zeromorph/zeromorph_mapping_tutorial.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Select SageMath Kernel First"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "vscode": {
     "languageId": "python"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Defining X0, X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, X30, X31, X32, X33, X34, X35, X36, X37, X38, X39, X40, X41, X42, X43, X44, X45, X46, X47, X48, X49, X50, X51, X52, X53, X54, X55, X56, X57, X58, X59, X60, X61, X62, X63, X64, X65, X66, X67, X68, X69, X70, X71, X72, X73, X74, X75, X76, X77, X78, X79, X80, X81, X82, X83, X84, X85, X86, X87, X88, X89, X90, X91, X92, X93, X94, X95, X96, X97, X98, X99, X\n"
     ]
    }
   ],
   "source": [
    "F193.<a> = GF(193)\n",
    "Fp=F193\n",
    "\n",
    "var_names = ['X' + str(i) for i in range(100)] + ['X']\n",
    "R = PolynomialRing(QQ, var_names)\n",
    "R.inject_variables()\n",
    "global_vars = R.gens()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "vscode": {
     "languageId": "python"
    }
   },
   "outputs": [],
   "source": [
    "class MLE:\n",
    "    def __init__(self, poly):\n",
    "        self.vars = global_vars\n",
    "        self.mle = None\n",
    "        # if type of poly is list, then it's a list of evals\n",
    "        if isinstance(poly, list):\n",
    "            if len(poly) == 0:\n",
    "                self.k = 0\n",
    "                return\n",
    "            self.k = log(len(poly), 2)\n",
    "            self.from_evals(poly)\n",
    "        else:\n",
    "            self.k = len(poly.variables())\n",
    "            self.mle = poly\n",
    "\n",
    "    def __repr__(self):\n",
    "        return str(self.mle)\n",
    "    \n",
    "    # fallback to mle's method\n",
    "    def __getattr__(self, name):\n",
    "        return getattr(self.mle, name)\n",
    "\n",
    "    # eval means f(0,0,...,0) = evals[0]\n",
    "    #            f(0,0,...,1) = evals[1]\n",
    "    #            ...\n",
    "    #            f(1,1,...,1) = evals[2^k-1]\n",
    "    # f(X0, X1) = (1-X0)(1-X1)evals[0] + X0(1-X1)evals[1] + (1-X0)X1*evals[2] + X0X1*evals[3]\n",
    "    def from_evals(self, evals):\n",
    "        vars = self.vars[:len(evals)]\n",
    "        bit_len = self.k\n",
    "        mle_poly = 0\n",
    "        for i in range(len(evals)):\n",
    "            mask = MLE.get_bit_mask(i, bit_len)\n",
    "            prods = prod([var if enabled else (1-var) for var, enabled in zip(self.vars, mask)])\n",
    "            mle_poly += evals[i] * prods\n",
    "        self.mle = mle_poly\n",
    "        return self\n",
    "\n",
    "    def get_evals(self):\n",
    "        vars = self.vars[:2**self.k]\n",
    "        evals = []\n",
    "        k = self.k\n",
    "        if k == 0:\n",
    "            return [self.mle]\n",
    "\n",
    "        for i in range(2**k):\n",
    "            mask = self.get_bit_mask(i, k)\n",
    "            mle_tmp = self.mle\n",
    "            for j in range(k):\n",
    "                mle_tmp = mle_tmp.subs({vars[j]: mask[j]})\n",
    "            evals.append(mle_tmp)\n",
    "        return evals\n",
    "\n",
    "    def get_coeffs(self):\n",
    "        coeffs = []\n",
    "        vars = self.vars[:2**self.k]\n",
    "        k = self.k\n",
    "        if k == 0:\n",
    "            return [self.mle]\n",
    "        for i in range(2**k):\n",
    "            mask = self.get_bit_mask(i, k)\n",
    "            monomial = R(prod([var if bit else 1 for var, bit in zip(vars, mask)]))\n",
    "            coeff = self.mle.monomial_coefficient(monomial)\n",
    "            coeffs.append(coeff)\n",
    "        return coeffs\n",
    "\n",
    "    def from_coeffs(self, coeffs):\n",
    "        self.k = log(len(coeffs), 2)\n",
    "        self.mle = 0\n",
    "\n",
    "        vars = self.vars[:2**self.k]\n",
    "        k = self.k\n",
    "        for i in range(2**k):\n",
    "            mask = self.get_bit_mask(i, k)\n",
    "            monomial = R(prod([var if bit else 1 for var, bit in zip(vars, mask)]))\n",
    "            coeff = coeffs[i]\n",
    "            self.mle += coeff * monomial\n",
    "        return self\n",
    "\n",
    "    @staticmethod\n",
    "    def get_bit_mask(num, bits):\n",
    "        mask = []\n",
    "        for i in range(bits):\n",
    "            mask.append(num & 1)\n",
    "            num >>= 1\n",
    "        return mask\n",
    "\n",
    "    @staticmethod\n",
    "    def evals_to_univar(evals):\n",
    "        uni_poly = 0\n",
    "        for i in range(len(evals)):\n",
    "            uni_poly += evals[i] * X^i\n",
    "        return uni_poly\n",
    "\n",
    "    def extend(self, n):\n",
    "        k = self.k\n",
    "        evals = self.get_evals()\n",
    "        # extend evals to n\n",
    "        for i in range(n - k):\n",
    "            evals = evals + evals\n",
    "        return MLE(evals)\n",
    "\n",
    "    def to_uni(self):\n",
    "        evals = self.get_evals()\n",
    "        return MLE.evals_to_univar(evals)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# ZeroMorph\n",
    "\n",
    "While KZG is a powerful commitment scheme, it's limited to univariate polynomials. However, for techniques like sumcheck, we require multivariate polynomials (hereafter referred to as MLE - Multilinear Extension).\n",
    "\n",
    "KZG proves the existence of:\n",
    "q(X) = (f(X) - f(a)) / (X - a)\n",
    "\n",
    "In MLE, we have a similar quotient theorem. For example:\n",
    "f(X0, X1, X2) = (X0-a)*q0 + (X1-b)*q1(X0) + (X2-c)*q2(X0, X1)\n",
    "\n",
    "To leverage the power of KZG for MLE, we need to find a way to treat the MLE as a univariate polynomial. This requires a proper mapping from MLE to a univariate form.\n",
    "\n",
    "ZeroMorph provides this crucial mapping.\n",
    "\n",
    "Before delving into the specifics of this mapping, let's first examine how MLE encodes values.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following image demonstrates MLE encoding of 8 numbers on a 3D cube.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "vscode": {
     "languageId": "python"
    }
   },
   "outputs": [
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       "        var height = fontSize;\n",
       "\n",
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       "\n",
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       "\n",
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       "            // Setting opacity=1 would cause the texture&#x27;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 === &#x27;orthographic&#x27; ) {\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",
       "    // camera is positioned so that the line from the camera to the center\n",
       "    // of the bounding sphere of the objects makes an angle of 60 degrees with x-axis\n",
       "    // and an angle of 30 degrees with z-axis and the field of view of the camera looking\n",
       "    // at the center has an angle of 45 degrees.\n",
       "    const sin8 = Math.sin(Math.PI / 8);\n",
       "    const sin5 = Math.sin(Math.PI / 5);\n",
       "    const cos5 = Math.cos(Math.PI / 5);\n",
       "    const sin3 = Math.sin(Math.PI / 3);\n",
       "    const cos3 = Math.cos(Math.PI / 3);\n",
       "    var r = midToCorner / sin8;\n",
       "    var offset = new THREE.Vector3( r * sin3 * cos5, r * sin3 * sin5, r * cos3 );\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 === &#x27;orthographic&#x27; ) {\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&#x27;s diagonal so that the entire plot fits\n",
       "            // within at the default zoom level and camera position.\n",
       "            if ( aspect &gt; 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 &lt; 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 === &#x27;camera&#x27; ) {\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( &#x27;change&#x27;, function() { if ( !animate ) render(); } );\n",
       "\n",
       "    window.addEventListener( &#x27;resize&#x27;, 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 = [{&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;(0,0,0)=1&quot;, &quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;(0,0,1)=1&quot;, &quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 1.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;(0,1,0)=2&quot;, &quot;x&quot;: 0.0, &quot;y&quot;: 1.0, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;(0,1,1)=3&quot;, &quot;x&quot;: 0.0, &quot;y&quot;: 1.0, &quot;z&quot;: 1.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;(1,0,0)=5&quot;, &quot;x&quot;: 1.0, &quot;y&quot;: 0.0, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;(1,0,1)=8&quot;, &quot;x&quot;: 1.0, &quot;y&quot;: 0.0, &quot;z&quot;: 1.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;(1,1,0)=13&quot;, &quot;x&quot;: 1.0, &quot;y&quot;: 1.0, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;(1,1,1)=21&quot;, &quot;x&quot;: 1.0, &quot;y&quot;: 1.0, &quot;z&quot;: 1.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;This 3D cube visualization represents a multilinear extension, with each vertex storing the corresponding function value.&quot;, &quot;x&quot;: 0.5, &quot;y&quot;: 0.5, &quot;z&quot;: -0.5}];\n",
       "    for ( var i=0 ; i &lt; 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 &lt; 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( &#x27;canvas&#x27; );\n",
       "        canvas.width = 128;\n",
       "        canvas.height = 128;\n",
       "\n",
       "        var context = canvas.getContext( &#x27;2d&#x27; );\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 &lt; 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;: [[0.0, 0.0, 0.0], [0.0, 0.0, 1.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[0.0, 1.0, 0.0], [0.0, 1.0, 1.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[1.0, 0.0, 0.0], [1.0, 0.0, 1.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[1.0, 1.0, 0.0], [1.0, 1.0, 1.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[0.0, 0.0, 0.0], [0.0, 1.0, 0.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[0.0, 0.0, 1.0], [0.0, 1.0, 1.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[1.0, 0.0, 0.0], [1.0, 1.0, 0.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[1.0, 0.0, 1.0], [1.0, 1.0, 1.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[0.0, 0.0, 0.0], [1.0, 0.0, 0.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[0.0, 0.0, 1.0], [1.0, 0.0, 1.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[0.0, 1.0, 0.0], [1.0, 1.0, 0.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[0.0, 1.0, 1.0], [1.0, 1.0, 1.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}];\n",
       "    for ( var i=0 ; i &lt; lines.length ; i++ ) addLine( lines[i] );\n",
       "\n",
       "    function addLine( json ) {\n",
       "\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i &lt; 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 &lt; 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 &gt; 1 &amp;&amp; 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 &lt; surfaces.length ; i++ ) addSurface( surfaces[i] );\n",
       "\n",
       "    function addSurface( json ) {\n",
       "\n",
       "        var useFaceColors = &#x27;faceColors&#x27; in json ? true : false;\n",
       "\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i &lt; 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 &lt; json.faces.length ; i++ ) {\n",
       "            var f = json.faces[i];\n",
       "            for ( var j=0 ; j &lt; 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 &lt; 1 ? true : false;\n",
       "        var depthWrite = &#x27;depthWrite&#x27; 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 ? &#x27;white&#x27; : 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 &amp;&amp; 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 &lt; json.faces.length ; i++ ) {\n",
       "            var f = json.faces[i];\n",
       "            for ( var j=0 ; j &lt; 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] &lt; 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 === &#x27;dark&#x27; ? &#x27;white&#x27; : &#x27;black&#x27;;\n",
       "        var linewidth = json.linewidth || 1;\n",
       "        var materialOptions = { color: gridColor, linewidth: linewidth };\n",
       "\n",
       "        var mesh;\n",
       "        if ( linewidth &gt; 1 &amp;&amp; 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( &#x27;menu-content&#x27; );\n",
       "        if ( m.style.display === &#x27;block&#x27; ) m.style.display = &#x27;none&#x27;\n",
       "        else m.style.display = &#x27;block&#x27;;\n",
       "\n",
       "    }\n",
       "\n",
       "\n",
       "    function saveAsPNG() {\n",
       "\n",
       "        var a = document.body.appendChild( document.createElement( &#x27;a&#x27; ) );\n",
       "        a.href = renderer.domElement.toDataURL( &#x27;image/png&#x27; );\n",
       "        a.download = &#x27;screenshot&#x27;;\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( [ &#x27;&lt;!DOCTYPE html&gt;\\n&#x27; + document.documentElement.outerHTML ] );\n",
       "        var a = document.body.appendChild( document.createElement( &#x27;a&#x27; ) );\n",
       "        a.href = window.URL.createObjectURL( blob );\n",
       "        a.download = suggestFilename();\n",
       "        a.click();\n",
       "\n",
       "        function suggestFilename() {\n",
       "            if ( !document.title ) {\n",
       "                return &#x27;graphic.html&#x27;;\n",
       "            } else if ( /\\.html?$/i.test( document.title ) ) {\n",
       "                return document.title; // already ends in .htm or .html\n",
       "            } else {\n",
       "                return document.title + &#x27;.html&#x27;;\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( &#x27;textarea&#x27; );\n",
       "        textArea.textContent = JSON.stringify( axis ) + &#x27;,&#x27; + angle;\n",
       "        textArea.style.csstext = &#x27;position: absolute; top: -100%&#x27;;\n",
       "        document.body.append( textArea );\n",
       "        textArea.select();\n",
       "        document.execCommand( &#x27;copy&#x27; );\n",
       "\n",
       "        var m = document.getElementById( &#x27;menu-message&#x27; );\n",
       "        m.innerHTML = &#x27;Viewpoint copied to clipboard&#x27;;\n",
       "        m.style.display = &#x27;block&#x27;;\n",
       "        setTimeout( function() { m.style.display = &#x27;none&#x27;; }, 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(&#x27;textarea&#x27;);\n",
       "        var cam_position = JSON.stringify(pos_r);\n",
       "        textArea.textContent = &#x27;,camera_position=&#x27; + cam_position;\n",
       "        textArea.style.csstext = &#x27;position: absolute; top: -100%&#x27;;\n",
       "        document.body.append( textArea );\n",
       "        textArea.select();\n",
       "        document.execCommand( &#x27;copy&#x27; );\n",
       "\n",
       "        var m = document.getElementById( &#x27;menu-message&#x27; );\n",
       "        m.innerHTML = &#x27;Camera position &#x27;+ cam_position+&#x27; copied to clipboard&#x27;;\n",
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   "source": [
    "mle_evals = [1, 1, 2, 3, 5, 8, 13, 21]\n",
    "\n",
    "vertices = [\n",
    "    ([0,0,0], mle_evals[0]), ([0,0,1], mle_evals[1]), ([0,1,0], mle_evals[2]), ([0,1,1], mle_evals[3]),\n",
    "    ([1,0,0], mle_evals[4]), ([1,0,1], mle_evals[5]), ([1,1,0], mle_evals[6]), ([1,1,1], mle_evals[7])\n",
    "]\n",
    "\n",
    "g = Graphics()\n",
    "\n",
    "for x in [0,1]:\n",
    "    for y in [0,1]:\n",
    "        g += line3d([(x,y,0), (x,y,1)], color='black')\n",
    "for x in [0,1]:\n",
    "    for z in [0,1]:\n",
    "        g += line3d([(x,0,z), (x,1,z)], color='black')\n",
    "for y in [0,1]:\n",
    "    for z in [0,1]:\n",
    "        g += line3d([(0,y,z), (1,y,z)], color='black')\n",
    "\n",
    "for (x,y,z), value in vertices:\n",
    "    g += text3d(f\"({x},{y},{z})={value}\", (x,y,z), fontsize=15)\n",
    "\n",
    "note = \"This 3D cube visualization represents a multilinear extension, with each vertex storing the corresponding function value.\"\n",
    "text_position = (0.5, 0.5, -0.5)\n",
    "g += text3d(note, text_position, fontsize=15)\n",
    "show(g, aspect_ratio=(1, 1, 1), ticks=[[0,2,1], [0,2,1], [0,2,1]], frame=False)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This cube shows the MLE evaluation form. Then we convert it into coefficient form and map it into the Uni-Variate polynomial.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "vscode": {
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mle: 4*X0*X1*X2 + X0*X1 + 3*X0*X2 + 7*X1*X2 + X1 + 4*X2 + 1\n",
      "mle  evals: [1, 1, 2, 3, 5, 8, 13, 21]\n",
      "U(mle): 21*X^7 + 13*X^6 + 8*X^5 + 5*X^4 + 3*X^3 + 2*X^2 + X + 1\n"
     ]
    }
   ],
   "source": [
    "mle = MLE(mle_evals)\n",
    "print(f\"mle: {mle}\")\n",
    "print(f\"mle  evals: {mle.get_evals()}\")\n",
    "print(f\"U(mle): {mle.to_uni()}\")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then we commit this polynomial. The quotient polynomial also applies the same procedure.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "vscode": {
     "languageId": "python"
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "q2: 4*X0*X1 + 3*X0 + 7*X1 + 4\n",
      "r2: 13*X0*X1 + 9*X0 + 22*X1 + 13\n",
      "q1: 13*X0 + 22\n",
      "r1: 35*X0 + 57\n",
      "q0: 35\n",
      "r0: 92\n",
      "mle - v:           4*X0*X1*X2 + X0*X1 + 3*X0*X2 + 7*X1*X2 + X1 + 4*X2 - 91\n",
      "sigma((Xk-uk)*qi): 4*X0*X1*X2 + X0*X1 + 3*X0*X2 + 7*X1*X2 + X1 + 4*X2 - 91\n",
      "q2_uni: 18*X^3 + 11*X^2 + 7*X + 4\n",
      "q1_uni: 35*X + 22\n",
      "q0_uni: 35\n"
     ]
    }
   ],
   "source": [
    "point = [1, 2, 3]\n",
    "\n",
    "q2, r2 = mle.quo_rem((X2-point[2]))\n",
    "print(f\"q2: {q2}\")\n",
    "print(f\"r2: {r2}\")\n",
    "\n",
    "q1, r1 = r2.quo_rem((X1-point[1]))\n",
    "print(f\"q1: {q1}\")\n",
    "print(f\"r1: {r1}\")\n",
    "\n",
    "q0, r0 = r1.quo_rem((X0-point[0]))\n",
    "print(f\"q0: {q0}\")\n",
    "print(f\"r0: {r0}\")\n",
    "\n",
    "v = r0\n",
    "\n",
    "# check if mle - v == q0*(X0-point[0]) + q1*(X1-point[1]) + q2*(X2-point[2])\n",
    "print(f\"mle - v:           {mle - v}\")\n",
    "print(f\"sigma((Xk-uk)*qi): {q0*(X0-point[0]) + q1*(X1-point[1]) + q2*(X2-point[2])}\")\n",
    "\n",
    "q2_uni = MLE(q2).to_uni()\n",
    "q1_uni = MLE(q1).to_uni()\n",
    "q0_uni = MLE(q0).to_uni()\n",
    "\n",
    "print(f\"q2_uni: {q2_uni}\")\n",
    "print(f\"q1_uni: {q1_uni}\")\n",
    "print(f\"q0_uni: {q0_uni}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Verifier's Equation Check\n",
    "The verifier needs to check if the equation holds using the Commitments of ${U}(mle)$, ${U}(q_0)$, ${U}(q_1)$, ${U}(q_2)$, and the plain point, v.\n",
    "\n",
    "## Equation Transformation\n",
    "Let's transform the equation:\n",
    "${U}(mle - v) = {U}(q_0 \\cdot (X_0 - point[0]) + q_1 \\cdot (X_1 - point[1]) + q_2 \\cdot (X_2 - point[2]))$\n",
    "\n",
    "${U}(mle) - {U}(v) = {U}(q_0X_0 - q_0 \\cdot point[0] + q_1X_1 - q_1 \\cdot point[1] + q_2X_2 - q_2 \\cdot point[2])$\n",
    "\n",
    "$\\quad = {U}(q_0X_0) - {U}(q_0 \\cdot point[0]) + {U}(q_1X_1) - {U}(q_1 \\cdot point[1]) + {U}(q_2X_2) - {U}(q_2 \\cdot point[2])$\n",
    "\n",
    "## Challenge: Aligning the Dimension of U(MLE) without the Original MLE\n",
    "\n",
    "We face a significant challenge in this process: we need to find a method to increase the dimension of ${U}(q)$, but the verifier only has access to ${U}(q)$, not the original $ q$ itself. This limitation requires us to develop a approach to upscale the dimension while working solely with the committed values.\n",
    "\n",
    "## ZeroMorph's U Mapping\n",
    "\n",
    "ZeroMorph's U mapping defines all dimensions in F[X] 2^n. We need to extend any lower degree items from k-dim to the n-dim vector space.\n",
    "\n",
    "### Example: ${U}(q_0X_0)$\n",
    "Let's examine what's happening with ${U}(q_0X_0)$:\n",
    "\n",
    "- It means a 0-dim $q_0$ multiplies a variable, making it 1-dim\n",
    "- When $X_0=0$, ${U}(q_0 \\cdot 0) = 0$\n",
    "- When $X_0=1$, ${U}(q_0 \\cdot 1) = {U}(q_0)$\n",
    "This appears to be zero-padding before $q_0$ on 1-dim.\n",
    "\n",
    "## Extending to Higher Dimensions\n",
    "To fulfill the space requirements, we need to extend the 1-dimensional representation into a 3-dimensional one.\n",
    "\n",
    "Consider the following scenario:\n",
    "1. The 1-dimensional representation is augmented with two additional variables that do not affect the outcome.\n",
    "2. Regardless of the values chosen for X1 and X2, only X0 has an impact on the result.\n",
    "\n",
    "To achieve this extension, we can:\n",
    "1. Replicate the 1-dimensional representation to create a 2-dimensional structure.\n",
    "2. Subsequently, expand the 2-dimensional structure into a 3-dimensional representation.\n",
    "\n",
    "This approach allows us to maintain the integrity of the original information while extending it to the required higher-dimensional space.\n"
   ]
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  {
   "cell_type": "code",
   "execution_count": 6,
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       "        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 === &#x27;orthographic&#x27; ) {\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",
       "    // camera is positioned so that the line from the camera to the center\n",
       "    // of the bounding sphere of the objects makes an angle of 60 degrees with x-axis\n",
       "    // and an angle of 30 degrees with z-axis and the field of view of the camera looking\n",
       "    // at the center has an angle of 45 degrees.\n",
       "    const sin8 = Math.sin(Math.PI / 8);\n",
       "    const sin5 = Math.sin(Math.PI / 5);\n",
       "    const cos5 = Math.cos(Math.PI / 5);\n",
       "    const sin3 = Math.sin(Math.PI / 3);\n",
       "    const cos3 = Math.cos(Math.PI / 3);\n",
       "    var r = midToCorner / sin8;\n",
       "    var offset = new THREE.Vector3( r * sin3 * cos5, r * sin3 * sin5, r * cos3 );\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 === &#x27;orthographic&#x27; ) {\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&#x27;s diagonal so that the entire plot fits\n",
       "            // within at the default zoom level and camera position.\n",
       "            if ( aspect &gt; 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 &lt; 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 === &#x27;camera&#x27; ) {\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( &#x27;change&#x27;, function() { if ( !animate ) render(); } );\n",
       "\n",
       "    window.addEventListener( &#x27;resize&#x27;, 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 = [{&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;(0)=0&quot;, &quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;(1)=35&quot;, &quot;x&quot;: 1.0, &quot;y&quot;: 0.0, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;This line visualization represents a multilinear extension, with each vertex storing the corresponding function value.&quot;, &quot;x&quot;: 0.5, &quot;y&quot;: 0.5, &quot;z&quot;: -0.5}];\n",
       "    for ( var i=0 ; i &lt; 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 &lt; 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( &#x27;canvas&#x27; );\n",
       "        canvas.width = 128;\n",
       "        canvas.height = 128;\n",
       "\n",
       "        var context = canvas.getContext( &#x27;2d&#x27; );\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 &lt; 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;: [[0.0, 0.0, 0.0], [1.0, 0.0, 0.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[0.0, 0.0, 0.0], [1.0, 0.0, 0.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}];\n",
       "    for ( var i=0 ; i &lt; lines.length ; i++ ) addLine( lines[i] );\n",
       "\n",
       "    function addLine( json ) {\n",
       "\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i &lt; 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 &lt; 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 &gt; 1 &amp;&amp; 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 &lt; surfaces.length ; i++ ) addSurface( surfaces[i] );\n",
       "\n",
       "    function addSurface( json ) {\n",
       "\n",
       "        var useFaceColors = &#x27;faceColors&#x27; in json ? true : false;\n",
       "\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i &lt; 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 &lt; json.faces.length ; i++ ) {\n",
       "            var f = json.faces[i];\n",
       "            for ( var j=0 ; j &lt; 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 &lt; 1 ? true : false;\n",
       "        var depthWrite = &#x27;depthWrite&#x27; 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 ? &#x27;white&#x27; : 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 &amp;&amp; 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 &lt; json.faces.length ; i++ ) {\n",
       "            var f = json.faces[i];\n",
       "            for ( var j=0 ; j &lt; 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] &lt; 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 === &#x27;dark&#x27; ? &#x27;white&#x27; : &#x27;black&#x27;;\n",
       "        var linewidth = json.linewidth || 1;\n",
       "        var materialOptions = { color: gridColor, linewidth: linewidth };\n",
       "\n",
       "        var mesh;\n",
       "        if ( linewidth &gt; 1 &amp;&amp; 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( &#x27;menu-content&#x27; );\n",
       "        if ( m.style.display === &#x27;block&#x27; ) m.style.display = &#x27;none&#x27;\n",
       "        else m.style.display = &#x27;block&#x27;;\n",
       "\n",
       "    }\n",
       "\n",
       "\n",
       "    function saveAsPNG() {\n",
       "\n",
       "        var a = document.body.appendChild( document.createElement( &#x27;a&#x27; ) );\n",
       "        a.href = renderer.domElement.toDataURL( &#x27;image/png&#x27; );\n",
       "        a.download = &#x27;screenshot&#x27;;\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( [ &#x27;&lt;!DOCTYPE html&gt;\\n&#x27; + document.documentElement.outerHTML ] );\n",
       "        var a = document.body.appendChild( document.createElement( &#x27;a&#x27; ) );\n",
       "        a.href = window.URL.createObjectURL( blob );\n",
       "        a.download = suggestFilename();\n",
       "        a.click();\n",
       "\n",
       "        function suggestFilename() {\n",
       "            if ( !document.title ) {\n",
       "                return &#x27;graphic.html&#x27;;\n",
       "            } else if ( /\\.html?$/i.test( document.title ) ) {\n",
       "                return document.title; // already ends in .htm or .html\n",
       "            } else {\n",
       "                return document.title + &#x27;.html&#x27;;\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( &#x27;textarea&#x27; );\n",
       "        textArea.textContent = JSON.stringify( axis ) + &#x27;,&#x27; + angle;\n",
       "        textArea.style.csstext = &#x27;position: absolute; top: -100%&#x27;;\n",
       "        document.body.append( textArea );\n",
       "        textArea.select();\n",
       "        document.execCommand( &#x27;copy&#x27; );\n",
       "\n",
       "        var m = document.getElementById( &#x27;menu-message&#x27; );\n",
       "        m.innerHTML = &#x27;Viewpoint copied to clipboard&#x27;;\n",
       "        m.style.display = &#x27;block&#x27;;\n",
       "        setTimeout( function() { m.style.display = &#x27;none&#x27;; }, 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(&#x27;textarea&#x27;);\n",
       "        var cam_position = JSON.stringify(pos_r);\n",
       "        textArea.textContent = &#x27;,camera_position=&#x27; + cam_position;\n",
       "        textArea.style.csstext = &#x27;position: absolute; top: -100%&#x27;;\n",
       "        document.body.append( textArea );\n",
       "        textArea.select();\n",
       "        document.execCommand( &#x27;copy&#x27; );\n",
       "\n",
       "        var m = document.getElementById( &#x27;menu-message&#x27; );\n",
       "        m.innerHTML = &#x27;Camera position &#x27;+ cam_position+&#x27; copied to clipboard&#x27;;\n",
       "        m.style.display = &#x27;block&#x27;;\n",
       "        setTimeout( function() { m.style.display = &#x27;none&#x27;; }, 2000 );\n",
       "\n",
       "    }\n",
       "\n",
       "&lt;/script&gt;\n",
       "\n",
       "&lt;div id=&quot;menu-container&quot; onclick=&quot;toggleMenu()&quot;&gt;&amp;#x24d8;\n",
       "&lt;div id=&quot;menu-message&quot;&gt;&lt;/div&gt;\n",
       "&lt;div id=&quot;menu-content&quot;&gt;\n",
       "&lt;div onclick=&quot;saveAsPNG()&quot;&gt;Save as PNG&lt;/div&gt;\n",
       "&lt;div onclick=&quot;saveAsHTML()&quot;&gt;Save as HTML&lt;/div&gt;\n",
       "&lt;div onclick=&quot;getCamera()&quot;&gt;Get camera&lt;/div&gt;\n",
       "&lt;div onclick=&quot;getViewpoint()&quot;&gt;Get viewpoint&lt;/div&gt;\n",
       "&lt;div&gt;Close Menu&lt;/div&gt;\n",
       "&lt;/div&gt;&lt;/div&gt;\n",
       "\n",
       "\n",
       "&lt;/body&gt;\n",
       "&lt;/html&gt;\n",
       "\"\n",
       "        width=\"100%\"\n",
       "        height=\"400\"\n",
       "        style=\"border: 0;\">\n",
       "</iframe>\n"
      ],
      "text/plain": [
       "Graphics3d Object"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "\n",
       "<iframe srcdoc=\"&lt;!DOCTYPE html&gt;\n",
       "&lt;html&gt;\n",
       "&lt;head&gt;\n",
       "&lt;title&gt;&lt;/title&gt;\n",
       "&lt;meta charset=&quot;utf-8&quot;&gt;\n",
       "&lt;meta name=viewport content=&quot;width=device-width, user-scalable=no, minimum-scale=1.0, maximum-scale=1.0&quot;&gt;\n",
       "&lt;style&gt;\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",
       "&lt;/style&gt;\n",
       "\n",
       "&lt;/head&gt;\n",
       "\n",
       "&lt;body&gt;\n",
       "\n",
       "&lt;script src=&quot;/nbextensions/threejs-sage/r122/three.min.js&quot;&gt;&lt;/script&gt;\n",
       "&lt;script&gt;\n",
       "  if ( !window.THREE ) document.write(&#x27; \\\n",
       "&lt;script src=&quot;https://cdn.jsdelivr.net/gh/sagemath/threejs-sage@r122/build/three.min.js&quot;&gt;&lt;\\/script&gt; \\\n",
       "            &#x27;);\n",
       "&lt;/script&gt;\n",
       "        \n",
       "&lt;script&gt;\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 === &#x27;dark&#x27; )\n",
       "        document.body.className = &#x27;dark-theme&#x27;;\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 === &#x27;dark&#x27; ? 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;:-0.5}, {&quot;x&quot;:1.0, &quot;y&quot;:1.0, &quot;z&quot;:0.0}]; // 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 &gt; autoAspect * rxRange &amp;&amp; autoScaling[0] ) a[0] = autoAspect * rxRange / xRange;\n",
       "    if ( yRange &gt; autoAspect * ryRange &amp;&amp; autoScaling[1] ) a[1] = autoAspect * ryRange / yRange;\n",
       "    if ( zRange &gt; autoAspect * rzRange &amp;&amp; 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 === &#x27;dark&#x27; ? &#x27;white&#x27; : &#x27;black&#x27;;\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] + &#x27;=&#x27; + 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*$/.test(ym) ) ym = ym.substr(1);\n",
       "        addLabel( al[1] + &#x27;=&#x27; + ym, a[0]*b[1].x+offset, a[1]*yMid, a[2]*b[0].z, als[1] );\n",
       "        addLabel( ( b[0].y ).toFixed(d), a[0]*b[1].x+offset, a[1]*b[0].y, a[2]*b[0].z, als[1] );\n",
       "        addLabel( ( b[1].y ).toFixed(d), a[0]*b[1].x+offset, a[1]*b[1].y, a[2]*b[0].z, als[1] );\n",
       "\n",
       "        var offset = offsetRatio * a[1]*( b[1].y - b[0].y );\n",
       "        var zm = zMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(zm) ) zm = zm.substr(1);\n",
       "        addLabel( al[2] + &#x27;=&#x27; + 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 || &#x27;black&#x27;;\n",
       "        var fontSize = style.fontSize || 14;\n",
       "        var fontFamily = style.fontFamily || &#x27;monospace&#x27;;\n",
       "        var fontStyle = style.fontStyle || &#x27;normal&#x27;;\n",
       "        var fontWeight = style.fontWeight || &#x27;normal&#x27;;\n",
       "        var opacity = style.opacity || 1;\n",
       "\n",
       "        if ( options.theme === &#x27;dark&#x27; )\n",
       "            if ( color === &#x27;black&#x27; || color === &#x27;#000000&#x27; )\n",
       "                color = &#x27;white&#x27;;\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 ) ? &#x27;&quot;&#x27; + f + &#x27;&quot;&#x27; : f;\n",
       "            }).join(&#x27;, &#x27;);\n",
       "        }\n",
       "\n",
       "        var canvas = document.createElement( &#x27;canvas&#x27; );\n",
       "        var context = canvas.getContext( &#x27;2d&#x27; );\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 + &#x27;px&#x27;, fontFamily].join(&#x27; &#x27;);\n",
       "\n",
       "        context.font = font;\n",
       "        var width = context.measureText( text ).width;\n",
       "        var height = fontSize;\n",
       "\n",
       "        // The dimensions of the canvas&#x27;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 + &#x27;px&#x27;;\n",
       "        canvas.style.height = height + &#x27;px&#x27;;\n",
       "\n",
       "        context.scale( pixelRatio, pixelRatio );\n",
       "        context.fillStyle = color;\n",
       "        context.font = font; // Must be set again after measureText.\n",
       "        context.textAlign = &#x27;center&#x27;;\n",
       "        context.textBaseline = &#x27;middle&#x27;;\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 &lt; 1 ) {\n",
       "            // Setting opacity=1 would cause the texture&#x27;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 === &#x27;orthographic&#x27; ) {\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",
       "    // camera is positioned so that the line from the camera to the center\n",
       "    // of the bounding sphere of the objects makes an angle of 60 degrees with x-axis\n",
       "    // and an angle of 30 degrees with z-axis and the field of view of the camera looking\n",
       "    // at the center has an angle of 45 degrees.\n",
       "    const sin8 = Math.sin(Math.PI / 8);\n",
       "    const sin5 = Math.sin(Math.PI / 5);\n",
       "    const cos5 = Math.cos(Math.PI / 5);\n",
       "    const sin3 = Math.sin(Math.PI / 3);\n",
       "    const cos3 = Math.cos(Math.PI / 3);\n",
       "    var r = midToCorner / sin8;\n",
       "    var offset = new THREE.Vector3( r * sin3 * cos5, r * sin3 * sin5, r * cos3 );\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 === &#x27;orthographic&#x27; ) {\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&#x27;s diagonal so that the entire plot fits\n",
       "            // within at the default zoom level and camera position.\n",
       "            if ( aspect &gt; 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 &lt; 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 === &#x27;camera&#x27; ) {\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( &#x27;change&#x27;, function() { if ( !animate ) render(); } );\n",
       "\n",
       "    window.addEventListener( &#x27;resize&#x27;, 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 = [{&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;(0,0)=0&quot;, &quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;(1,0)=35&quot;, &quot;x&quot;: 1.0, &quot;y&quot;: 0.0, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;(0,1)=0&quot;, &quot;x&quot;: 0.0, &quot;y&quot;: 1.0, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;(1,1)=35&quot;, &quot;x&quot;: 1.0, &quot;y&quot;: 1.0, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;The second line is the copy of the first line, now it&#x27;s a 2D plane&quot;, &quot;x&quot;: 0.5, &quot;y&quot;: 0.5, &quot;z&quot;: -0.5}];\n",
       "    for ( var i=0 ; i &lt; 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 &lt; 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( &#x27;canvas&#x27; );\n",
       "        canvas.width = 128;\n",
       "        canvas.height = 128;\n",
       "\n",
       "        var context = canvas.getContext( &#x27;2d&#x27; );\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 &lt; 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;: [[1.0, 0.0, 0.0], [0.0, 0.0, 0.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[1.0, 1.0, 0.0], [0.0, 1.0, 0.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[0.0, 0.0, 0.0], [1.0, 0.0, 0.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[0.0, 1.0, 0.0], [1.0, 1.0, 0.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[0.0, 1.0, 0.0], [0.0, 0.0, 0.0]], &quot;color&quot;: &quot;#ff0000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[0.0, 0.0, 0.0], [0.0, 1.0, 0.0]], &quot;color&quot;: &quot;#ff0000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[1.0, 1.0, 0.0], [1.0, 0.0, 0.0]], &quot;color&quot;: &quot;#ff0000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[1.0, 0.0, 0.0], [1.0, 1.0, 0.0]], &quot;color&quot;: &quot;#ff0000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}];\n",
       "    for ( var i=0 ; i &lt; lines.length ; i++ ) addLine( lines[i] );\n",
       "\n",
       "    function addLine( json ) {\n",
       "\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i &lt; 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 &lt; 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 &gt; 1 &amp;&amp; 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 &lt; surfaces.length ; i++ ) addSurface( surfaces[i] );\n",
       "\n",
       "    function addSurface( json ) {\n",
       "\n",
       "        var useFaceColors = &#x27;faceColors&#x27; in json ? true : false;\n",
       "\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i &lt; 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 &lt; json.faces.length ; i++ ) {\n",
       "            var f = json.faces[i];\n",
       "            for ( var j=0 ; j &lt; 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 &lt; 1 ? true : false;\n",
       "        var depthWrite = &#x27;depthWrite&#x27; 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 ? &#x27;white&#x27; : 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 &amp;&amp; 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 &lt; json.faces.length ; i++ ) {\n",
       "            var f = json.faces[i];\n",
       "            for ( var j=0 ; j &lt; 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] &lt; 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 === &#x27;dark&#x27; ? &#x27;white&#x27; : &#x27;black&#x27;;\n",
       "        var linewidth = json.linewidth || 1;\n",
       "        var materialOptions = { color: gridColor, linewidth: linewidth };\n",
       "\n",
       "        var mesh;\n",
       "        if ( linewidth &gt; 1 &amp;&amp; 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( &#x27;menu-content&#x27; );\n",
       "        if ( m.style.display === &#x27;block&#x27; ) m.style.display = &#x27;none&#x27;\n",
       "        else m.style.display = &#x27;block&#x27;;\n",
       "\n",
       "    }\n",
       "\n",
       "\n",
       "    function saveAsPNG() {\n",
       "\n",
       "        var a = document.body.appendChild( document.createElement( &#x27;a&#x27; ) );\n",
       "        a.href = renderer.domElement.toDataURL( &#x27;image/png&#x27; );\n",
       "        a.download = &#x27;screenshot&#x27;;\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( [ &#x27;&lt;!DOCTYPE html&gt;\\n&#x27; + document.documentElement.outerHTML ] );\n",
       "        var a = document.body.appendChild( document.createElement( &#x27;a&#x27; ) );\n",
       "        a.href = window.URL.createObjectURL( blob );\n",
       "        a.download = suggestFilename();\n",
       "        a.click();\n",
       "\n",
       "        function suggestFilename() {\n",
       "            if ( !document.title ) {\n",
       "                return &#x27;graphic.html&#x27;;\n",
       "            } else if ( /\\.html?$/i.test( document.title ) ) {\n",
       "                return document.title; // already ends in .htm or .html\n",
       "            } else {\n",
       "                return document.title + &#x27;.html&#x27;;\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( &#x27;textarea&#x27; );\n",
       "        textArea.textContent = JSON.stringify( axis ) + &#x27;,&#x27; + angle;\n",
       "        textArea.style.csstext = &#x27;position: absolute; top: -100%&#x27;;\n",
       "        document.body.append( textArea );\n",
       "        textArea.select();\n",
       "        document.execCommand( &#x27;copy&#x27; );\n",
       "\n",
       "        var m = document.getElementById( &#x27;menu-message&#x27; );\n",
       "        m.innerHTML = &#x27;Viewpoint copied to clipboard&#x27;;\n",
       "        m.style.display = &#x27;block&#x27;;\n",
       "        setTimeout( function() { m.style.display = &#x27;none&#x27;; }, 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(&#x27;textarea&#x27;);\n",
       "        var cam_position = JSON.stringify(pos_r);\n",
       "        textArea.textContent = &#x27;,camera_position=&#x27; + cam_position;\n",
       "        textArea.style.csstext = &#x27;position: absolute; top: -100%&#x27;;\n",
       "        document.body.append( textArea );\n",
       "        textArea.select();\n",
       "        document.execCommand( &#x27;copy&#x27; );\n",
       "\n",
       "        var m = document.getElementById( &#x27;menu-message&#x27; );\n",
       "        m.innerHTML = &#x27;Camera position &#x27;+ cam_position+&#x27; copied to clipboard&#x27;;\n",
       "        m.style.display = &#x27;block&#x27;;\n",
       "        setTimeout( function() { m.style.display = &#x27;none&#x27;; }, 2000 );\n",
       "\n",
       "    }\n",
       "\n",
       "&lt;/script&gt;\n",
       "\n",
       "&lt;div id=&quot;menu-container&quot; onclick=&quot;toggleMenu()&quot;&gt;&amp;#x24d8;\n",
       "&lt;div id=&quot;menu-message&quot;&gt;&lt;/div&gt;\n",
       "&lt;div id=&quot;menu-content&quot;&gt;\n",
       "&lt;div onclick=&quot;saveAsPNG()&quot;&gt;Save as PNG&lt;/div&gt;\n",
       "&lt;div onclick=&quot;saveAsHTML()&quot;&gt;Save as HTML&lt;/div&gt;\n",
       "&lt;div onclick=&quot;getCamera()&quot;&gt;Get camera&lt;/div&gt;\n",
       "&lt;div onclick=&quot;getViewpoint()&quot;&gt;Get viewpoint&lt;/div&gt;\n",
       "&lt;div&gt;Close Menu&lt;/div&gt;\n",
       "&lt;/div&gt;&lt;/div&gt;\n",
       "\n",
       "\n",
       "&lt;/body&gt;\n",
       "&lt;/html&gt;\n",
       "\"\n",
       "        width=\"100%\"\n",
       "        height=\"400\"\n",
       "        style=\"border: 0;\">\n",
       "</iframe>\n"
      ],
      "text/plain": [
       "Graphics3d Object"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "\n",
       "<iframe srcdoc=\"&lt;!DOCTYPE html&gt;\n",
       "&lt;html&gt;\n",
       "&lt;head&gt;\n",
       "&lt;title&gt;&lt;/title&gt;\n",
       "&lt;meta charset=&quot;utf-8&quot;&gt;\n",
       "&lt;meta name=viewport content=&quot;width=device-width, user-scalable=no, minimum-scale=1.0, maximum-scale=1.0&quot;&gt;\n",
       "&lt;style&gt;\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",
       "&lt;/style&gt;\n",
       "\n",
       "&lt;/head&gt;\n",
       "\n",
       "&lt;body&gt;\n",
       "\n",
       "&lt;script src=&quot;/nbextensions/threejs-sage/r122/three.min.js&quot;&gt;&lt;/script&gt;\n",
       "&lt;script&gt;\n",
       "  if ( !window.THREE ) document.write(&#x27; \\\n",
       "&lt;script src=&quot;https://cdn.jsdelivr.net/gh/sagemath/threejs-sage@r122/build/three.min.js&quot;&gt;&lt;\\/script&gt; \\\n",
       "            &#x27;);\n",
       "&lt;/script&gt;\n",
       "        \n",
       "&lt;script&gt;\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 === &#x27;dark&#x27; )\n",
       "        document.body.className = &#x27;dark-theme&#x27;;\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 === &#x27;dark&#x27; ? 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;:-0.5}, {&quot;x&quot;:1.0, &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 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 &gt; autoAspect * rxRange &amp;&amp; autoScaling[0] ) a[0] = autoAspect * rxRange / xRange;\n",
       "    if ( yRange &gt; autoAspect * ryRange &amp;&amp; autoScaling[1] ) a[1] = autoAspect * ryRange / yRange;\n",
       "    if ( zRange &gt; autoAspect * rzRange &amp;&amp; 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 === &#x27;dark&#x27; ? &#x27;white&#x27; : &#x27;black&#x27;;\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] + &#x27;=&#x27; + 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*$/.test(ym) ) ym = ym.substr(1);\n",
       "        addLabel( al[1] + &#x27;=&#x27; + ym, a[0]*b[1].x+offset, a[1]*yMid, a[2]*b[0].z, als[1] );\n",
       "        addLabel( ( b[0].y ).toFixed(d), a[0]*b[1].x+offset, a[1]*b[0].y, a[2]*b[0].z, als[1] );\n",
       "        addLabel( ( b[1].y ).toFixed(d), a[0]*b[1].x+offset, a[1]*b[1].y, a[2]*b[0].z, als[1] );\n",
       "\n",
       "        var offset = offsetRatio * a[1]*( b[1].y - b[0].y );\n",
       "        var zm = zMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(zm) ) zm = zm.substr(1);\n",
       "        addLabel( al[2] + &#x27;=&#x27; + 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 || &#x27;black&#x27;;\n",
       "        var fontSize = style.fontSize || 14;\n",
       "        var fontFamily = style.fontFamily || &#x27;monospace&#x27;;\n",
       "        var fontStyle = style.fontStyle || &#x27;normal&#x27;;\n",
       "        var fontWeight = style.fontWeight || &#x27;normal&#x27;;\n",
       "        var opacity = style.opacity || 1;\n",
       "\n",
       "        if ( options.theme === &#x27;dark&#x27; )\n",
       "            if ( color === &#x27;black&#x27; || color === &#x27;#000000&#x27; )\n",
       "                color = &#x27;white&#x27;;\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 ) ? &#x27;&quot;&#x27; + f + &#x27;&quot;&#x27; : f;\n",
       "            }).join(&#x27;, &#x27;);\n",
       "        }\n",
       "\n",
       "        var canvas = document.createElement( &#x27;canvas&#x27; );\n",
       "        var context = canvas.getContext( &#x27;2d&#x27; );\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 + &#x27;px&#x27;, fontFamily].join(&#x27; &#x27;);\n",
       "\n",
       "        context.font = font;\n",
       "        var width = context.measureText( text ).width;\n",
       "        var height = fontSize;\n",
       "\n",
       "        // The dimensions of the canvas&#x27;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 + &#x27;px&#x27;;\n",
       "        canvas.style.height = height + &#x27;px&#x27;;\n",
       "\n",
       "        context.scale( pixelRatio, pixelRatio );\n",
       "        context.fillStyle = color;\n",
       "        context.font = font; // Must be set again after measureText.\n",
       "        context.textAlign = &#x27;center&#x27;;\n",
       "        context.textBaseline = &#x27;middle&#x27;;\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 &lt; 1 ) {\n",
       "            // Setting opacity=1 would cause the texture&#x27;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 === &#x27;orthographic&#x27; ) {\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",
       "    // camera is positioned so that the line from the camera to the center\n",
       "    // of the bounding sphere of the objects makes an angle of 60 degrees with x-axis\n",
       "    // and an angle of 30 degrees with z-axis and the field of view of the camera looking\n",
       "    // at the center has an angle of 45 degrees.\n",
       "    const sin8 = Math.sin(Math.PI / 8);\n",
       "    const sin5 = Math.sin(Math.PI / 5);\n",
       "    const cos5 = Math.cos(Math.PI / 5);\n",
       "    const sin3 = Math.sin(Math.PI / 3);\n",
       "    const cos3 = Math.cos(Math.PI / 3);\n",
       "    var r = midToCorner / sin8;\n",
       "    var offset = new THREE.Vector3( r * sin3 * cos5, r * sin3 * sin5, r * cos3 );\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 === &#x27;orthographic&#x27; ) {\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&#x27;s diagonal so that the entire plot fits\n",
       "            // within at the default zoom level and camera position.\n",
       "            if ( aspect &gt; 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 &lt; 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 === &#x27;camera&#x27; ) {\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( &#x27;change&#x27;, function() { if ( !animate ) render(); } );\n",
       "\n",
       "    window.addEventListener( &#x27;resize&#x27;, 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 = [{&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;(0,0,0)=0&quot;, &quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;(1,0,0)=35&quot;, &quot;x&quot;: 1.0, &quot;y&quot;: 0.0, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;(0,1,0)=0&quot;, &quot;x&quot;: 0.0, &quot;y&quot;: 1.0, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;(1,1,0)=35&quot;, &quot;x&quot;: 1.0, &quot;y&quot;: 1.0, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;(0,0,1)=0&quot;, &quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 1.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;(1,0,1)=35&quot;, &quot;x&quot;: 1.0, &quot;y&quot;: 0.0, &quot;z&quot;: 1.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;(0,1,1)=0&quot;, &quot;x&quot;: 0.0, &quot;y&quot;: 1.0, &quot;z&quot;: 1.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;(1,1,1)=35&quot;, &quot;x&quot;: 1.0, &quot;y&quot;: 1.0, &quot;z&quot;: 1.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 15.0, &quot;fontFamily&quot;: [&quot;monospace&quot;], &quot;fontStyle&quot;: &quot;normal&quot;, &quot;fontWeight&quot;: &quot;normal&quot;, &quot;opacity&quot;: 1.0, &quot;text&quot;: &quot;The top plane is the copy of the bottom plane, now it&#x27;s a 3D cube&quot;, &quot;x&quot;: 0.5, &quot;y&quot;: 0.5, &quot;z&quot;: -0.5}];\n",
       "    for ( var i=0 ; i &lt; 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 &lt; 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( &#x27;canvas&#x27; );\n",
       "        canvas.width = 128;\n",
       "        canvas.height = 128;\n",
       "\n",
       "        var context = canvas.getContext( &#x27;2d&#x27; );\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 &lt; 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;: [[0.0, 0.0, 0.0], [0.0, 0.0, 1.0]], &quot;color&quot;: &quot;#ff0000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[0.0, 1.0, 0.0], [0.0, 1.0, 1.0]], &quot;color&quot;: &quot;#ff0000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[1.0, 0.0, 0.0], [1.0, 0.0, 1.0]], &quot;color&quot;: &quot;#ff0000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[1.0, 1.0, 0.0], [1.0, 1.0, 1.0]], &quot;color&quot;: &quot;#ff0000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[0.0, 0.0, 0.0], [0.0, 1.0, 0.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[0.0, 0.0, 1.0], [0.0, 1.0, 1.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[1.0, 0.0, 0.0], [1.0, 1.0, 0.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[1.0, 0.0, 1.0], [1.0, 1.0, 1.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[0.0, 0.0, 0.0], [1.0, 0.0, 0.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[0.0, 0.0, 1.0], [1.0, 0.0, 1.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[0.0, 1.0, 0.0], [1.0, 1.0, 0.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}, {&quot;points&quot;: [[0.0, 1.0, 1.0], [1.0, 1.0, 1.0]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0, &quot;linewidth&quot;: 1.0}];\n",
       "    for ( var i=0 ; i &lt; lines.length ; i++ ) addLine( lines[i] );\n",
       "\n",
       "    function addLine( json ) {\n",
       "\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i &lt; 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 &lt; 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 &gt; 1 &amp;&amp; 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 &lt; surfaces.length ; i++ ) addSurface( surfaces[i] );\n",
       "\n",
       "    function addSurface( json ) {\n",
       "\n",
       "        var useFaceColors = &#x27;faceColors&#x27; in json ? true : false;\n",
       "\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i &lt; 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 &lt; json.faces.length ; i++ ) {\n",
       "            var f = json.faces[i];\n",
       "            for ( var j=0 ; j &lt; 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 &lt; 1 ? true : false;\n",
       "        var depthWrite = &#x27;depthWrite&#x27; 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 ? &#x27;white&#x27; : 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 &amp;&amp; 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 &lt; json.faces.length ; i++ ) {\n",
       "            var f = json.faces[i];\n",
       "            for ( var j=0 ; j &lt; 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] &lt; 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 === &#x27;dark&#x27; ? &#x27;white&#x27; : &#x27;black&#x27;;\n",
       "        var linewidth = json.linewidth || 1;\n",
       "        var materialOptions = { color: gridColor, linewidth: linewidth };\n",
       "\n",
       "        var mesh;\n",
       "        if ( linewidth &gt; 1 &amp;&amp; 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( &#x27;menu-content&#x27; );\n",
       "        if ( m.style.display === &#x27;block&#x27; ) m.style.display = &#x27;none&#x27;\n",
       "        else m.style.display = &#x27;block&#x27;;\n",
       "\n",
       "    }\n",
       "\n",
       "\n",
       "    function saveAsPNG() {\n",
       "\n",
       "        var a = document.body.appendChild( document.createElement( &#x27;a&#x27; ) );\n",
       "        a.href = renderer.domElement.toDataURL( &#x27;image/png&#x27; );\n",
       "        a.download = &#x27;screenshot&#x27;;\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( [ &#x27;&lt;!DOCTYPE html&gt;\\n&#x27; + document.documentElement.outerHTML ] );\n",
       "        var a = document.body.appendChild( document.createElement( &#x27;a&#x27; ) );\n",
       "        a.href = window.URL.createObjectURL( blob );\n",
       "        a.download = suggestFilename();\n",
       "        a.click();\n",
       "\n",
       "        function suggestFilename() {\n",
       "            if ( !document.title ) {\n",
       "                return &#x27;graphic.html&#x27;;\n",
       "            } else if ( /\\.html?$/i.test( document.title ) ) {\n",
       "                return document.title; // already ends in .htm or .html\n",
       "            } else {\n",
       "                return document.title + &#x27;.html&#x27;;\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( &#x27;textarea&#x27; );\n",
       "        textArea.textContent = JSON.stringify( axis ) + &#x27;,&#x27; + angle;\n",
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       "        var m = document.getElementById( &#x27;menu-message&#x27; );\n",
       "        m.innerHTML = &#x27;Viewpoint copied to clipboard&#x27;;\n",
       "        m.style.display = &#x27;block&#x27;;\n",
       "        setTimeout( function() { m.style.display = &#x27;none&#x27;; }, 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(&#x27;textarea&#x27;);\n",
       "        var cam_position = JSON.stringify(pos_r);\n",
       "        textArea.textContent = &#x27;,camera_position=&#x27; + cam_position;\n",
       "        textArea.style.csstext = &#x27;position: absolute; top: -100%&#x27;;\n",
       "        document.body.append( textArea );\n",
       "        textArea.select();\n",
       "        document.execCommand( &#x27;copy&#x27; );\n",
       "\n",
       "        var m = document.getElementById( &#x27;menu-message&#x27; );\n",
       "        m.innerHTML = &#x27;Camera position &#x27;+ cam_position+&#x27; copied to clipboard&#x27;;\n",
       "        m.style.display = &#x27;block&#x27;;\n",
       "        setTimeout( function() { m.style.display = &#x27;none&#x27;; }, 2000 );\n",
       "\n",
       "    }\n",
       "\n",
       "&lt;/script&gt;\n",
       "\n",
       "&lt;div id=&quot;menu-container&quot; onclick=&quot;toggleMenu()&quot;&gt;&amp;#x24d8;\n",
       "&lt;div id=&quot;menu-message&quot;&gt;&lt;/div&gt;\n",
       "&lt;div id=&quot;menu-content&quot;&gt;\n",
       "&lt;div onclick=&quot;saveAsPNG()&quot;&gt;Save as PNG&lt;/div&gt;\n",
       "&lt;div onclick=&quot;saveAsHTML()&quot;&gt;Save as HTML&lt;/div&gt;\n",
       "&lt;div onclick=&quot;getCamera()&quot;&gt;Get camera&lt;/div&gt;\n",
       "&lt;div onclick=&quot;getViewpoint()&quot;&gt;Get viewpoint&lt;/div&gt;\n",
       "&lt;div&gt;Close Menu&lt;/div&gt;\n",
       "&lt;/div&gt;&lt;/div&gt;\n",
       "\n",
       "\n",
       "&lt;/body&gt;\n",
       "&lt;/html&gt;\n",
       "\"\n",
       "        width=\"100%\"\n",
       "        height=\"400\"\n",
       "        style=\"border: 0;\">\n",
       "</iframe>\n"
      ],
      "text/plain": [
       "Graphics3d Object"
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   ],
   "source": [
    "# To explain the expansion of the k-dim mle to n-dim mle by using evals\n",
    "# We can use a 1D example to explain\n",
    "# the evals is [0, 35] which lying on the 1D line\n",
    "mle_evals = [0, 35]\n",
    "\n",
    "vertices = [\n",
    "    ([0,0,0], mle_evals[0]), ([1,0,0], mle_evals[1]),\n",
    "]\n",
    "\n",
    "g = Graphics()\n",
    "\n",
    "for x in [0,1]:\n",
    "    g += line3d([(0,0,0), (1,0,0)], color='black')\n",
    "\n",
    "for (x,y,z), value in vertices:\n",
    "    g += text3d(f\"({x})={value}\", (x,y,z), fontsize=15)\n",
    "\n",
    "note = \"This line visualization represents a multilinear extension, with each vertex storing the corresponding function value.\"\n",
    "text_position = (0.5, 0.5, -0.5)  # x和y居中，z在立方体下方\n",
    "g += text3d(note, text_position, fontsize=15)\n",
    "show(g, aspect_ratio=(1, 1, 1), ticks=[[0,2,1], [0,2,1], [0,2,1]], frame=False)\n",
    "\n",
    "# Then expand the 1D mle to 2D mle, copy the 1D evals to next dimension\n",
    "vertices = [\n",
    "    ([0,0,0], mle_evals[0]), ([1,0,0], mle_evals[1]),\n",
    "    ([0,1,0], mle_evals[0]), ([1,1,0], mle_evals[1]),\n",
    "]\n",
    "\n",
    "g = Graphics()\n",
    "\n",
    "for x in [0,1]:\n",
    "    for y in [0,1]:\n",
    "        g += line3d([(1-x,y,0), (x,y,0)], color='black')\n",
    "\n",
    "for x in [0,1]:\n",
    "    for y in [0,1]:\n",
    "        g += line3d([(x,1-y,0), (x,y,0)], color='red', linestyle='dotted')\n",
    "\n",
    "for (x,y,z), value in vertices:\n",
    "    g += text3d(f\"({x},{y})={value}\", (x,y,z), fontsize=15)\n",
    "\n",
    "note = \"The second line is the copy of the first line, now it's a 2D plane\"\n",
    "text_position = (0.5, 0.5, -0.5)  # x和y居中，z在立方体下方\n",
    "g += text3d(note, text_position, fontsize=15)\n",
    "show(g, aspect_ratio=(1, 1, 1), ticks=[[0,2,1], [0,2,1], [0,2,1]], frame=False)\n",
    "\n",
    "# Then expand the 2D plane to 3D cube by copying the 2D plane to next dimension\n",
    "vertices = [\n",
    "    ([0,0,0], mle_evals[0]), ([1,0,0], mle_evals[1]),\n",
    "    ([0,1,0], mle_evals[0]), ([1,1,0], mle_evals[1]),\n",
    "    ([0,0,1], mle_evals[0]), ([1,0,1], mle_evals[1]),\n",
    "    ([0,1,1], mle_evals[0]), ([1,1,1], mle_evals[1]),\n",
    "]\n",
    "\n",
    "g = Graphics()\n",
    "\n",
    "for x in [0,1]:\n",
    "    for y in [0,1]:\n",
    "        g += line3d([(x,y,0), (x,y,1)], color='red')\n",
    "for x in [0,1]:\n",
    "    for z in [0,1]:\n",
    "        g += line3d([(x,0,z), (x,1,z)], color='black')\n",
    "for y in [0,1]:\n",
    "    for z in [0,1]:\n",
    "        g += line3d([(0,y,z), (1,y,z)], color='black')\n",
    "\n",
    "\n",
    "for (x,y,z), value in vertices:\n",
    "    g += text3d(f\"({x},{y},{z})={value}\", (x,y,z), fontsize=15)\n",
    "\n",
    "note = \"The top plane is the copy of the bottom plane, now it's a 3D cube\"\n",
    "text_position = (0.5, 0.5, -0.5)  # x和y居中，z在立方体下方\n",
    "g += text3d(note, text_position, fontsize=15)\n",
    "show(g, aspect_ratio=(1, 1, 1), ticks=[[0,2,1], [0,2,1], [0,2,1]], frame=False)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's examine $q_0$. The verifier has the commitment of $U(q_0)$, but how can they obtain $U(q_0X_0)$ independently?\n",
    "\n",
    "$U(q_0)$ is a 0-dimensional number. Multiplying it by $X_0$ is equivalent to padding $U(q_0)$ with a leading zero.\n",
    "\n",
    "At this point, the verifier has $[0, U(q_0)]$.\n",
    "\n",
    "Now, let's extend this to 3 dimensions: $[0, U(q_0), 0, U(q_0), 0, U(q_0), 0, U(q_0)]$\n",
    "Notice the pattern? It's constructed through padding and duplication. We'll call this the period function $\\Phi(X)$.\n",
    "\n",
    "This function transforms $0 + U(q_0)X$ into $0 + U(q_0)X + 0 + U(q_0)X^3 + 0 + U(q_0)X^5 + 0 + U(q_0)X^7$\n",
    "We can rewrite this equation as: $(0 + U(q_0)X) (1 + X^2 + X^4 + X^6)$\n",
    "\n",
    "The $U(q_0)X$ term causes a right shift for $U(q_0)$, resulting in a leading zero padding. For a k-dimensional polynomial, this shift would be represented by $X^{2^k}$.\n",
    "\n",
    "We call this expression $(1 + X^2 + X^4 + X^6)$ the period function, denoted as $\\Phi(X)$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "vscode": {
     "languageId": "python"
    }
   },
   "outputs": [],
   "source": [
    "def Phi(x, k, n):\n",
    "    result = 0\n",
    "    x = x^(2^(k))\n",
    "    for i in range(2**(n-k)):\n",
    "        result = result + x^(i)\n",
    "    return result\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "vscode": {
     "languageId": "python"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "U(q0X0)_k: 35*X\n",
      "U(q0X0)_n: 35*X^7 + 35*X^5 + 35*X^3 + 35*X\n",
      "U(q0X0)_n/U(q0)_k: X^7 + X^5 + X^3 + X\n",
      "Phi(X, 1, 3): X^6 + X^4 + X^2 + 1\n",
      "Shift(X) * Phi(X, 1, 3): X^7 + X^5 + X^3 + X\n",
      "U(q1X1)_k: 35*X^3 + 22*X^2\n",
      "U(q1X1)_n: 35*X^7 + 22*X^6 + 35*X^3 + 22*X^2\n",
      "U(q1X1)_n/U(q1)_k: X^6 + X^2\n",
      "Phi(X, 2, 3): X^4 + 1\n",
      "Shift(X^2) * Phi(X, 2, 3): X^6 + X^2\n"
     ]
    }
   ],
   "source": [
    "\n",
    "q0X0 = MLE(q0*X0)\n",
    "q0X0_uni = q0X0.to_uni()\n",
    "print(f\"U(q0X0)_k: {q0X0_uni}\")\n",
    "q0X0_ext_3_uni = q0X0.extend(3).to_uni()\n",
    "print(f\"U(q0X0)_n: {q0X0_ext_3_uni}\")\n",
    "print(f\"U(q0X0)_n/U(q0)_k: {q0X0_ext_3_uni/q0_uni}\")\n",
    "print(f\"Phi(X, 1, 3): {Phi(X, 1, 3)}\")\n",
    "print(f\"Shift(X) * Phi(X, 1, 3): {X * Phi(X, 1, 3)}\")\n",
    "\n",
    "q1X1 = MLE(q1*X1)\n",
    "q1X1_uni = q1X1.to_uni()\n",
    "print(f\"U(q1X1)_k: {q1X1_uni}\")\n",
    "q1X1_ext_3_uni = q1X1.extend(3).to_uni()\n",
    "print(f\"U(q1X1)_n: {q1X1_ext_3_uni}\")\n",
    "print(f\"U(q1X1)_n/U(q1)_k: {q1X1_ext_3_uni/q1_uni}\")\n",
    "print(f\"Phi(X, 2, 3): {Phi(X, 2, 3)}\")\n",
    "print(f\"Shift(X^2) * Phi(X, 2, 3): {X^2 * Phi(X, 2, 3)}\")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The verifier can now construct the transformation by applying a shift of $X^d$ and duplicating with $Phi(X)$ on $U(q)$.\n",
    "\n",
    "This process mirrors the padding and duplication performed on the original Multi-Linear Extension (MLE). As a result, the verifier can recover the original extension by applying the shift with $X^d$ and $Phi(X)$ on $U(q)$, which is equivalent to the original MLE.\n",
    "\n",
    "With this understanding of the mapping from MLE to Univariate Polynomial and how the verifier can construct the dimension extension without knowing the original MLE, we can now proceed to apply the Kate-Zaverucha-Goldberg (KZG) commitment scheme to it.\n"
   ]
  }
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