Sage_v_Numpy_calculation_tests.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      ],
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       "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[0.540302305868140, 0.5403023058681398\\right]$$"
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      "text/plain": [
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "x1 = cos(1).n() # Sage/Python standard Cosine function at 1 radian\n",
    "x2 = np.cos(1) # Numpy Cosine function at 1 radian\n",
    "\n",
    "y1 = sin(1).n() # Sage/Python standard Sine function at 1 radian\n",
    "y2 = np.sin(1) # Numpy Sine function at 1 radian\n",
    "\n",
    "L = [x1,x2]\n",
    "\n",
    "show(L)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[0.291926581726429, 0.2919265817264289\\right]</script></html>"
      ],
      "text/latex": [
       "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[0.291926581726429, 0.2919265817264289\\right]$$"
      ],
      "text/plain": [
       "[0.291926581726429, 0.2919265817264289]"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "M = [x1**2, np.square(x2)] # Sage/Python square function vs. Numpy square function\n",
    "\n",
    "show(M)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[0.814477216699512, 0.8144772166995121\\right]</script></html>"
      ],
      "text/latex": [
       "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[0.814477216699512, 0.8144772166995121\\right]$$"
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      "text/plain": [
       "[0.814477216699512, 0.8144772166995121]"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "N = [x1**(1/3), np.cbrt(x2)] # Sage/Python cube root function vs. Numpy cube root function.\n",
    "# Observe this is cubic roots of positive numbers.  See below for cubic roots of negative numbers.\n",
    "\n",
    "show(N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[0.358422035314132, 0.3584220353141324\\right]</script></html>"
      ],
      "text/latex": [
       "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[0.358422035314132, 0.3584220353141324\\right]$$"
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       "[0.358422035314132, 0.3584220353141324]"
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     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "P = [x1**(5/3), np.power(np.cbrt(x2),5)] # Sage/Python rational power (5/3) vs. Numpy rational power (5/3)\n",
    "\n",
    "show(P)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[0.750006871533950, 0.7500068715339497\\right]</script></html>"
      ],
      "text/latex": [
       "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[0.750006871533950, 0.7500068715339497\\right]$$"
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      "text/plain": [
       "[0.750006871533950, 0.7500068715339497]"
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     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Q = [y1**(5/3), np.power(np.cbrt(y2),5)]\n",
    "\n",
    "show(Q)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[1.05281950345160, 1.0528195034516041\\right]</script></html>"
      ],
      "text/latex": [
       "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[1.05281950345160, 1.0528195034516041\\right]$$"
      ],
      "text/plain": [
       "[1.05281950345160, 1.0528195034516041]"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "r1 = sqrt(P[0]+Q[0]) # Radius of curve <x1,y1> calculated with Sage/Python decimals\n",
    "r2 = np.sqrt(P[1]+Q[1]) # Radius of curve <x2,y2> calculated with Numpy decimals\n",
    "\n",
    "R = [r1, r2]\n",
    "\n",
    "show(R)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[-0.814477216699512, -0.8144772166995121\\right]</script></html>"
      ],
      "text/latex": [
       "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[-0.814477216699512, -0.8144772166995121\\right]$$"
      ],
      "text/plain": [
       "[-0.814477216699512, -0.8144772166995121]"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "negative_cube_root = real_nth_root(-1*x1,3) # Calculates the real cubic (1/3) root of a negative number\n",
    "# Sage requires this method for odd real roots of negative numbers\n",
    "\n",
    "np_negative_cube_root = np.cbrt(-1*x2.item(0)) # Numpy cubic root of a negative number\n",
    "# x2.item(0) calls the \"first\" item stored in the singleton array x2\n",
    "\n",
    "C = [negative_cube_root, np_negative_cube_root]\n",
    "\n",
    "show(C)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[1.41421356237310, 1.4142135623730951\\right]</script></html>"
      ],
      "text/latex": [
       "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[1.41421356237310, 1.4142135623730951\\right]$$"
      ],
      "text/plain": [
       "[1.41421356237310, 1.4142135623730951]"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sqrt2 = sqrt(2).n() # Sage/Python standard square root of 2\n",
    "np_sqrt2 = np.sqrt(2) # Numpy square root of 2\n",
    "\n",
    "S = [sqrt2, np_sqrt2]\n",
    "\n",
    "show(S)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-193.640098389547</script></html>"
      ],
      "text/latex": [
       "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-193.640098389547$$"
      ],
      "text/plain": [
       "-193.640098389547"
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     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
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\n",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "show((real_nth_root(28*sin(-1).n(),3))**5)\n",
    "\n",
    "parametric_plot([16*real_nth_root(cos(u),6)**5,-28*real_nth_root(sin(-u),6)**5],(u,-pi/2,0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "Graphics object consisting of 2 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#u = -1\n",
    "#np.power(np.cbrt(np.cos(u)),5)\n",
    "\n",
    "u = var ('u')\n",
    "parametric_plot([u,-28*(np.power(2*(np.sin(u)),5))], (u,0,1000*pi))\n",
    "\n",
    "parametric_plot([u,np.sin(u)], (u,0,pi))\n",
    "\n",
    "# parametric_plot([u,np.cbrt(u)], (u,0,pi))\n",
    "\n",
    "sage_real_nth_root = parametric_plot([u,real_nth_root(np.sin(u),3)], (u,0,pi/2))\n",
    "\n",
    "# x2.item(0)\n",
    "\n",
    "numpy_cbrt = parametric_plot([lambda u: u, lambda u: np.cbrt(np.sin(u))], (0, pi/2), color=\"green\")\n",
    "\n",
    "show(sage_real_nth_root+numpy_cbrt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "np.float_power(np.sin(3),0.5)\n",
    "\n",
    "numpy_float_power = parametric_plot([lambda u: u, lambda u: np.float_power(np.sin(u),1/6)], (0, pi/2), color=\"green\")\n",
    "\n",
    "show(numpy_float_power)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "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/build/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",
       "        \n",
       "<script>\n",
       "\n",
       "    var options = {&quot;animate&quot;: false, &quot;animationControls&quot;: true, &quot;aspectRatio&quot;: [1.0, 1.0, 1.0], &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;:0.0, &quot;y&quot;:0.0, &quot;z&quot;:0.0}, {&quot;x&quot;:1.5707963267948966, &quot;y&quot;:1.0, &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 rRange = 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",
       "    var autoAspect = 2.5;\n",
       "    if ( zRange > autoAspect * rRange && a[2] === 1 ) a[2] = autoAspect * rRange / 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 = [];\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;: 0.0}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.02564102564102564, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.02564102564102564, &quot;z&quot;: 0.0017635330198443977}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.0, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.05128205128205128, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.05128205128205128, &quot;z&quot;: 0.0035270660396887954}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.07692307692307693, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.07692307692307693, &quot;z&quot;: 0.0052905990595331935}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.10256410256410256, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.10256410256410256, &quot;z&quot;: 0.007054132079377591}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.1282051282051282, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.1282051282051282, &quot;z&quot;: 0.008817665099221987}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.15384615384615383, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.15384615384615383, &quot;z&quot;: 0.010581198119066385}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.17948717948717946, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.17948717948717946, &quot;z&quot;: 0.012344731138910782}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.2051282051282051, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.2051282051282051, &quot;z&quot;: 0.01410826415875518}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.23076923076923073, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.23076923076923073, &quot;z&quot;: 0.015871797178599576}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.2564102564102564, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.2564102564102564, &quot;z&quot;: 0.017635330198443974}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.28205128205128205, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.28205128205128205, &quot;z&quot;: 0.019398863218288376}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.3076923076923077, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.3076923076923077, &quot;z&quot;: 0.021162396238132774}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.33333333333333337, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.33333333333333337, &quot;z&quot;: 0.022925929257977172}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.35897435897435903, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.35897435897435903, &quot;z&quot;: 0.02468946227782157}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.3846153846153847, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.3846153846153847, &quot;z&quot;: 0.02645299529766597}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.41025641025641035, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.41025641025641035, &quot;z&quot;: 0.02821652831751037}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.435897435897436, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.435897435897436, &quot;z&quot;: 0.029980061337354768}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.4615384615384617, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.4615384615384617, &quot;z&quot;: 0.031743594357199166}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.48717948717948734, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.48717948717948734, &quot;z&quot;: 0.03350712737704357}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.512820512820513, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.512820512820513, &quot;z&quot;: 0.03527066039688796}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.5384615384615387, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.5384615384615387, &quot;z&quot;: 0.037034193416732364}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.5641025641025643, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.5641025641025643, &quot;z&quot;: 0.038797726436576765}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.58974358974359, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.58974358974359, &quot;z&quot;: 0.04056125945642116}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.6153846153846156, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.6153846153846156, &quot;z&quot;: 0.04232479247626556}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.6410256410256413, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.6410256410256413, &quot;z&quot;: 0.04408832549610996}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.666666666666667, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.666666666666667, &quot;z&quot;: 0.04585185851595436}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.6923076923076926, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.6923076923076926, &quot;z&quot;: 0.04761539153579876}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.7179487179487183, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.7179487179487183, &quot;z&quot;: 0.04937892455564316}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.743589743589744, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.743589743589744, &quot;z&quot;: 0.051142457575487556}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.7692307692307696, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.7692307692307696, &quot;z&quot;: 0.05290599059533196}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.7948717948717953, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.04027682889217683, &quot;y&quot;: 0.7948717948717953, &quot;z&quot;: 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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], [1361, 1360, 1400, 1401], [1360, 1362, 1402, 1400], [1362, 1363, 1403, 1402], [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], [1401, 1400, 1440, 1441], [1400, 1402, 1442, 1440], [1402, 1403, 1443, 1442], [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], [1441, 1440, 1480, 1481], [1440, 1442, 1482, 1480], [1442, 1443, 1483, 1482], [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], [1481, 1480, 1520, 1521], [1480, 1482, 1522, 1520], [1482, 1483, 1523, 1522], [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], [1521, 1520, 1560, 1561], [1520, 1522, 1562, 1560], [1522, 1523, 1563, 1562], [1523, 1524, 1564, 1563], [1524, 1525, 1565, 1564], [1525, 1526, 1566, 1565], [1526, 1527, 1567, 1566], [1527, 1528, 1568, 1567], [1528, 1529, 1569, 1568], [1529, 1530, 1570, 1569], [1530, 1531, 1571, 1570], [1531, 1532, 1572, 1571], [1532, 1533, 1573, 1572], [1533, 1534, 1574, 1573], [1534, 1535, 1575, 1574], [1535, 1536, 1576, 1575], [1536, 1537, 1577, 1576], [1537, 1538, 1578, 1577], [1538, 1539, 1579, 1578], [1539, 1540, 1580, 1579], [1540, 1541, 1581, 1580], [1541, 1542, 1582, 1581], [1542, 1543, 1583, 1582], [1543, 1544, 1584, 1583], [1544, 1545, 1585, 1584], [1545, 1546, 1586, 1585], [1546, 1547, 1587, 1586], [1547, 1548, 1588, 1587], [1548, 1549, 1589, 1588], [1549, 1550, 1590, 1589], [1550, 1551, 1591, 1590], [1551, 1552, 1592, 1591], [1552, 1553, 1593, 1592], [1553, 1554, 1594, 1593], [1554, 1555, 1595, 1594], [1555, 1556, 1596, 1595], [1556, 1557, 1597, 1596], [1557, 1558, 1598, 1597], [1558, 1559, 1599, 1598]], &quot;color&quot;: &quot;#008000&quot;, &quot;opacity&quot;: 1.0}];\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",
       "</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;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": [
    "(np.float_power(np.cos(1),5/6)) \n",
    "\n",
    "\n",
    "numpy_float_power = parametric_plot3d([lambda u, v: u, lambda u, v: v, lambda u, v: v*np.float_power(np.sin(u),5/6)], (0, pi/2), (0,1), color=\"green\")\n",
    "\n",
    "show(numpy_float_power)\n",
    "# u = -3\n",
    "# (-28*(np.float_power(np.sin(-u),5/6)))"
   ]
  }
 ],
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  "kernelspec": {
   "display_name": "SageMath 9.3",
   "language": "sage",
   "name": "sagemath"
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   "codemirror_mode": {
    "name": "ipython",
    "version": 3
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   "file_extension": ".py",
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   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
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