, 2000 );\n",
       "\n",
       "    }\n",
       "                                       \n",
       "</script>\n",
       "\n",
       "<div id=&quot;menu-container&quot; onclick=&quot;toggleMenu()&quot;>&#x24d8;\n",
       "<div id=&quot;menu-message&quot;></div>\n",
       "<div id=&quot;menu-content&quot;>\n",
       "<div onclick=&quot;saveAsPNG()&quot;>Save as PNG</div>\n",
       "<div onclick=&quot;saveAsHTML()&quot;>Save as HTML</div>\n",
       "<div onclick=&quot;getCamera()&quot;>Get camera</div>\n",
       "<div onclick=&quot;getViewpoint()&quot;>Get viewpoint</div>\n",
       "<div>Close Menu</div>\n",
       "</div></div>\n",
       "\n",
       "\n",
       "</body>\n",
       "</html>\n",
       "\"\n",
       "        width=\"100%\"\n",
       "        height=\"400\"\n",
       "        style=\"border: 0;\">\n",
       "</iframe>\n"
      ],
      "text/plain": [
       "Graphics3d Object"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "J = manifolds.OpenInterval(0, 2*pi)      # open interval\n",
    "c.<t>=J.chart()                          # coordinate t on c\n",
    "R3 = Manifold(3, 'R^3')                  # manifold R^3\n",
    "X.<x,y,z> = R3.chart()                   # Cartesian coord. on R^3\n",
    "                                         # curve c in R^3:\n",
    "c = R3.curve({X:[cos(3*t),sin(3*t),2*t]},(t,-pi/2,+pi/2),name='c')\n",
    "                                         # plot c:\n",
    "p = c.plot(X,color='grey',thickness=4,plot_points=200,label_axes=False)\n",
    "p.show(aspect_ratio=[1,1,0.3])           # show plot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "**Example 7.4**\n",
    "\n",
    "Define a  curve on the sphere  $S^2$, using spherical coordinates $\\ \\theta=t,\\ \\phi=\\cos 2t$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<iframe srcdoc=\"<!DOCTYPE html>\n",
       "<html>\n",
       "<head>\n",
       "<title></title>\n",
       "<meta charset=&quot;utf-8&quot;>\n",
       "<meta name=viewport content=&quot;width=device-width, user-scalable=no, minimum-scale=1.0, maximum-scale=1.0&quot;>\n",
       "<style>\n",
       "\n",
       "    body { margin: 0px; overflow: hidden; }\n",
       "\n",
       "    #menu-container { position: absolute; bottom: 30px; right: 40px; cursor: default; }\n",
       "\n",
       "    #menu-message { position: absolute; bottom: 0px; right: 0px; white-space: nowrap;\n",
       "                    display: none; background-color: #F5F5F5; padding: 10px; }\n",
       "\n",
       "    #menu-content { position: absolute; bottom: 0px; right: 0px;\n",
       "                    display: none; background-color: #F5F5F5; border-bottom: 1px solid black;\n",
       "                    border-right: 1px solid black; border-left: 1px solid black; }\n",
       "\n",
       "    #menu-content div { border-top: 1px solid black; padding: 10px; white-space: nowrap; }\n",
       "\n",
       "    #menu-content div:hover { background-color: #FEFEFE; }\n",
       "\n",
       "    .dark-theme #menu-container { color: white; }\n",
       "\n",
       "    .dark-theme #menu-message { background-color: #181818; }\n",
       "\n",
       "    .dark-theme #menu-content { background-color: #181818; border-color: white; }\n",
       "\n",
       "    .dark-theme #menu-content div { border-color: white; }\n",
       "\n",
       "    .dark-theme #menu-content div:hover { background-color: #303030; }\n",
       "\n",
       "</style>\n",
       "\n",
       "</head>\n",
       "\n",
       "<body>\n",
       "\n",
       "<script src=&quot;/nbextensions/threejs-sage/r122/three.min.js&quot;></script>\n",
       "<script>\n",
       "  if ( !window.THREE ) document.write(' \\\n",
       "<script src=&quot;https://cdn.jsdelivr.net/gh/sagemath/threejs-sage@r122/build/three.min.js&quot;><\\/script> \\\n",
       "            ');\n",
       "</script>\n",
       "        <script>// fat_lines.js\n",
       "\n",
       "var _fatLines = [];\n",
       "\n",
       "function _createFatLine( lineStrip, geometry, materialOptions ) {\n",
       "\n",
       "    var vertexCount = geometry.vertices.length;\n",
       "    var positions = [];\n",
       "    for ( var i=0 ; i < vertexCount ; i++ ) {\n",
       "        var v = geometry.vertices[i];\n",
       "        positions.push( v.x, v.y, v.z );\n",
       "        // For a line strip, duplicate all but the first and last vertices.\n",
       "        if ( lineStrip && i > 0 && i < vertexCount - 1 ) {\n",
       "            positions.push( v.x, v.y, v.z );\n",
       "        }\n",
       "    }\n",
       "\n",
       "    geometry = new THREE.LineSegmentsGeometry();\n",
       "    geometry.setPositions( positions );\n",
       "\n",
       "    var material = new THREE.LineMaterial( materialOptions );\n",
       "    material.resolution = new THREE.Vector2( window.innerWidth, window.innerHeight );\n",
       "\n",
       "    var line = new THREE.LineSegments2( geometry, material );\n",
       "    line.computeLineDistances();\n",
       "    line.scale.set( 1, 1, 1 );\n",
       "\n",
       "    _fatLines.push( line );\n",
       "\n",
       "    return line;\n",
       "\n",
       "}\n",
       "\n",
       "function createFatLineStrip( geometry, materialOptions ) {\n",
       "    return _createFatLine( true, geometry, materialOptions );\n",
       "}\n",
       "\n",
       "function createFatLineSegments( geometry, materialOptions ) {\n",
       "    return _createFatLine( false, geometry, materialOptions );\n",
       "}\n",
       "\n",
       "function rescaleFatLines() {\n",
       "    var res = new THREE.Vector2( window.innerWidth, window.innerHeight );\n",
       "    var n = _fatLines.length;\n",
       "    for ( var i=0 ; i < n ; i++ ) {\n",
       "        _fatLines[i].material.resolution = res;\n",
       "    }\n",
       "}\n",
       "</script>\n",
       "<script>\n",
       "\n",
       "    var options = {&quot;animate&quot;: false, &quot;animationControls&quot;: true, &quot;aspectRatio&quot;: [1.0, 1.0, 1.0], &quot;autoScaling&quot;: [false, false, false], &quot;autoPlay&quot;: true, &quot;axes&quot;: false, &quot;axesLabels&quot;: false, &quot;axesLabelsStyle&quot;: null, &quot;decimals&quot;: 2, &quot;delay&quot;: 20, &quot;frame&quot;: false, &quot;loop&quot;: true, &quot;projection&quot;: &quot;perspective&quot;, &quot;theme&quot;: &quot;light&quot;, &quot;viewpoint&quot;: false};\n",
       "    var animate = options.animate;\n",
       "\n",
       "    if ( options.theme === 'dark' )\n",
       "        document.body.className = 'dark-theme';\n",
       "\n",
       "    var scene = new THREE.Scene();\n",
       "\n",
       "    var renderer = new THREE.WebGLRenderer( { antialias: true, preserveDrawingBuffer: true } );\n",
       "    renderer.setPixelRatio( window.devicePixelRatio );\n",
       "    renderer.setSize( window.innerWidth, window.innerHeight );\n",
       "    renderer.setClearColor( options.theme === 'dark' ? 0 : 0xffffff, 1 );\n",
       "    document.body.appendChild( renderer.domElement );\n",
       "\n",
       "    var b = [{&quot;x&quot;:-0.9983786540671052, &quot;y&quot;:-0.9991889981715698, &quot;z&quot;:-1.0}, {&quot;x&quot;:0.9983786540671052, &quot;y&quot;:0.9959489361348121, &quot;z&quot;:1.0}]; // bounds\n",
       "\n",
       "    if ( b[0].x === b[1].x ) {\n",
       "        b[0].x -= 1;\n",
       "        b[1].x += 1;\n",
       "    }\n",
       "    if ( b[0].y === b[1].y ) {\n",
       "        b[0].y -= 1;\n",
       "        b[1].y += 1;\n",
       "    }\n",
       "    if ( b[0].z === b[1].z ) {\n",
       "        b[0].z -= 1;\n",
       "        b[1].z += 1;\n",
       "    }\n",
       "\n",
       "    var rxRange = Math.sqrt( Math.pow( b[1].z - b[0].z, 2 )\n",
       "                            + Math.pow( b[1].y - b[0].y, 2 ) );\n",
       "    var ryRange = Math.sqrt( Math.pow( b[1].x - b[0].x, 2 )\n",
       "                            + Math.pow( b[1].z - b[0].z, 2 ) );\n",
       "    var rzRange = Math.sqrt( Math.pow( b[1].x - b[0].x, 2 )\n",
       "                            + Math.pow( b[1].y - b[0].y, 2 ) );\n",
       "    var xRange = b[1].x - b[0].x;\n",
       "    var yRange = b[1].y - b[0].y;\n",
       "    var zRange = b[1].z - b[0].z;\n",
       "\n",
       "    var ar = options.aspectRatio;\n",
       "    var a = [ ar[0], ar[1], ar[2] ]; // aspect multipliers\n",
       "\n",
       "    var autoScaling = options.autoScaling;\n",
       "    var autoAspect = 2.5;\n",
       "    if ( xRange > autoAspect * rxRange && autoScaling[0] ) a[0] = autoAspect * rxRange / xRange;\n",
       "    if ( yRange > autoAspect * ryRange && autoScaling[1] ) a[1] = autoAspect * ryRange / yRange;\n",
       "    if ( zRange > autoAspect * rzRange && autoScaling[2] ) a[2] = autoAspect * rzRange / zRange;\n",
       "\n",
       "    // Distance from (xMid,yMid,zMid) to any corner of the bounding box, after applying aspectRatio\n",
       "    var midToCorner = Math.sqrt( a[0]*a[0]*xRange*xRange + a[1]*a[1]*yRange*yRange + a[2]*a[2]*zRange*zRange ) / 2;\n",
       "\n",
       "    var xMid = ( b[0].x + b[1].x ) / 2;\n",
       "    var yMid = ( b[0].y + b[1].y ) / 2;\n",
       "    var zMid = ( b[0].z + b[1].z ) / 2;\n",
       "\n",
       "    var box = new THREE.Geometry();\n",
       "    box.vertices.push( new THREE.Vector3( a[0]*b[0].x, a[1]*b[0].y, a[2]*b[0].z ) );\n",
       "    box.vertices.push( new THREE.Vector3( a[0]*b[1].x, a[1]*b[1].y, a[2]*b[1].z ) );\n",
       "    var boxMesh = new THREE.Line( box );\n",
       "    var boxColor = options.theme === 'dark' ? 'white' : 'black';\n",
       "    if ( options.frame ) scene.add( new THREE.BoxHelper( boxMesh, boxColor ) );\n",
       "\n",
       "    if ( options.axesLabels ) {\n",
       "\n",
       "        var d = options.decimals; // decimals\n",
       "        var offsetRatio = 0.1;\n",
       "        var al = options.axesLabels;\n",
       "        var als = options.axesLabelsStyle || [{}, {}, {}];\n",
       "\n",
       "        var offset = offsetRatio * a[1]*( b[1].y - b[0].y );\n",
       "        var xm = xMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(xm) ) xm = xm.substr(1);\n",
       "        addLabel( al[0] + '=' + xm, a[0]*xMid, a[1]*b[1].y+offset, a[2]*b[0].z, als[0] );\n",
       "        addLabel( ( b[0].x ).toFixed(d), a[0]*b[0].x, a[1]*b[1].y+offset, a[2]*b[0].z, als[0] );\n",
       "        addLabel( ( b[1].x ).toFixed(d), a[0]*b[1].x, a[1]*b[1].y+offset, a[2]*b[0].z, als[0] );\n",
       "\n",
       "        var offset = offsetRatio * a[0]*( b[1].x - b[0].x );\n",
       "        var ym = yMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(ym) ) ym = ym.substr(1);\n",
       "        addLabel( al[1] + '=' + ym, a[0]*b[1].x+offset, a[1]*yMid, a[2]*b[0].z, als[1] );\n",
       "        addLabel( ( b[0].y ).toFixed(d), a[0]*b[1].x+offset, a[1]*b[0].y, a[2]*b[0].z, als[1] );\n",
       "        addLabel( ( b[1].y ).toFixed(d), a[0]*b[1].x+offset, a[1]*b[1].y, a[2]*b[0].z, als[1] );\n",
       "\n",
       "        var offset = offsetRatio * a[1]*( b[1].y - b[0].y );\n",
       "        var zm = zMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(zm) ) zm = zm.substr(1);\n",
       "        addLabel( al[2] + '=' + zm, a[0]*b[1].x, a[1]*b[0].y-offset, a[2]*zMid, als[2] );\n",
       "        addLabel( ( b[0].z ).toFixed(d), a[0]*b[1].x, a[1]*b[0].y-offset, a[2]*b[0].z, als[2] );\n",
       "        addLabel( ( b[1].z ).toFixed(d), a[0]*b[1].x, a[1]*b[0].y-offset, a[2]*b[1].z, als[2] );\n",
       "\n",
       "    }\n",
       "\n",
       "    function addLabel( text, x, y, z, style ) {\n",
       "\n",
       "        var color = style.color || 'black';\n",
       "        var fontSize = style.fontSize || 14;\n",
       "        var fontFamily = style.fontFamily || 'monospace';\n",
       "        var fontStyle = style.fontStyle || 'normal';\n",
       "        var fontWeight = style.fontWeight || 'normal';\n",
       "        var opacity = style.opacity || 1;\n",
       "\n",
       "        if ( options.theme === 'dark' )\n",
       "            if ( color === 'black' || color === '#000000' )\n",
       "                color = 'white';\n",
       "\n",
       "        if ( Array.isArray( fontStyle ) ) {\n",
       "            fontFamily = fontFamily.map( function( f ) {\n",
       "                // Need to put quotes around fonts that have whitespace in their names.\n",
       "                return /\\s/.test( f ) ? '&quot;' + f + '&quot;' : f;\n",
       "            }).join(', ');\n",
       "        }\n",
       "\n",
       "        var canvas = document.createElement( 'canvas' );\n",
       "        var context = canvas.getContext( '2d' );\n",
       "        var pixelRatio = Math.round( window.devicePixelRatio );\n",
       "\n",
       "        // For example: italic bold 20px &quot;Times New Roman&quot;, Georgia, serif\n",
       "        var font = [fontStyle, fontWeight, fontSize + 'px', fontFamily].join(' ');\n",
       "\n",
       "        context.font = font;\n",
       "        var width = context.measureText( text ).width;\n",
       "        var height = fontSize;\n",
       "\n",
       "        // The dimensions of the canvas's underlying image data need to be powers\n",
       "        // of two in order for the resulting texture to support mipmapping.\n",
       "        canvas.width = THREE.MathUtils.ceilPowerOfTwo( width * pixelRatio );\n",
       "        canvas.height = THREE.MathUtils.ceilPowerOfTwo( height * pixelRatio );\n",
       "\n",
       "        // Re-compute the unscaled dimensions after the power of two conversion.\n",
       "        width = canvas.width / pixelRatio;\n",
       "        height = canvas.height / pixelRatio;\n",
       "\n",
       "        canvas.style.width = width + 'px';\n",
       "        canvas.style.height = height + 'px';\n",
       "\n",
       "        context.scale( pixelRatio, pixelRatio );\n",
       "        context.fillStyle = color;\n",
       "        context.font = font; // Must be set again after measureText.\n",
       "        context.textAlign = 'center';\n",
       "        context.textBaseline = 'middle';\n",
       "        context.fillText( text, width/2, height/2 );\n",
       "\n",
       "        var texture = new THREE.Texture( canvas );\n",
       "        texture.needsUpdate = true;\n",
       "\n",
       "        var materialOptions = { map: texture, sizeAttenuation: false, depthWrite: false };\n",
       "        if ( opacity < 1 ) {\n",
       "            // Setting opacity=1 would cause the texture's alpha component to be\n",
       "            // discarded, giving the text a black background instead of the\n",
       "            // background being transparent.\n",
       "            materialOptions.opacity = opacity;\n",
       "        }\n",
       "        var sprite = new THREE.Sprite( new THREE.SpriteMaterial( materialOptions ) );\n",
       "        sprite.position.set( x, y, z );\n",
       "\n",
       "        // Scaling factor, chosen somewhat arbitrarily so that the size of the text\n",
       "        // is consistent with previously generated plots.\n",
       "        var scale = 1/625;\n",
       "        if ( options.projection === 'orthographic' ) {\n",
       "            scale = midToCorner/256; // Needs to scale along with the plot itself.\n",
       "        }\n",
       "        sprite.scale.set( scale * width, scale * height, 1 );\n",
       "\n",
       "        scene.add( sprite );\n",
       "\n",
       "        return sprite;\n",
       "\n",
       "    }\n",
       "\n",
       "    if ( options.axes ) scene.add( new THREE.AxesHelper( Math.min( a[0]*b[1].x, a[1]*b[1].y, a[2]*b[1].z ) ) );\n",
       "\n",
       "    var camera = createCamera();\n",
       "    camera.up.set( 0, 0, 1 );\n",
       "    camera.position.set( a[0]*xMid, a[1]*yMid, a[2]*zMid );\n",
       "\n",
       "    var offset = new THREE.Vector3( a[0]*xRange, a[1]*yRange, a[2]*zRange );\n",
       "\n",
       "    if ( options.viewpoint ) {\n",
       "\n",
       "        var aa = options.viewpoint;\n",
       "        var axis = new THREE.Vector3( aa[0][0], aa[0][1], aa[0][2] ).normalize();\n",
       "        var angle = aa[1] * Math.PI / 180;\n",
       "        var q = new THREE.Quaternion().setFromAxisAngle( axis, angle ).inverse();\n",
       "\n",
       "        offset.set( 0, 0, offset.length() );\n",
       "        offset.applyQuaternion( q );\n",
       "\n",
       "    }\n",
       "\n",
       "    camera.position.add( offset );\n",
       "\n",
       "    function createCamera() {\n",
       "\n",
       "        var aspect = window.innerWidth / window.innerHeight;\n",
       "\n",
       "        // Scale the near and far clipping planes along with the overall plot size.\n",
       "        var nearClip = 0.01 * midToCorner;\n",
       "        var farClip = 100 * midToCorner;\n",
       "\n",
       "        if ( options.projection === 'orthographic' ) {\n",
       "            var camera = new THREE.OrthographicCamera( -1, 1, 1, -1, -farClip, farClip );\n",
       "            updateCameraAspect( camera, aspect );\n",
       "            return camera;\n",
       "        }\n",
       "\n",
       "        return new THREE.PerspectiveCamera( 45, aspect, nearClip, farClip );\n",
       "\n",
       "    }\n",
       "\n",
       "    function updateCameraAspect( camera, aspect ) {\n",
       "\n",
       "        if ( camera.isPerspectiveCamera ) {\n",
       "            camera.aspect = aspect;\n",
       "        } else if ( camera.isOrthographicCamera ) {\n",
       "            // Fit the camera frustum to the bounding box's diagonal so that the entire plot fits\n",
       "            // within at the default zoom level and camera position.\n",
       "            if ( aspect > 1 ) { // Wide window\n",
       "                camera.top = midToCorner;\n",
       "                camera.right = midToCorner * aspect;\n",
       "            } else { // Tall or square window\n",
       "                camera.top = midToCorner / aspect;\n",
       "                camera.right = midToCorner;\n",
       "            }\n",
       "            camera.bottom = -camera.top;\n",
       "            camera.left = -camera.right;\n",
       "        }\n",
       "\n",
       "        camera.updateProjectionMatrix();\n",
       "\n",
       "    }\n",
       "\n",
       "    var lights = [{&quot;x&quot;:-5, &quot;y&quot;:3, &quot;z&quot;:0, &quot;color&quot;:&quot;#7f7f7f&quot;, &quot;parent&quot;:&quot;camera&quot;}];\n",
       "    for ( var i=0 ; i < lights.length ; i++ ) {\n",
       "        var light = new THREE.DirectionalLight( lights[i].color, 1 );\n",
       "        light.position.set( a[0]*lights[i].x, a[1]*lights[i].y, a[2]*lights[i].z );\n",
       "        if ( lights[i].parent === 'camera' ) {\n",
       "            light.target.position.set( a[0]*xMid, a[1]*yMid, a[2]*zMid );\n",
       "            scene.add( light.target );\n",
       "            camera.add( light );\n",
       "        } else scene.add( light );\n",
       "    }\n",
       "    scene.add( camera );\n",
       "\n",
       "    var ambient = {&quot;color&quot;:&quot;#7f7f7f&quot;};\n",
       "    scene.add( new THREE.AmbientLight( ambient.color, 1 ) );\n",
       "\n",
       "    var controls = new THREE.OrbitControls( camera, renderer.domElement );\n",
       "    controls.target.set( a[0]*xMid, a[1]*yMid, a[2]*zMid );\n",
       "    controls.addEventListener( 'change', function() { if ( !animate ) render(); } );\n",
       "\n",
       "    window.addEventListener( 'resize', function() {\n",
       "\n",
       "        renderer.setSize( window.innerWidth, window.innerHeight );\n",
       "        updateCameraAspect( camera, window.innerWidth / window.innerHeight );\n",
       "        if ( window.rescaleFatLines ) rescaleFatLines();\n",
       "        if ( !animate ) render();\n",
       "\n",
       "    } );\n",
       "\n",
       "    var texts = [];\n",
       "    for ( var i=0 ; i < texts.length ; i++ ) addText( texts[i] );\n",
       "\n",
       "    function addText( json ) {\n",
       "        var sprite = addLabel( json.text, a[0]*json.x, a[1]*json.y, a[2]*json.z, json );\n",
       "        sprite.userData = json;\n",
       "    }\n",
       "\n",
       "    var points = [];\n",
       "    for ( var i=0 ; i < points.length ; i++ ) addPoint( points[i] );\n",
       "\n",
       "    function addPoint( json ) {\n",
       "\n",
       "        var geometry = new THREE.Geometry();\n",
       "        var v = json.point;\n",
       "        geometry.vertices.push( new THREE.Vector3( a[0]*v[0], a[1]*v[1], a[2]*v[2] ) );\n",
       "\n",
       "        var canvas = document.createElement( 'canvas' );\n",
       "        canvas.width = 128;\n",
       "        canvas.height = 128;\n",
       "\n",
       "        var context = canvas.getContext( '2d' );\n",
       "        context.arc( 64, 64, 64, 0, 2 * Math.PI );\n",
       "        context.fillStyle = json.color;\n",
       "        context.fill();\n",
       "\n",
       "        var texture = new THREE.Texture( canvas );\n",
       "        texture.needsUpdate = true;\n",
       "\n",
       "        var transparent = json.opacity < 1 ? true : false;\n",
       "        var size = camera.isOrthographicCamera ? json.size : json.size/100;\n",
       "        var material = new THREE.PointsMaterial( { size: size, map: texture,\n",
       "                                                   transparent: transparent, opacity: json.opacity,\n",
       "                                                   alphaTest: .1 } );\n",
       "\n",
       "        var c = new THREE.Vector3();\n",
       "        geometry.computeBoundingBox();\n",
       "        geometry.boundingBox.getCenter( c );\n",
       "        geometry.translate( -c.x, -c.y, -c.z );\n",
       "\n",
       "        var mesh = new THREE.Points( geometry, material );\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        mesh.userData = json;\n",
       "        scene.add( mesh );\n",
       "\n",
       "    }\n",
       "\n",
       "    var lines = [{&quot;points&quot;: [[0.8091436649589344, 0.5693121021184392, -0.14550003380861354], [0.8349932931269609, 0.5463395188371165, -0.06556928083958899], [0.8411427327965988, 0.5406110463424716, 0.014785115351851574], [0.827808231263924, 0.5529015944883772, 0.09504398489070044], [0.7945409241324839, 0.5815398109612372, 0.17468877509979636], [0.7405273265103555, 0.6225002464795619, 0.253204900869454], [0.6652609800538656, 0.6696802784775959, 0.3300850693926338], [0.5694144707460289, 0.7154563303673062, 0.4048325577845336], [0.4556568021934493, 0.7515862015564374, 0.47696442240989645], [0.3291258315629206, 0.7704182128149182, 0.5460146191825668], [0.19731722138939195, 0.7662169365284682, 0.611537014677034], [0.06930473964593388, 0.7362894212256024, 0.6731082685971678], [-0.04557321560553228, 0.6815719639443061, 0.7303305689785093], [-0.1392230990891027, 0.606454895397346, 0.7828342024519752], [-0.20607352968203446, 0.5178502840383405, 0.8302799429624913], 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[0.24376570321371596, -0.4237500947021666, 0.8723612435090337], [0.20607352968201634, -0.5178502840383723, 0.8302799429624759], [0.13922309908907504, -0.6064548953973744, 0.7828342024519581], [0.04557321560549614, -0.6815719639443286, 0.7303305689784906], [-0.06930473964597601, -0.736289421225617, 0.6731082685971473], [-0.19731722138943703, -0.7662169365284739, 0.6115370146770123], [-0.329125831562965, -0.7704182128149155, 0.5460146191825438], [-0.45565680219349075, -0.7515862015564276, 0.47696442240987225], [-0.5694144707460649, -0.7154563303672916, 0.40483255778450844], [-0.665260980053895, -0.6696802784775796, 0.33008506939260784], [-0.7405273265103777, -0.6225002464795465, 0.2532049008694274], [-0.7945409241324988, -0.581539810961225, 0.17468877509976927], [-0.827808231263932, -0.5529015944883701, 0.09504398489067303], [-0.8411427327966, -0.5406110463424705, 0.014785115351824044], [-0.8349932931269557, -0.5463395188371216, -0.06556928083961647], [-0.8091436649589224, -0.5693121021184494, -0.14550003380863988]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 3.0}];\n",
       "    for ( var i=0 ; i < lines.length ; i++ ) addLine( lines[i] );\n",
       "\n",
       "    function addLine( json ) {\n",
       "\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i < json.points.length ; i++ ) {\n",
       "            var v = json.points[i];\n",
       "            geometry.vertices.push( new THREE.Vector3( a[0]*v[0], a[1]*v[1], a[2]*v[2] ) );\n",
       "        }\n",
       "\n",
       "        var c = new THREE.Vector3();\n",
       "        geometry.computeBoundingBox();\n",
       "        geometry.boundingBox.getCenter( c );\n",
       "        geometry.translate( -c.x, -c.y, -c.z );\n",
       "\n",
       "        var transparent = json.opacity < 1 ? true : false;\n",
       "        var materialOptions = { color: json.color, linewidth: json.linewidth,\n",
       "                                transparent: transparent, opacity: json.opacity };\n",
       "\n",
       "        var mesh;\n",
       "        if ( json.linewidth > 1 && window.createFatLineStrip ) {\n",
       "            mesh = createFatLineStrip( geometry, materialOptions );\n",
       "        } else {\n",
       "            var material = new THREE.LineBasicMaterial( materialOptions );\n",
       "            mesh = new THREE.Line( geometry, material );\n",
       "        }\n",
       "\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        mesh.userData = json;\n",
       "        scene.add( mesh );\n",
       "\n",
       "    }\n",
       "\n",
       "    var surfaces = [{&quot;vertices&quot;: [{&quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 1.0}, {&quot;x&quot;: 0.012907745354078983, &quot;y&quot;: -0.07942454778541594, &quot;z&quot;: 0.99675730813421}, {&quot;x&quot;: 4.927156290865444e-18, &quot;y&quot;: -0.08046656871672588, &quot;z&quot;: 0.99675730813421}, {&quot;x&quot;: 0.02548118688361391, &quot;y&quot;: -0.07632547278625865, &quot;z&quot;: 0.99675730813421}, {&quot;x&quot;: 0.03739467905751473, &quot;y&quot;: -0.07124960813386301, &quot;z&quot;: 0.99675730813421}, {&quot;x&quot;: 0.04833966868634682, &quot;y&quot;: -0.06432841605649706, &quot;z&quot;: 0.99675730813421}, {&quot;x&quot;: 0.05803268628786192, &quot;y&quot;: -0.055741151793429686, &quot;z&quot;: 0.99675730813421}, {&quot;x&quot;: 0.0662226877976886, &quot;y&quot;: -0.04571022097839203, &quot;z&quot;: 0.99675730813421}, {&quot;x&quot;: 0.07269755647873218, &quot;y&quot;: -0.0344954194504881, &quot;z&quot;: 0.99675730813421}, {&quot;x&quot;: 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1262, 1261], [1222, 1223, 1263, 1262], [1223, 1224, 1264, 1263], [1224, 1225, 1265, 1264], [1225, 1226, 1266, 1265], [1226, 1227, 1267, 1266], [1227, 1228, 1268, 1267], [1228, 1229, 1269, 1268], [1229, 1230, 1270, 1269], [1230, 1231, 1271, 1270], [1231, 1232, 1272, 1271], [1232, 1233, 1273, 1272], [1233, 1234, 1274, 1273], [1234, 1235, 1275, 1274], [1235, 1236, 1276, 1275], [1236, 1237, 1277, 1276], [1237, 1238, 1278, 1277], [1238, 1239, 1279, 1278], [1239, 1240, 1280, 1279], [1242, 1241, 1281, 1282], [1241, 1243, 1283, 1281], [1243, 1244, 1284, 1283], [1244, 1245, 1285, 1284], [1245, 1246, 1286, 1285], [1246, 1247, 1287, 1286], [1247, 1248, 1288, 1287], [1248, 1249, 1289, 1288], [1249, 1250, 1290, 1289], [1250, 1251, 1291, 1290], [1251, 1252, 1292, 1291], [1252, 1253, 1293, 1292], [1253, 1254, 1294, 1293], [1254, 1255, 1295, 1294], [1255, 1256, 1296, 1295], [1256, 1257, 1297, 1296], [1257, 1258, 1298, 1297], [1258, 1259, 1299, 1298], [1259, 1260, 1300, 1299], [1260, 1261, 1301, 1300], [1261, 1262, 1302, 1301], [1262, 1263, 1303, 1302], [1263, 1264, 1304, 1303], [1264, 1265, 1305, 1304], [1265, 1266, 1306, 1305], [1266, 1267, 1307, 1306], [1267, 1268, 1308, 1307], [1268, 1269, 1309, 1308], [1269, 1270, 1310, 1309], [1270, 1271, 1311, 1310], [1271, 1272, 1312, 1311], [1272, 1273, 1313, 1312], [1273, 1274, 1314, 1313], [1274, 1275, 1315, 1314], [1275, 1276, 1316, 1315], [1276, 1277, 1317, 1316], [1277, 1278, 1318, 1317], [1278, 1279, 1319, 1318], [1279, 1280, 1320, 1319], [1282, 1281, 1321, 1322], [1281, 1283, 1323, 1321], [1283, 1284, 1324, 1323], [1284, 1285, 1325, 1324], [1285, 1286, 1326, 1325], [1286, 1287, 1327, 1326], [1287, 1288, 1328, 1327], [1288, 1289, 1329, 1328], [1289, 1290, 1330, 1329], [1290, 1291, 1331, 1330], [1291, 1292, 1332, 1331], [1292, 1293, 1333, 1332], [1293, 1294, 1334, 1333], [1294, 1295, 1335, 1334], [1295, 1296, 1336, 1335], [1296, 1297, 1337, 1336], [1297, 1298, 1338, 1337], [1298, 1299, 1339, 1338], [1299, 1300, 1340, 1339], [1300, 1301, 1341, 1340], [1301, 1302, 1342, 1341], [1302, 1303, 1343, 1342], [1303, 1304, 1344, 1343], [1304, 1305, 1345, 1344], [1305, 1306, 1346, 1345], [1306, 1307, 1347, 1346], [1307, 1308, 1348, 1347], [1308, 1309, 1349, 1348], [1309, 1310, 1350, 1349], [1310, 1311, 1351, 1350], [1311, 1312, 1352, 1351], [1312, 1313, 1353, 1352], [1313, 1314, 1354, 1353], [1314, 1315, 1355, 1354], [1315, 1316, 1356, 1355], [1316, 1317, 1357, 1356], [1317, 1318, 1358, 1357], [1318, 1319, 1359, 1358], [1319, 1320, 1360, 1359], [1322, 1321, 1361, 1362], [1321, 1323, 1363, 1361], [1323, 1324, 1364, 1363], [1324, 1325, 1365, 1364], [1325, 1326, 1366, 1365], [1326, 1327, 1367, 1366], [1327, 1328, 1368, 1367], [1328, 1329, 1369, 1368], [1329, 1330, 1370, 1369], [1330, 1331, 1371, 1370], [1331, 1332, 1372, 1371], [1332, 1333, 1373, 1372], [1333, 1334, 1374, 1373], [1334, 1335, 1375, 1374], [1335, 1336, 1376, 1375], [1336, 1337, 1377, 1376], [1337, 1338, 1378, 1377], [1338, 1339, 1379, 1378], [1339, 1340, 1380, 1379], [1340, 1341, 1381, 1380], [1341, 1342, 1382, 1381], [1342, 1343, 1383, 1382], [1343, 1344, 1384, 1383], [1344, 1345, 1385, 1384], [1345, 1346, 1386, 1385], [1346, 1347, 1387, 1386], [1347, 1348, 1388, 1387], [1348, 1349, 1389, 1388], [1349, 1350, 1390, 1389], [1350, 1351, 1391, 1390], [1351, 1352, 1392, 1391], [1352, 1353, 1393, 1392], [1353, 1354, 1394, 1393], [1354, 1355, 1395, 1394], [1355, 1356, 1396, 1395], [1356, 1357, 1397, 1396], [1357, 1358, 1398, 1397], [1358, 1359, 1399, 1398], [1359, 1360, 1400, 1399], [1362, 1361, 1401, 1402], [1361, 1363, 1403, 1401], [1363, 1364, 1404, 1403], [1364, 1365, 1405, 1404], [1365, 1366, 1406, 1405], [1366, 1367, 1407, 1406], [1367, 1368, 1408, 1407], [1368, 1369, 1409, 1408], [1369, 1370, 1410, 1409], [1370, 1371, 1411, 1410], [1371, 1372, 1412, 1411], [1372, 1373, 1413, 1412], [1373, 1374, 1414, 1413], [1374, 1375, 1415, 1414], [1375, 1376, 1416, 1415], [1376, 1377, 1417, 1416], [1377, 1378, 1418, 1417], [1378, 1379, 1419, 1418], [1379, 1380, 1420, 1419], [1380, 1381, 1421, 1420], [1381, 1382, 1422, 1421], [1382, 1383, 1423, 1422], [1383, 1384, 1424, 1423], [1384, 1385, 1425, 1424], [1385, 1386, 1426, 1425], [1386, 1387, 1427, 1426], [1387, 1388, 1428, 1427], [1388, 1389, 1429, 1428], [1389, 1390, 1430, 1429], [1390, 1391, 1431, 1430], [1391, 1392, 1432, 1431], [1392, 1393, 1433, 1432], [1393, 1394, 1434, 1433], [1394, 1395, 1435, 1434], [1395, 1396, 1436, 1435], [1396, 1397, 1437, 1436], [1397, 1398, 1438, 1437], [1398, 1399, 1439, 1438], [1399, 1400, 1440, 1439], [1402, 1401, 1441, 1442], [1401, 1403, 1443, 1441], [1403, 1404, 1444, 1443], [1404, 1405, 1445, 1444], [1405, 1406, 1446, 1445], [1406, 1407, 1447, 1446], [1407, 1408, 1448, 1447], [1408, 1409, 1449, 1448], [1409, 1410, 1450, 1449], [1410, 1411, 1451, 1450], [1411, 1412, 1452, 1451], [1412, 1413, 1453, 1452], [1413, 1414, 1454, 1453], [1414, 1415, 1455, 1454], [1415, 1416, 1456, 1455], [1416, 1417, 1457, 1456], [1417, 1418, 1458, 1457], [1418, 1419, 1459, 1458], [1419, 1420, 1460, 1459], [1420, 1421, 1461, 1460], [1421, 1422, 1462, 1461], [1422, 1423, 1463, 1462], [1423, 1424, 1464, 1463], [1424, 1425, 1465, 1464], [1425, 1426, 1466, 1465], [1426, 1427, 1467, 1466], [1427, 1428, 1468, 1467], [1428, 1429, 1469, 1468], [1429, 1430, 1470, 1469], [1430, 1431, 1471, 1470], [1431, 1432, 1472, 1471], [1432, 1433, 1473, 1472], [1433, 1434, 1474, 1473], [1434, 1435, 1475, 1474], [1435, 1436, 1476, 1475], [1436, 1437, 1477, 1476], [1437, 1438, 1478, 1477], [1438, 1439, 1479, 1478], [1439, 1440, 1480, 1479], [1442, 1441, 1481, 1482], [1441, 1443, 1483, 1481], [1443, 1444, 1484, 1483], [1444, 1445, 1485, 1484], [1445, 1446, 1486, 1485], [1446, 1447, 1487, 1486], [1447, 1448, 1488, 1487], [1448, 1449, 1489, 1488], [1449, 1450, 1490, 1489], [1450, 1451, 1491, 1490], [1451, 1452, 1492, 1491], [1452, 1453, 1493, 1492], [1453, 1454, 1494, 1493], [1454, 1455, 1495, 1494], [1455, 1456, 1496, 1495], [1456, 1457, 1497, 1496], [1457, 1458, 1498, 1497], [1458, 1459, 1499, 1498], [1459, 1460, 1500, 1499], [1460, 1461, 1501, 1500], [1461, 1462, 1502, 1501], [1462, 1463, 1503, 1502], [1463, 1464, 1504, 1503], [1464, 1465, 1505, 1504], [1465, 1466, 1506, 1505], [1466, 1467, 1507, 1506], [1467, 1468, 1508, 1507], [1468, 1469, 1509, 1508], [1469, 1470, 1510, 1509], [1470, 1471, 1511, 1510], [1471, 1472, 1512, 1511], [1472, 1473, 1513, 1512], [1473, 1474, 1514, 1513], [1474, 1475, 1515, 1514], [1475, 1476, 1516, 1515], [1476, 1477, 1517, 1516], [1477, 1478, 1518, 1517], [1478, 1479, 1519, 1518], [1479, 1480, 1520, 1519], [1482, 1481, 1521, 1522], [1481, 1483, 1523, 1521], [1483, 1484, 1524, 1523], [1484, 1485, 1525, 1524], [1485, 1486, 1526, 1525], [1486, 1487, 1527, 1526], [1487, 1488, 1528, 1527], [1488, 1489, 1529, 1528], [1489, 1490, 1530, 1529], [1490, 1491, 1531, 1530], [1491, 1492, 1532, 1531], [1492, 1493, 1533, 1532], [1493, 1494, 1534, 1533], [1494, 1495, 1535, 1534], [1495, 1496, 1536, 1535], [1496, 1497, 1537, 1536], [1497, 1498, 1538, 1537], [1498, 1499, 1539, 1538], [1499, 1500, 1540, 1539], [1500, 1501, 1541, 1540], [1501, 1502, 1542, 1541], [1502, 1503, 1543, 1542], [1503, 1504, 1544, 1543], [1504, 1505, 1545, 1544], [1505, 1506, 1546, 1545], [1506, 1507, 1547, 1546], [1507, 1508, 1548, 1547], [1508, 1509, 1549, 1548], [1509, 1510, 1550, 1549], [1510, 1511, 1551, 1550], [1511, 1512, 1552, 1551], [1512, 1513, 1553, 1552], [1513, 1514, 1554, 1553], [1514, 1515, 1555, 1554], [1515, 1516, 1556, 1555], [1516, 1517, 1557, 1556], [1517, 1518, 1558, 1557], [1518, 1519, 1559, 1558], [1519, 1520, 1560, 1559]], &quot;color&quot;: &quot;#d3d3d3&quot;, &quot;opacity&quot;: 0.7}];\n",
       "    for ( var i=0 ; i < surfaces.length ; i++ ) addSurface( surfaces[i] );\n",
       "\n",
       "    function addSurface( json ) {\n",
       "\n",
       "        var useFaceColors = 'faceColors' in json ? true : false;\n",
       "\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i < json.vertices.length ; i++ ) {\n",
       "            var v = json.vertices[i];\n",
       "            geometry.vertices.push( new THREE.Vector3( a[0]*v.x, a[1]*v.y, a[2]*v.z ) );\n",
       "        }\n",
       "        for ( var i=0 ; i < json.faces.length ; i++ ) {\n",
       "            var f = json.faces[i];\n",
       "            for ( var j=0 ; j < f.length - 2 ; j++ ) {\n",
       "                var face = new THREE.Face3( f[0], f[j+1], f[j+2] );\n",
       "                if ( useFaceColors ) face.color.set( json.faceColors[i] );\n",
       "                geometry.faces.push( face );\n",
       "            }\n",
       "        }\n",
       "        geometry.computeVertexNormals();\n",
       "\n",
       "        var side = json.singleSide ? THREE.FrontSide : THREE.DoubleSide;\n",
       "        var transparent = json.opacity < 1 ? true : false;\n",
       "        var depthWrite = 'depthWrite' in json ? json.depthWrite : !transparent;\n",
       "        var flatShading = json.useFlatShading ? json.useFlatShading : false;\n",
       "\n",
       "        var material = new THREE.MeshPhongMaterial( { side: side,\n",
       "                                     color: useFaceColors ? 'white' : json.color,\n",
       "                                     vertexColors: useFaceColors ? THREE.FaceColors : THREE.NoColors,\n",
       "                                     transparent: transparent, opacity: json.opacity,\n",
       "                                     shininess: 20, flatShading: flatShading,\n",
       "                                     depthWrite: depthWrite } );\n",
       "\n",
       "        var c = new THREE.Vector3();\n",
       "        geometry.computeBoundingBox();\n",
       "        geometry.boundingBox.getCenter( c );\n",
       "        geometry.translate( -c.x, -c.y, -c.z );\n",
       "\n",
       "        var mesh = new THREE.Mesh( geometry, material );\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        if ( transparent && json.renderOrder ) mesh.renderOrder = json.renderOrder;\n",
       "        mesh.userData = json;\n",
       "        scene.add( mesh );\n",
       "\n",
       "        if ( json.showMeshGrid ) addSurfaceMeshGrid( json );\n",
       "\n",
       "    }\n",
       "\n",
       "    function addSurfaceMeshGrid( json ) {\n",
       "\n",
       "        var geometry = new THREE.Geometry();\n",
       "\n",
       "        for ( var i=0 ; i < json.faces.length ; i++ ) {\n",
       "            var f = json.faces[i];\n",
       "            for ( var j=0 ; j < f.length ; j++ ) {\n",
       "                var k = j === f.length-1 ? 0 : j+1;\n",
       "                var v1 = json.vertices[f[j]];\n",
       "                var v2 = json.vertices[f[k]];\n",
       "                // vertices in opposite directions on neighboring faces\n",
       "                var nudge = f[j] < f[k] ? .0005*zRange : -.0005*zRange;\n",
       "                geometry.vertices.push( new THREE.Vector3( a[0]*v1.x, a[1]*v1.y, a[2]*(v1.z+nudge) ) );\n",
       "                geometry.vertices.push( new THREE.Vector3( a[0]*v2.x, a[1]*v2.y, a[2]*(v2.z+nudge) ) );\n",
       "            }\n",
       "        }\n",
       "\n",
       "        var c = new THREE.Vector3();\n",
       "        geometry.computeBoundingBox();\n",
       "        geometry.boundingBox.getCenter( c );\n",
       "        geometry.translate( -c.x, -c.y, -c.z );\n",
       "\n",
       "        var gridColor = options.theme === 'dark' ? 'white' : 'black';\n",
       "        var linewidth = json.linewidth || 1;\n",
       "        var materialOptions = { color: gridColor, linewidth: linewidth };\n",
       "\n",
       "        var mesh;\n",
       "        if ( linewidth > 1 && window.createFatLineSegments ) {\n",
       "            mesh = createFatLineSegments( geometry, materialOptions );\n",
       "        } else {\n",
       "            var material = new THREE.LineBasicMaterial( materialOptions );\n",
       "            mesh = new THREE.LineSegments( geometry, material );\n",
       "        }\n",
       "\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        mesh.userData = json;\n",
       "        scene.add( mesh );\n",
       "\n",
       "    }\n",
       "\n",
       "    function render() {\n",
       "\n",
       "        if ( window.updateAnimation ) animate = updateAnimation();\n",
       "        if ( animate ) requestAnimationFrame( render );\n",
       "\n",
       "        renderer.render( scene, camera );\n",
       "\n",
       "    }\n",
       "\n",
       "    render();\n",
       "    controls.update();\n",
       "    if ( !animate ) render();\n",
       "\n",
       "\n",
       "    // menu functions\n",
       "\n",
       "    function toggleMenu() {\n",
       "\n",
       "        var m = document.getElementById( 'menu-content' );\n",
       "        if ( m.style.display === 'block' ) m.style.display = 'none'\n",
       "        else m.style.display = 'block';\n",
       "\n",
       "    }\n",
       "\n",
       "\n",
       "    function saveAsPNG() {\n",
       "\n",
       "        var a = document.body.appendChild( document.createElement( 'a' ) );\n",
       "        a.href = renderer.domElement.toDataURL( 'image/png' );\n",
       "        a.download = 'screenshot';\n",
       "        a.click();\n",
       "\n",
       "    }\n",
       "\n",
       "    function saveAsHTML() {\n",
       "\n",
       "        toggleMenu(); // otherwise visible in output\n",
       "        event.stopPropagation();\n",
       "\n",
       "        var blob = new Blob( [ '<!DOCTYPE html>\\n' + document.documentElement.outerHTML ] );\n",
       "        var a = document.body.appendChild( document.createElement( 'a' ) );\n",
       "        a.href = window.URL.createObjectURL( blob );\n",
       "        a.download = suggestFilename();\n",
       "        a.click();\n",
       "\n",
       "        function suggestFilename() {\n",
       "            if ( !document.title ) {\n",
       "                return 'graphic.html';\n",
       "            } else if ( /\\.html?$/i.test( document.title ) ) {\n",
       "                return document.title; // already ends in .htm or .html\n",
       "            } else {\n",
       "                return document.title + '.html';\n",
       "            }\n",
       "        }\n",
       "\n",
       "    }\n",
       "\n",
       "    function getViewpoint() {\n",
       "\n",
       "        function roundTo( x, n ) { return +x.toFixed(n); }\n",
       "\n",
       "        var v = camera.quaternion.inverse();\n",
       "        var r = Math.sqrt( v.x*v.x + v.y*v.y + v.z*v.z );\n",
       "        var axis = [ roundTo( v.x / r, 4 ), roundTo( v.y / r, 4 ), roundTo( v.z / r, 4 ) ];\n",
       "        var angle = roundTo( 2 * Math.atan2( r, v.w ) * 180 / Math.PI, 2 );\n",
       "\n",
       "        var textArea = document.createElement( 'textarea' );\n",
       "        textArea.textContent = JSON.stringify( axis ) + ',' + angle;\n",
       "        textArea.style.csstext = 'position: absolute; top: -100%';\n",
       "        document.body.append( textArea );\n",
       "        textArea.select();\n",
       "        document.execCommand( 'copy' );\n",
       "\n",
       "        var m = document.getElementById( 'menu-message' );\n",
       "        m.innerHTML = 'Viewpoint copied to clipboard';\n",
       "        m.style.display = 'block';\n",
       "        setTimeout( function() { m.style.display = 'none'; }, 2000 );\n",
       "\n",
       "    }\n",
       "\n",
       "    function getCamera() {\n",
       "\n",
       "        function roundTo( x, n ) { return +x.toFixed(n); }\n",
       "\n",
       "        var pos = camera.position;\n",
       "        var pos_r = [ roundTo( pos.x, 4 ), roundTo( pos.y, 4 ), roundTo( pos.z, 4 ) ];\n",
       "   //     var up = camera.up; // up is always (0,0,1)\n",
       "        var textArea = document.createElement('textarea');\n",
       "        var cam_position = JSON.stringify(pos_r);\n",
       "        textArea.textContent = ',camera_position=' + cam_position;\n",
       "        textArea.style.csstext = 'position: absolute; top: -100%';\n",
       "        document.body.append( textArea );\n",
       "        textArea.select();\n",
       "        document.execCommand( 'copy' );\n",
       "\n",
       "        var m = document.getElementById( 'menu-message' );\n",
       "        m.innerHTML = 'Camera position '+ cam_position+' copied to clipboard';\n",
       "        m.style.display = 'block';\n",
       "        setTimeout( function() { m.style.display = 'none'; }, 2000 );\n",
       "\n",
       "    }\n",
       "                                       \n",
       "</script>\n",
       "\n",
       "<div id=&quot;menu-container&quot; onclick=&quot;toggleMenu()&quot;>&#x24d8;\n",
       "<div id=&quot;menu-message&quot;></div>\n",
       "<div id=&quot;menu-content&quot;>\n",
       "<div onclick=&quot;saveAsPNG()&quot;>Save as PNG</div>\n",
       "<div onclick=&quot;saveAsHTML()&quot;>Save as HTML</div>\n",
       "<div onclick=&quot;getCamera()&quot;>Get camera</div>\n",
       "<div onclick=&quot;getViewpoint()&quot;>Get viewpoint</div>\n",
       "<div>Close Menu</div>\n",
       "</div></div>\n",
       "\n",
       "\n",
       "</body>\n",
       "</html>\n",
       "\"\n",
       "        width=\"100%\"\n",
       "        height=\"400\"\n",
       "        style=\"border: 0;\">\n",
       "</iframe>\n"
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       "Graphics3d Object"
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   "source": [
    "%display latex\n",
    "S2=manifolds.Sphere(2)                 # sphere S^2\n",
    "Phi=S2.embedding()                     # embedding S^2-> E^3\n",
    "sph.<th,ph>=S2.spherical_coordinates() # spherical coord. on S^2\n",
    "E=S2.ambient()                         # Euclidean space E^3\n",
    "cart.<x,y,z> = E.cartesian_coordinates() # Cartesian coord. on E^3\n",
    "fun=Phi.expr()                         # functions defining Phi\n",
    "R.<t> = manifolds.RealLine()           # Real line as manifold\n",
    "c = S2.curve({sph: [t, cos(2*t)]}, (t, -oo, +oo),\n",
    "             name='c')                 # curve c\n",
    "                                       # plot c:\n",
    "p1=c.plot(chart=cart,mapping=Phi,thickness=3,color='black',\n",
    "          plot_points=200, aspect_ratio=1,label_axes=False)\n",
    "p2=parametric_plot3d(fun,(th,0,pi),(ph,0,2*pi),\n",
    "        color='lightgrey',opacity=0.7) # plot sphere\n",
    "(p1+p2).rotateZ(pi/2).show(frame=False,label_axes=False) # show plot"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "### Tangent vector to a smooth curve\n",
    "\n",
    "<br>\n",
    "\n",
    "If $\\gamma$ is a smooth curve in $M$ and $f ∈ C^∞ (M),$ then $\\gamma^∗ f = f ◦ \\gamma$ is a smooth function from an open interval $I ⊂ R$ into $R$. If $t_0 ∈ I$ , the\n",
    "**tangent vector to  $\\gamma$ at the point $\\gamma(t_0 )$**, denoted by $\\gamma'_{t_0}$ , is the map $C^\\infty(M)\\to R$ defined by\n",
    "\\begin{equation}\n",
    "\\gamma'_{t_0}(f) =\\frac{d}{dt}(\\gamma^*f)\\Big|_{t_0}=\n",
    "\\frac{d}{dt}(f\\circ\\gamma)\\Big|_{t_0},\\quad f\\in C^\\infty(M).\n",
    "\\label{}\\tag{7.6}\n",
    "\\end{equation}\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using (7.3), (7.4) we can check that $\\gamma'_{t_0}:C^∞ (M)\\to R$ has the properties\n",
    "$$\\gamma'_{t_0}(af+bg)=\\frac{d}{dt}(\\gamma^*(af+bg))\\Big|_{t_0}=\\frac{d}{dt}(a\\gamma^*(f)+b\\gamma^*(g))\\Big|_{t_0}\\\\\n",
    "=a\\frac{d}{dt}(\\gamma^*f)\\Big|_{t_0}+\n",
    "b\\frac{d}{dt}(\\gamma^*g)\\Big|_{t_0}=\n",
    "a\\gamma'_{t_0}(f)+b\\gamma'_{t_0}(f),\n",
    "$$\n",
    "and\n",
    "$$\\gamma'_{t_0}(fg)=\\frac{d}{dt}(\\gamma^*(fg))\\Big|_{t_0}=\\frac{d}{dt}(\\gamma^*(f)\\gamma^*(g))\\Big|_{t_0}\\\\\n",
    "=f (\\gamma(t_0 ))\\frac{d}{dt}(\\gamma^*(g))\\Big|_{t_0}\n",
    "+g (\\gamma(t_0 ))\\frac{d}{dt}(\\gamma^*(f))\\Big|_{t_0}\\\\\n",
    "=f (\\gamma(t_0 ))\\gamma'_{t_0}(g)+\n",
    " g (\\gamma(t_0 ))\\gamma'_{t_0}(f),\n",
    "$$<br>\n",
    "for $f, g ∈ C^∞ (M)$ and $a,b\\in R$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## What's next?\n",
    "\n",
    "Take a look at the notebook [Tangent spaces](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/08Manifold_Tangent_Vect.ipynb)."
   ]
  }
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