{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "0d38eab6-bff0-4243-9a86-0d5c7278681c",
   "metadata": {},
   "source": [
    "# Working with Vectors in Sage"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3d04f269-03ed-4b3d-a406-04f68776fb0a",
   "metadata": {},
   "source": [
    "## Find the angle and radius of a vector in component form."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "a04f588f-ae21-4231-95b9-027cae7da5f5",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "For the position vector <3,4>, the angle with the x axis is 0.93 radians or 53.13 degrees\n"
     ]
    }
   ],
   "source": [
    "from sympy import deg\n",
    "angle_rad = N(atan(4/3))\n",
    "angle_deg = N(deg(angle_rad))\n",
    "\n",
    "# Vector <3,4>, angle\n",
    "print(\"For the position vector <3,4>, the angle with the x axis is {0:.2f} radians or {1:.2f} degrees\".format(angle_rad, angle_deg))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "cc7f3091-3b73-45f8-bd93-fed5fc936725",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The radius of the position vector <3, 4> is  5\n"
     ]
    }
   ],
   "source": [
    "# For the radius, simply use the pythagorean theorem: works for \n",
    "\n",
    "r = sqrt(3**2 + 4**2)\n",
    "print(\"The radius of the position vector <3, 4> is \", r)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "563c767d-303e-448d-9f99-17a74bdac232",
   "metadata": {},
   "source": [
    "Try a problem (Moer 11.1, Question 5)\n",
    "\n",
    "Given a vector $\\vec{u} = \\langle 2, -2\\rangle$, find the magnitude and the angle (in radians):\n",
    "\n",
    "Magnitude is $\\sqrt{2^2 + 2^2} = \\sqrt{8} = 2 \\sqrt{2}$  $\\tan\\theta = \\frac{b}{a} = \\tan-\\frac{2}{2}$\n",
    "\n",
    "Angle = $\\frac{7pi}{4}$\n",
    "\n",
    "And Moer says we're good!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "75ae2c49-4b0c-48f8-8cca-08566d0583b0",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-1/4*pi\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# No help!\n",
    "print(atan(-1))\n",
    "\n",
    "# plot it -- better!\n",
    "v1 = vector((2, -2))\n",
    "plot(v1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "15f8f0fb-e8f2-4124-9955-e74f295d35ee",
   "metadata": {},
   "source": [
    "# From \"polar form to component form\n",
    "\n",
    "Given a vector $\\vec{v}$ with an magnititude $\\|\\vec{v}\\|$ making an angle $\\theta$ with the positive x axis, then:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\boldsymbol{\\vec{v} = \\langle\\|\\vec{v}\\|\\cos\\theta, |\\vec{v}\\|\\sin\\theta\\rangle = |\\vec{v}\\| \\langle\\cos\\theta, \\sin\\theta\\rangle}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Here let's work a problem with that, Moer #11.1, Question 4\n",
    "\n",
    "A vector with magnitude 10 points in a direction 340 degrees counterclockwise from the positive x axis."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "26ebdb8d-7f1a-4ae4-81bb-ca6f8d34b7b7",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "17/9*pi\n",
      "Answer is <9.4, -3.42>\n"
     ]
    }
   ],
   "source": [
    "# Get degrees in radians:\n",
    "from sympy import rad\n",
    "angle = rad(340)\n",
    "print(angle)\n",
    "horiz = round((N(cos(angle)) * 10), 2)\n",
    "vert = round((N(sin(angle)) * 10), 2)\n",
    "print(\"Answer is <{0}, {1}>\".format(horiz, vert))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "079f5dc0-7c82-46ac-9682-89d83ec49a87",
   "metadata": {},
   "source": [
    "Check if it makes sense visually (spoiler alert, Moer marked it correct)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "d8abb597-4eb6-4941-bcdc-95aa67a921b2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = plot(vector((horiz, vert)))\n",
    "p.show(aspect_ratio=1, xmin=-4, xmax=12, ymin=-12, ymax=4)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8794f089-3670-4438-953b-d477380b6f18",
   "metadata": {},
   "source": [
    "# Finding unit vector in a given direction\n",
    "\n",
    "Given a vector $\\vec{v} = \\langle u,v \\rangle$\n",
    "\n",
    "1. Find the magnitiude, $\\|\\vec{v}\\|$\n",
    "2. Then find our new vector with scalar multiplication: vector $\\vec{u} =  \\frac{1}{\\|\\vec{v}\\|} \\cdot \\vec{v}$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "cab33796-3522-4fa1-a835-624aab7b94cb",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Terms:   -4 2\n",
      "2*sqrt(5)\n",
      "1/10*sqrt(5)\n",
      "1\n",
      "1\n"
     ]
    }
   ],
   "source": [
    "# Do Moer 11.1 Q8:  \n",
    "# Find the unit vector in the direction of <−4, 2>\n",
    "# My answer differed from what I found by hand, but only because of \n",
    "# Sage moving the radical to the numerator.\n",
    "\n",
    "v1 = vector((-4,2))\n",
    "print(\"Terms:  \", v1[0],  v1[1])\n",
    "magnitude = sqrt((v1[0])**2 + (v1[1])**2)\n",
    "print(magnitude)\n",
    "inverse = 1/magnitude\n",
    "print(inverse)\n",
    "v2 = inverse * v1\n",
    "v2\n",
    "\n",
    "\n",
    "# Check magnitude of result:\n",
    "v2_mag = sqrt((v2[0])**2 + (v2[1])**2)\n",
    "print(v2_mag)\n",
    "\n",
    "# Check answer done by hand\n",
    "v3 = vector((-2/sqrt(5), 1/sqrt(5)))\n",
    "v3_mag = sqrt((v3[0])**2 + (v3[1])**2)\n",
    "print(v3_mag)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "741b8f50-905e-4263-8186-fdbd782aa666",
   "metadata": {},
   "source": [
    "### Do Moer 11.1 Q8 by hand  \n",
    "Find the unit vector in the direction of <−4, 2>\n",
    "\n",
    "$\\sqrt{20} = 2 \\sqrt{5}$\n",
    "\n",
    "$\\frac{1}{2 \\sqrt{5}} \\cdot -4 = \\frac{-2}{\\sqrt{5}}$ \n",
    "\n",
    "$\\frac{1}{2 \\sqrt{5}} \\cdot 2 = \\frac{1}{\\sqrt{5}}$ \n",
    "\n",
    "So the unit vecor is as shown below:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\langle - \\frac{2}{\\sqrt{5}}. \\frac{1}{\\sqrt{5}} \\rangle\n",
    "\\end{align}\n",
    "$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0c0620f3-3337-49ff-8ea7-178132adb43e",
   "metadata": {},
   "source": [
    "## Instantiating vectors in Sage, finding dot product and cross product."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "52f68e46-f607-48c0-bc6e-4a05bf77dc0b",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True\n",
      "Dot product of V1 and V10:  26\n",
      "Cross product of V1 and V10:  (8, -4, 0)\n"
     ]
    }
   ],
   "source": [
    "V1 = vector([1,2,3])\n",
    "\n",
    "# Can instantiate using tuple or list, among other iterables.\n",
    "V10 = vector([1,2,7])\n",
    "V11 = vector((1,2,7))\n",
    "print(V10 == V11)\n",
    "\n",
    "print(\"Dot product of V1 and V10: \", V1 * V10)\n",
    "print(\"Cross product of V1 and V10: \", V1.cross_product(V10))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "1ff7b94a-1a32-41f8-923f-9a8c0049bbce",
   "metadata": {},
   "outputs": [
    {
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       "\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;:0.0}, {&quot;x&quot;:2.0, &quot;y&quot;:2.0, &quot;z&quot;:4.0}]; // 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 = [];\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 = [{&quot;point&quot;: [0.0, 0.0, 0.0], &quot;size&quot;: 30.0, &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}];\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 = [];\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 = [{&quot;vertices&quot;: [{&quot;x&quot;: 0.9759464882278817, &quot;y&quot;: 1.9518929764557633, &quot;z&quot;: 2.9278394646836454}, {&quot;x&quot;: 0.9848350148430511, &quot;y&quot;: 1.9537390693039487, &quot;z&quot;: 2.9236458939131325}, {&quot;x&quot;: 0.9751536231308308, &quot;y&quot;: 1.9603072462616615, &quot;z&quot;: 2.9224942398453972}, {&quot;x&quot;: 0.9822327648830154, &quot;y&quot;: 1.9446196547764318, &quot;z&quot;: 2.9305929202514887}, {&quot;x&quot;: 0.9709430942482699, &quot;y&quot;: 1.9455517235986397, &quot;z&quot;: 2.9337347645815988}, {&quot;x&quot;: 0.9665679440342413, &quot;y&quot;: 1.955247188338135, &quot;z&quot;: 2.9287295048266113}, {&quot;x&quot;: 0.008888526615169437, &quot;y&quot;: 0.0018460928481852752, &quot;z&quot;: -0.004193570770513328}, {&quot;x&quot;: -0.0007928650970509072, &quot;y&quot;: 0.008414269805898185, &quot;z&quot;: -0.005345224838248487}, {&quot;x&quot;: 0.006286276655133676, &quot;y&quot;: -0.007273321679331586, &quot;z&quot;: 0.002753455567843164}, {&quot;x&quot;: -0.005003393979611823, &quot;y&quot;: -0.006341252857123658, &quot;z&quot;: 0.005895299897953045}, {&quot;x&quot;: -0.00937854419364038, &quot;y&quot;: 0.0033542118823717826, &quot;z&quot;: 0.000890040142965605}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 0.0}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 2, 5], [2, 1, 6, 7], [1, 3, 8, 6], [3, 4, 9, 8], [4, 5, 10, 9], [5, 2, 7, 10], [7, 6, 11], [6, 8, 11], [8, 9, 11], [9, 10, 11], [10, 7, 11]], &quot;color&quot;: &quot;#6666ff&quot;, &quot;opacity&quot;: 1.0}, {&quot;vertices&quot;: [{&quot;x&quot;: 0.9999999999999999, &quot;y&quot;: 1.9999999999999998, &quot;z&quot;: 3.0}, {&quot;x&quot;: 1.0026120680733899, &quot;y&quot;: 1.9574312550003192, &quot;z&quot;: 2.9152587523721056}, {&quot;x&quot;: 0.973567892936729, &quot;y&quot;: 1.977135785873458, &quot;z&quot;: 2.9118037901689}, {&quot;x&quot;: 0.9948053181932827, &quot;y&quot;: 1.9300730114177687, &quot;z&quot;: 2.936099831387175}, {&quot;x&quot;: 0.9609363062890461, &quot;y&quot;: 1.9328692178843923, &quot;z&quot;: 2.9455253643775046}, {&quot;x&quot;: 0.9478108556469605, &quot;y&quot;: 1.9619556121028787, &quot;z&quot;: 2.9305095851125422}, {&quot;x&quot;: 0.9759464882278817, &quot;y&quot;: 1.9518929764557633, &quot;z&quot;: 2.9278394646836454}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 2, 5], [2, 1, 6], [1, 3, 6], [3, 4, 6], [4, 5, 6], [5, 2, 6]], &quot;color&quot;: &quot;#6666ff&quot;, &quot;opacity&quot;: 1.0}, {&quot;vertices&quot;: [{&quot;x&quot;: 1.9597507764050042, &quot;y&quot;: 0.0, &quot;z&quot;: 3.919501552810008}, {&quot;x&quot;: 1.9682572844885244, &quot;y&quot;: 0.0030901699437494747, &quot;z&quot;: 3.9152482987682475}, {&quot;x&quot;: 1.9597507764050042, &quot;y&quot;: 0.01, &quot;z&quot;: 3.919501552810008}, {&quot;x&quot;: 1.9650080875261955, &quot;y&quot;: -0.008090169943749474, &quot;z&quot;: 3.9168728972494122}, {&quot;x&quot;: 1.9544934652838128, &quot;y&quot;: -0.008090169943749474, &quot;z&quot;: 3.9221302083706036}, {&quot;x&quot;: 1.9512442683214837, &quot;y&quot;: 0.0030901699437494725, &quot;z&quot;: 3.923754806851768}, {&quot;x&quot;: 0.0085065080835204, &quot;y&quot;: 0.0030901699437494747, &quot;z&quot;: -0.0042532540417602}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.01, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.005257311121191337, &quot;y&quot;: -0.008090169943749474, &quot;z&quot;: -0.002628655560595669}, {&quot;x&quot;: -0.005257311121191335, &quot;y&quot;: -0.008090169943749474, &quot;z&quot;: 0.002628655560595668}, {&quot;x&quot;: -0.0085065080835204, &quot;y&quot;: 0.0030901699437494725, &quot;z&quot;: 0.0042532540417602}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 0.0}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 2, 5], [2, 1, 6, 7], [1, 3, 8, 6], [3, 4, 9, 8], [4, 5, 10, 9], [5, 2, 7, 10], [7, 6, 11], [6, 8, 11], [8, 9, 11], [9, 10, 11], [10, 7, 11]], &quot;color&quot;: &quot;#6666ff&quot;, &quot;opacity&quot;: 1.0}, {&quot;vertices&quot;: [{&quot;x&quot;: 2.0000000000000004, &quot;y&quot;: 0.0, &quot;z&quot;: 4.0}, {&quot;x&quot;: 1.9852703006555654, &quot;y&quot;: 0.009270509831248424, &quot;z&quot;: 3.9067417906847273}, {&quot;x&quot;: 1.9597507764050042, &quot;y&quot;: 0.03, &quot;z&quot;: 3.919501552810008}, {&quot;x&quot;: 1.9755227097685781, &quot;y&quot;: -0.02427050983124842, &quot;z&quot;: 3.911615586128221}, {&quot;x&quot;: 1.9439788430414302, &quot;y&quot;: -0.024270509831248423, &quot;z&quot;: 3.927387519491795}, {&quot;x&quot;: 1.934231252154443, &quot;y&quot;: 0.009270509831248417, &quot;z&quot;: 3.9322613149352885}, {&quot;x&quot;: 1.9597507764050042, &quot;y&quot;: 0.0, &quot;z&quot;: 3.919501552810008}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 2, 5], [2, 1, 6], [1, 3, 6], [3, 4, 6], [4, 5, 6], [5, 2, 6]], &quot;color&quot;: &quot;#6666ff&quot;, &quot;opacity&quot;: 1.0}];\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"
    }
   ],
   "source": [
    "# Lets get a simple plot going:\n",
    "V1 = vector([1,2,3])\n",
    "V2 = vector([2,0, 4])\n",
    "origin = point((0,0), color='black', size=30)\n",
    "pl = plot(V1) + plot(V2) + origin\n",
    "pl.show(aspect_ratio=1, xmax=10, xmin=-5, ymax=10, ymin=-5, zmax=10, zmin=-5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "fb363128-621b-4758-bceb-0fef999f8f74",
   "metadata": {},
   "outputs": [
    {
     "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;:-0.08774351209636412, &quot;y&quot;:-0.08108719811121771, &quot;z&quot;:-0.0899999999999997}, {&quot;x&quot;:5.3, &quot;y&quot;:5.3, &quot;z&quot;:5.3}]; // 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;: 14.0, &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;x&quot;, &quot;x&quot;: 5.3, &quot;y&quot;: 0.0, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 14.0, &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;y&quot;, &quot;x&quot;: 0.0, &quot;y&quot;: 5.3, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 14.0, &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;z&quot;, &quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 5.3}];\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 = [{&quot;point&quot;: [0.0, 0.0, 0.0], &quot;size&quot;: 30.0, &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}];\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 = [];\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 = [{&quot;vertices&quot;: [{&quot;x&quot;: 0.9759464882278817, &quot;y&quot;: 1.9518929764557633, &quot;z&quot;: 2.9278394646836454}, {&quot;x&quot;: 0.9848350148430511, &quot;y&quot;: 1.9537390693039487, &quot;z&quot;: 2.9236458939131325}, {&quot;x&quot;: 0.9751536231308308, &quot;y&quot;: 1.9603072462616615, &quot;z&quot;: 2.9224942398453972}, {&quot;x&quot;: 0.9822327648830154, &quot;y&quot;: 1.9446196547764318, &quot;z&quot;: 2.9305929202514887}, {&quot;x&quot;: 0.9709430942482699, &quot;y&quot;: 1.9455517235986397, &quot;z&quot;: 2.9337347645815988}, {&quot;x&quot;: 0.9665679440342413, &quot;y&quot;: 1.955247188338135, &quot;z&quot;: 2.9287295048266113}, {&quot;x&quot;: 0.008888526615169437, &quot;y&quot;: 0.0018460928481852752, &quot;z&quot;: -0.004193570770513328}, {&quot;x&quot;: -0.0007928650970509072, &quot;y&quot;: 0.008414269805898185, &quot;z&quot;: -0.005345224838248487}, {&quot;x&quot;: 0.006286276655133676, &quot;y&quot;: -0.007273321679331586, &quot;z&quot;: 0.002753455567843164}, {&quot;x&quot;: -0.005003393979611823, &quot;y&quot;: -0.006341252857123658, &quot;z&quot;: 0.005895299897953045}, {&quot;x&quot;: -0.00937854419364038, &quot;y&quot;: 0.0033542118823717826, &quot;z&quot;: 0.000890040142965605}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 0.0}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 2, 5], [2, 1, 6, 7], [1, 3, 8, 6], [3, 4, 9, 8], [4, 5, 10, 9], [5, 2, 7, 10], [7, 6, 11], [6, 8, 11], [8, 9, 11], [9, 10, 11], [10, 7, 11]], &quot;color&quot;: &quot;#0000ff&quot;, &quot;opacity&quot;: 1.0}, {&quot;vertices&quot;: [{&quot;x&quot;: 0.9999999999999999, &quot;y&quot;: 1.9999999999999998, &quot;z&quot;: 3.0}, {&quot;x&quot;: 1.0026120680733899, &quot;y&quot;: 1.9574312550003192, &quot;z&quot;: 2.9152587523721056}, {&quot;x&quot;: 0.973567892936729, &quot;y&quot;: 1.977135785873458, &quot;z&quot;: 2.9118037901689}, {&quot;x&quot;: 0.9948053181932827, &quot;y&quot;: 1.9300730114177687, &quot;z&quot;: 2.936099831387175}, {&quot;x&quot;: 0.9609363062890461, &quot;y&quot;: 1.9328692178843923, &quot;z&quot;: 2.9455253643775046}, {&quot;x&quot;: 0.9478108556469605, &quot;y&quot;: 1.9619556121028787, &quot;z&quot;: 2.9305095851125422}, {&quot;x&quot;: 0.9759464882278817, &quot;y&quot;: 1.9518929764557633, &quot;z&quot;: 2.9278394646836454}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 2, 5], [2, 1, 6], [1, 3, 6], [3, 4, 6], [4, 5, 6], [5, 2, 6]], &quot;color&quot;: &quot;#0000ff&quot;, &quot;opacity&quot;: 1.0}, {&quot;vertices&quot;: [{&quot;x&quot;: 1.9597507764050042, &quot;y&quot;: 0.0, &quot;z&quot;: 3.919501552810008}, {&quot;x&quot;: 1.9682572844885244, &quot;y&quot;: 0.0030901699437494747, &quot;z&quot;: 3.9152482987682475}, {&quot;x&quot;: 1.9597507764050042, &quot;y&quot;: 0.01, &quot;z&quot;: 3.919501552810008}, {&quot;x&quot;: 1.9650080875261955, &quot;y&quot;: -0.008090169943749474, &quot;z&quot;: 3.9168728972494122}, {&quot;x&quot;: 1.9544934652838128, &quot;y&quot;: -0.008090169943749474, &quot;z&quot;: 3.9221302083706036}, {&quot;x&quot;: 1.9512442683214837, &quot;y&quot;: 0.0030901699437494725, &quot;z&quot;: 3.923754806851768}, {&quot;x&quot;: 0.0085065080835204, &quot;y&quot;: 0.0030901699437494747, &quot;z&quot;: -0.0042532540417602}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.01, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.005257311121191337, &quot;y&quot;: -0.008090169943749474, &quot;z&quot;: -0.002628655560595669}, {&quot;x&quot;: -0.005257311121191335, &quot;y&quot;: -0.008090169943749474, &quot;z&quot;: 0.002628655560595668}, {&quot;x&quot;: -0.0085065080835204, &quot;y&quot;: 0.0030901699437494725, &quot;z&quot;: 0.0042532540417602}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 0.0}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 2, 5], [2, 1, 6, 7], [1, 3, 8, 6], [3, 4, 9, 8], [4, 5, 10, 9], [5, 2, 7, 10], [7, 6, 11], [6, 8, 11], [8, 9, 11], [9, 10, 11], [10, 7, 11]], &quot;color&quot;: &quot;#ff0000&quot;, &quot;opacity&quot;: 1.0}, {&quot;vertices&quot;: [{&quot;x&quot;: 2.0000000000000004, &quot;y&quot;: 0.0, &quot;z&quot;: 4.0}, {&quot;x&quot;: 1.9852703006555654, &quot;y&quot;: 0.009270509831248424, &quot;z&quot;: 3.9067417906847273}, {&quot;x&quot;: 1.9597507764050042, &quot;y&quot;: 0.03, &quot;z&quot;: 3.919501552810008}, {&quot;x&quot;: 1.9755227097685781, &quot;y&quot;: -0.02427050983124842, &quot;z&quot;: 3.911615586128221}, {&quot;x&quot;: 1.9439788430414302, &quot;y&quot;: -0.024270509831248423, &quot;z&quot;: 3.927387519491795}, {&quot;x&quot;: 1.934231252154443, &quot;y&quot;: 0.009270509831248417, &quot;z&quot;: 3.9322613149352885}, {&quot;x&quot;: 1.9597507764050042, &quot;y&quot;: 0.0, &quot;z&quot;: 3.919501552810008}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 2, 5], [2, 1, 6], [1, 3, 6], [3, 4, 6], [4, 5, 6], [5, 2, 6]], &quot;color&quot;: &quot;#ff0000&quot;, &quot;opacity&quot;: 1.0}, {&quot;vertices&quot;: [{&quot;x&quot;: 4.73, &quot;y&quot;: 0.0, &quot;z&quot;: 2.8962896799834905e-16}, {&quot;x&quot;: 4.73, &quot;y&quot;: 0.009270509831248424, &quot;z&quot;: -0.028531695488854317}, {&quot;x&quot;: 4.73, &quot;y&quot;: 0.03, &quot;z&quot;: 2.8962896799834905e-16}, {&quot;x&quot;: 4.73, &quot;y&quot;: -0.02427050983124842, &quot;z&quot;: -0.017633557568773908}, {&quot;x&quot;: 4.73, &quot;y&quot;: -0.024270509831248423, &quot;z&quot;: 0.017633557568774477}, {&quot;x&quot;: 4.73, &quot;y&quot;: 0.009270509831248417, &quot;z&quot;: 0.028531695488854897}, {&quot;x&quot;: 1.7470624777336386e-18, &quot;y&quot;: 0.009270509831248424, &quot;z&quot;: -0.028531695488854605}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.03, &quot;z&quot;: 0.0}, {&quot;x&quot;: 1.079743991708995e-18, &quot;y&quot;: -0.02427050983124842, &quot;z&quot;: -0.017633557568774196}, {&quot;x&quot;: -1.0797439917089947e-18, &quot;y&quot;: -0.024270509831248423, &quot;z&quot;: 0.01763355756877419}, {&quot;x&quot;: -1.7470624777336386e-18, &quot;y&quot;: 0.009270509831248417, &quot;z&quot;: 0.02853169548885461}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 0.0}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 2, 5], [2, 1, 6, 7], [1, 3, 8, 6], [3, 4, 9, 8], [4, 5, 10, 9], [5, 2, 7, 10], [7, 6, 11], [6, 8, 11], [8, 9, 11], [9, 10, 11], [10, 7, 11]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}, {&quot;vertices&quot;: [{&quot;x&quot;: 5.0, &quot;y&quot;: 0.0, &quot;z&quot;: 3.061616997868383e-16}, {&quot;x&quot;: 4.73, &quot;y&quot;: 0.05611408216728602, &quot;z&quot;: -0.0703648334221224}, {&quot;x&quot;: 4.73, &quot;y&quot;: 0.09, &quot;z&quot;: 2.8962896799834905e-16}, {&quot;x&quot;: 4.73, &quot;y&quot;: -0.02002688405606829, &quot;z&quot;: -0.08774351209636383}, {&quot;x&quot;: 4.73, &quot;y&quot;: -0.08108719811121771, &quot;z&quot;: -0.03904953652057995}, {&quot;x&quot;: 4.73, &quot;y&quot;: -0.08108719811121771, &quot;z&quot;: 0.03904953652058051}, {&quot;x&quot;: 4.73, &quot;y&quot;: -0.02002688405606831, &quot;z&quot;: 0.08774351209636441}, {&quot;x&quot;: 4.73, &quot;y&quot;: 0.056114082167286, &quot;z&quot;: 0.07036483342212298}, {&quot;x&quot;: 4.73, &quot;y&quot;: 0.0, &quot;z&quot;: 2.8962896799834905e-16}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 6, 5], [0, 7, 6], [0, 2, 7], [2, 1, 8], [1, 3, 8], [3, 4, 8], [4, 5, 8], [5, 6, 8], [6, 7, 8], [7, 2, 8]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}, {&quot;vertices&quot;: [{&quot;x&quot;: 0.0, &quot;y&quot;: 4.73, &quot;z&quot;: 2.8962896799834905e-16}, {&quot;x&quot;: 0.028531695488854605, &quot;y&quot;: 4.73, &quot;z&quot;: -0.009270509831248134}, {&quot;x&quot;: 0.0, &quot;y&quot;: 4.73, &quot;z&quot;: -0.02999999999999971}, {&quot;x&quot;: 0.017633557568774196, &quot;y&quot;: 4.73, &quot;z&quot;: 0.024270509831248708}, {&quot;x&quot;: -0.01763355756877419, &quot;y&quot;: 4.73, &quot;z&quot;: 0.02427050983124871}, {&quot;x&quot;: -0.02853169548885461, &quot;y&quot;: 4.73, &quot;z&quot;: -0.009270509831248127}, {&quot;x&quot;: 0.028531695488854605, &quot;y&quot;: 5.676550095651226e-19, &quot;z&quot;: -0.009270509831248424}, {&quot;x&quot;: 0.0, &quot;y&quot;: 1.8369701987210296e-18, &quot;z&quot;: -0.03}, {&quot;x&quot;: 0.017633557568774196, &quot;y&quot;: -1.4861401089256373e-18, &quot;z&quot;: 0.02427050983124842}, {&quot;x&quot;: -0.01763355756877419, &quot;y&quot;: -1.4861401089256375e-18, &quot;z&quot;: 0.024270509831248423}, {&quot;x&quot;: -0.02853169548885461, &quot;y&quot;: 5.676550095651222e-19, &quot;z&quot;: -0.009270509831248417}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 0.0}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 2, 5], [2, 1, 6, 7], [1, 3, 8, 6], [3, 4, 9, 8], [4, 5, 10, 9], [5, 2, 7, 10], [7, 6, 11], [6, 8, 11], [8, 9, 11], [9, 10, 11], [10, 7, 11]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}, {&quot;vertices&quot;: [{&quot;x&quot;: 0.0, &quot;y&quot;: 5.0, &quot;z&quot;: 3.061616997868383e-16}, {&quot;x&quot;: 0.07036483342212269, &quot;y&quot;: 4.73, &quot;z&quot;: -0.05611408216728573}, {&quot;x&quot;: 0.0, &quot;y&quot;: 4.73, &quot;z&quot;: -0.0899999999999997}, {&quot;x&quot;: 0.08774351209636412, &quot;y&quot;: 4.73, &quot;z&quot;: 0.02002688405606858}, {&quot;x&quot;: 0.03904953652058024, &quot;y&quot;: 4.73, &quot;z&quot;: 0.081087198111218}, {&quot;x&quot;: -0.03904953652058022, &quot;y&quot;: 4.73, &quot;z&quot;: 0.081087198111218}, {&quot;x&quot;: -0.08774351209636412, &quot;y&quot;: 4.73, &quot;z&quot;: 0.0200268840560686}, {&quot;x&quot;: -0.07036483342212269, &quot;y&quot;: 4.73, &quot;z&quot;: -0.05611408216728571}, {&quot;x&quot;: 0.0, &quot;y&quot;: 4.73, &quot;z&quot;: 2.8962896799834905e-16}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 6, 5], [0, 7, 6], [0, 2, 7], [2, 1, 8], [1, 3, 8], [3, 4, 8], [4, 5, 8], [5, 6, 8], [6, 7, 8], [7, 2, 8]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}, {&quot;vertices&quot;: [{&quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 4.73}, {&quot;x&quot;: 0.028531695488854605, &quot;y&quot;: 0.009270509831248424, &quot;z&quot;: 4.73}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.03, &quot;z&quot;: 4.73}, {&quot;x&quot;: 0.017633557568774196, &quot;y&quot;: -0.02427050983124842, &quot;z&quot;: 4.73}, {&quot;x&quot;: -0.01763355756877419, &quot;y&quot;: -0.024270509831248423, &quot;z&quot;: 4.73}, {&quot;x&quot;: -0.02853169548885461, &quot;y&quot;: 0.009270509831248417, &quot;z&quot;: 4.73}, {&quot;x&quot;: 0.028531695488854605, &quot;y&quot;: 0.009270509831248424, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.03, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.017633557568774196, &quot;y&quot;: -0.02427050983124842, &quot;z&quot;: 0.0}, {&quot;x&quot;: -0.01763355756877419, &quot;y&quot;: -0.024270509831248423, &quot;z&quot;: 0.0}, {&quot;x&quot;: -0.02853169548885461, &quot;y&quot;: 0.009270509831248417, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 0.0}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 2, 5], [2, 1, 6, 7], [1, 3, 8, 6], [3, 4, 9, 8], [4, 5, 10, 9], [5, 2, 7, 10], [7, 6, 11], [6, 8, 11], [8, 9, 11], [9, 10, 11], [10, 7, 11]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}, {&quot;vertices&quot;: [{&quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 5.0}, {&quot;x&quot;: 0.07036483342212269, &quot;y&quot;: 0.05611408216728602, &quot;z&quot;: 4.73}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.09, &quot;z&quot;: 4.73}, {&quot;x&quot;: 0.08774351209636412, &quot;y&quot;: -0.02002688405606829, &quot;z&quot;: 4.73}, {&quot;x&quot;: 0.03904953652058024, &quot;y&quot;: -0.08108719811121771, &quot;z&quot;: 4.73}, {&quot;x&quot;: -0.03904953652058022, &quot;y&quot;: -0.08108719811121771, &quot;z&quot;: 4.73}, {&quot;x&quot;: -0.08774351209636412, &quot;y&quot;: -0.02002688405606831, &quot;z&quot;: 4.73}, {&quot;x&quot;: -0.07036483342212269, &quot;y&quot;: 0.056114082167286, &quot;z&quot;: 4.73}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 4.73}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 6, 5], [0, 7, 6], [0, 2, 7], [2, 1, 8], [1, 3, 8], [3, 4, 8], [4, 5, 8], [5, 6, 8], [6, 7, 8], [7, 2, 8]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}];\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"
    }
   ],
   "source": [
    "# Here we asked Gemini to refine it to exclude the box but get our standard axis lines showing:\n",
    "\n",
    "# 1. Define your vectors and origin\n",
    "V1 = vector([1, 2, 3])\n",
    "V2 = vector([2, 0, 4])\n",
    "origin_point = point((0, 0, 0), color='black', size=30) # Corrected to 3D\n",
    "\n",
    "# 2. Create the plots for your vectors\n",
    "plot_v1 = plot(V1, color='blue', legend_label='V1 = [1, 2, 3]')\n",
    "plot_v2 = plot(V2, color='red', legend_label='V2 = [2, 0, 4]')\n",
    "\n",
    "# 3. Define and plot the axes manually\n",
    "axis_len = 5  # How long to draw the axes\n",
    "label_offset = 0.3 # How far to put the label from the arrowhead\n",
    "\n",
    "# Create arrows for X, Y, Z axes\n",
    "x_axis = arrow((0, 0, 0), (axis_len, 0, 0), color='black', width=1.5)\n",
    "y_axis = arrow((0, 0, 0), (0, axis_len, 0), color='black', width=1.5)\n",
    "z_axis = arrow((0, 0, 0), (0, 0, axis_len), color='black', width=1.5)\n",
    "\n",
    "# Create 3D text labels for the axes\n",
    "x_label = text3d(\"x\", (axis_len + label_offset, 0, 0), color='black', fontsize=14)\n",
    "y_label = text3d(\"y\", (0, axis_len + label_offset, 0), color='black', fontsize=14)\n",
    "z_label = text3d(\"z\", (0, 0, axis_len + label_offset), color='black', fontsize=14)\n",
    "\n",
    "# 4. Combine all the plot objects\n",
    "pl = plot_v1 + plot_v2 + origin_point + x_axis + y_axis + z_axis + x_label + y_label + z_label\n",
    "\n",
    "# 5. Show the final plot\n",
    "#    The key is frame=False\n",
    "pl.show(frame=False, aspect_ratio=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "d7931251-c7bf-4b32-86ce-6f2dc16131e8",
   "metadata": {},
   "outputs": [
    {
     "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;:-0.05706339097770922, &quot;y&quot;:-0.3, &quot;z&quot;:-0.0599999999999997}, {&quot;x&quot;:5.3, &quot;y&quot;:5.3, &quot;z&quot;:5.3}]; // 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;: 14.0, &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;x&quot;, &quot;x&quot;: 5.3, &quot;y&quot;: 0.0, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 14.0, &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;y&quot;, &quot;x&quot;: 0.0, &quot;y&quot;: 5.3, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 14.0, &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;z&quot;, &quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 5.3}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 10.0, &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&quot;, &quot;x&quot;: 1.0, &quot;y&quot;: -0.3, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 10.0, &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;2&quot;, &quot;x&quot;: 2.0, &quot;y&quot;: -0.3, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 10.0, &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;3&quot;, &quot;x&quot;: 3.0, &quot;y&quot;: -0.3, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 10.0, &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;4&quot;, &quot;x&quot;: 4.0, &quot;y&quot;: -0.3, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 10.0, &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;5&quot;, &quot;x&quot;: 5.0, &quot;y&quot;: -0.3, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 10.0, &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&quot;, &quot;x&quot;: 0.3, &quot;y&quot;: 1.0, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 10.0, &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;2&quot;, &quot;x&quot;: 0.3, &quot;y&quot;: 2.0, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 10.0, &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;3&quot;, &quot;x&quot;: 0.3, &quot;y&quot;: 3.0, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 10.0, &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;4&quot;, &quot;x&quot;: 0.3, &quot;y&quot;: 4.0, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 10.0, &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;5&quot;, &quot;x&quot;: 0.3, &quot;y&quot;: 5.0, &quot;z&quot;: 0.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 10.0, &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&quot;, &quot;x&quot;: 0.0, &quot;y&quot;: 0.3, &quot;z&quot;: 1.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 10.0, &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;2&quot;, &quot;x&quot;: 0.0, &quot;y&quot;: 0.3, &quot;z&quot;: 2.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 10.0, &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;3&quot;, &quot;x&quot;: 0.0, &quot;y&quot;: 0.3, &quot;z&quot;: 3.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 10.0, &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;4&quot;, &quot;x&quot;: 0.0, &quot;y&quot;: 0.3, &quot;z&quot;: 4.0}, {&quot;color&quot;: &quot;#000000&quot;, &quot;fontSize&quot;: 10.0, &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;5&quot;, &quot;x&quot;: 0.0, &quot;y&quot;: 0.3, &quot;z&quot;: 5.0}];\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 = [{&quot;point&quot;: [0.0, 0.0, 0.0], &quot;size&quot;: 30.0, &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}, {&quot;point&quot;: [1.0, 0.0, 0.0], &quot;size&quot;: 15.0, &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}, {&quot;point&quot;: [2.0, 0.0, 0.0], &quot;size&quot;: 15.0, &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}, {&quot;point&quot;: [3.0, 0.0, 0.0], &quot;size&quot;: 15.0, &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}, {&quot;point&quot;: [4.0, 0.0, 0.0], &quot;size&quot;: 15.0, &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}, {&quot;point&quot;: [5.0, 0.0, 0.0], &quot;size&quot;: 15.0, &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}, {&quot;point&quot;: [0.0, 1.0, 0.0], &quot;size&quot;: 15.0, &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}, {&quot;point&quot;: [0.0, 2.0, 0.0], &quot;size&quot;: 15.0, &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}, {&quot;point&quot;: [0.0, 3.0, 0.0], &quot;size&quot;: 15.0, &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}, {&quot;point&quot;: [0.0, 4.0, 0.0], &quot;size&quot;: 15.0, &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}, {&quot;point&quot;: [0.0, 5.0, 0.0], &quot;size&quot;: 15.0, &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}, {&quot;point&quot;: [0.0, 0.0, 1.0], &quot;size&quot;: 15.0, &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}, {&quot;point&quot;: [0.0, 0.0, 2.0], &quot;size&quot;: 15.0, &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}, {&quot;point&quot;: [0.0, 0.0, 3.0], &quot;size&quot;: 15.0, &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}, {&quot;point&quot;: [0.0, 0.0, 4.0], &quot;size&quot;: 15.0, &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}, {&quot;point&quot;: [0.0, 0.0, 5.0], &quot;size&quot;: 15.0, &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}];\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 = [];\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 = [{&quot;vertices&quot;: [{&quot;x&quot;: 0.9759464882278817, &quot;y&quot;: 1.9518929764557633, &quot;z&quot;: 2.9278394646836454}, {&quot;x&quot;: 0.9848350148430511, &quot;y&quot;: 1.9537390693039487, &quot;z&quot;: 2.9236458939131325}, {&quot;x&quot;: 0.9751536231308308, &quot;y&quot;: 1.9603072462616615, &quot;z&quot;: 2.9224942398453972}, {&quot;x&quot;: 0.9822327648830154, &quot;y&quot;: 1.9446196547764318, &quot;z&quot;: 2.9305929202514887}, {&quot;x&quot;: 0.9709430942482699, &quot;y&quot;: 1.9455517235986397, &quot;z&quot;: 2.9337347645815988}, {&quot;x&quot;: 0.9665679440342413, &quot;y&quot;: 1.955247188338135, &quot;z&quot;: 2.9287295048266113}, {&quot;x&quot;: 0.008888526615169437, &quot;y&quot;: 0.0018460928481852752, &quot;z&quot;: -0.004193570770513328}, {&quot;x&quot;: -0.0007928650970509072, &quot;y&quot;: 0.008414269805898185, &quot;z&quot;: -0.005345224838248487}, {&quot;x&quot;: 0.006286276655133676, &quot;y&quot;: -0.007273321679331586, &quot;z&quot;: 0.002753455567843164}, {&quot;x&quot;: -0.005003393979611823, &quot;y&quot;: -0.006341252857123658, &quot;z&quot;: 0.005895299897953045}, {&quot;x&quot;: -0.00937854419364038, &quot;y&quot;: 0.0033542118823717826, &quot;z&quot;: 0.000890040142965605}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 0.0}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 2, 5], [2, 1, 6, 7], [1, 3, 8, 6], [3, 4, 9, 8], [4, 5, 10, 9], [5, 2, 7, 10], [7, 6, 11], [6, 8, 11], [8, 9, 11], [9, 10, 11], [10, 7, 11]], &quot;color&quot;: &quot;#0000ff&quot;, &quot;opacity&quot;: 1.0}, {&quot;vertices&quot;: [{&quot;x&quot;: 0.9999999999999999, &quot;y&quot;: 1.9999999999999998, &quot;z&quot;: 3.0}, {&quot;x&quot;: 1.0026120680733899, &quot;y&quot;: 1.9574312550003192, &quot;z&quot;: 2.9152587523721056}, {&quot;x&quot;: 0.973567892936729, &quot;y&quot;: 1.977135785873458, &quot;z&quot;: 2.9118037901689}, {&quot;x&quot;: 0.9948053181932827, &quot;y&quot;: 1.9300730114177687, &quot;z&quot;: 2.936099831387175}, {&quot;x&quot;: 0.9609363062890461, &quot;y&quot;: 1.9328692178843923, &quot;z&quot;: 2.9455253643775046}, {&quot;x&quot;: 0.9478108556469605, &quot;y&quot;: 1.9619556121028787, &quot;z&quot;: 2.9305095851125422}, {&quot;x&quot;: 0.9759464882278817, &quot;y&quot;: 1.9518929764557633, &quot;z&quot;: 2.9278394646836454}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 2, 5], [2, 1, 6], [1, 3, 6], [3, 4, 6], [4, 5, 6], [5, 2, 6]], &quot;color&quot;: &quot;#0000ff&quot;, &quot;opacity&quot;: 1.0}, {&quot;vertices&quot;: [{&quot;x&quot;: 1.9597507764050042, &quot;y&quot;: 0.0, &quot;z&quot;: 3.919501552810008}, {&quot;x&quot;: 1.9682572844885244, &quot;y&quot;: 0.0030901699437494747, &quot;z&quot;: 3.9152482987682475}, {&quot;x&quot;: 1.9597507764050042, &quot;y&quot;: 0.01, &quot;z&quot;: 3.919501552810008}, {&quot;x&quot;: 1.9650080875261955, &quot;y&quot;: -0.008090169943749474, &quot;z&quot;: 3.9168728972494122}, {&quot;x&quot;: 1.9544934652838128, &quot;y&quot;: -0.008090169943749474, &quot;z&quot;: 3.9221302083706036}, {&quot;x&quot;: 1.9512442683214837, &quot;y&quot;: 0.0030901699437494725, &quot;z&quot;: 3.923754806851768}, {&quot;x&quot;: 0.0085065080835204, &quot;y&quot;: 0.0030901699437494747, &quot;z&quot;: -0.0042532540417602}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.01, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.005257311121191337, &quot;y&quot;: -0.008090169943749474, &quot;z&quot;: -0.002628655560595669}, {&quot;x&quot;: -0.005257311121191335, &quot;y&quot;: -0.008090169943749474, &quot;z&quot;: 0.002628655560595668}, {&quot;x&quot;: -0.0085065080835204, &quot;y&quot;: 0.0030901699437494725, &quot;z&quot;: 0.0042532540417602}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 0.0}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 2, 5], [2, 1, 6, 7], [1, 3, 8, 6], [3, 4, 9, 8], [4, 5, 10, 9], [5, 2, 7, 10], [7, 6, 11], [6, 8, 11], [8, 9, 11], [9, 10, 11], [10, 7, 11]], &quot;color&quot;: &quot;#ff0000&quot;, &quot;opacity&quot;: 1.0}, {&quot;vertices&quot;: [{&quot;x&quot;: 2.0000000000000004, &quot;y&quot;: 0.0, &quot;z&quot;: 4.0}, {&quot;x&quot;: 1.9852703006555654, &quot;y&quot;: 0.009270509831248424, &quot;z&quot;: 3.9067417906847273}, {&quot;x&quot;: 1.9597507764050042, &quot;y&quot;: 0.03, &quot;z&quot;: 3.919501552810008}, {&quot;x&quot;: 1.9755227097685781, &quot;y&quot;: -0.02427050983124842, &quot;z&quot;: 3.911615586128221}, {&quot;x&quot;: 1.9439788430414302, &quot;y&quot;: -0.024270509831248423, &quot;z&quot;: 3.927387519491795}, {&quot;x&quot;: 1.934231252154443, &quot;y&quot;: 0.009270509831248417, &quot;z&quot;: 3.9322613149352885}, {&quot;x&quot;: 1.9597507764050042, &quot;y&quot;: 0.0, &quot;z&quot;: 3.919501552810008}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 2, 5], [2, 1, 6], [1, 3, 6], [3, 4, 6], [4, 5, 6], [5, 2, 6]], &quot;color&quot;: &quot;#ff0000&quot;, &quot;opacity&quot;: 1.0}, {&quot;vertices&quot;: [{&quot;x&quot;: 4.82, &quot;y&quot;: 0.0, &quot;z&quot;: 2.9513987859451214e-16}, {&quot;x&quot;: 4.82, &quot;y&quot;: 0.006180339887498949, &quot;z&quot;: -0.019021130325902778}, {&quot;x&quot;: 4.82, &quot;y&quot;: 0.02, &quot;z&quot;: 2.9513987859451214e-16}, {&quot;x&quot;: 4.82, &quot;y&quot;: -0.016180339887498948, &quot;z&quot;: -0.01175570504584917}, {&quot;x&quot;: 4.82, &quot;y&quot;: -0.016180339887498948, &quot;z&quot;: 0.011755705045849756}, {&quot;x&quot;: 4.82, &quot;y&quot;: 0.006180339887498945, &quot;z&quot;: 0.019021130325903367}, {&quot;x&quot;: 1.1647083184890924e-18, &quot;y&quot;: 0.006180339887498949, &quot;z&quot;: -0.019021130325903073}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.02, &quot;z&quot;: 0.0}, {&quot;x&quot;: 7.198293278059968e-19, &quot;y&quot;: -0.016180339887498948, &quot;z&quot;: -0.011755705045849465}, {&quot;x&quot;: -7.198293278059966e-19, &quot;y&quot;: -0.016180339887498948, &quot;z&quot;: 0.011755705045849461}, {&quot;x&quot;: -1.1647083184890924e-18, &quot;y&quot;: 0.006180339887498945, &quot;z&quot;: 0.019021130325903073}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 0.0}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 2, 5], [2, 1, 6, 7], [1, 3, 8, 6], [3, 4, 9, 8], [4, 5, 10, 9], [5, 2, 7, 10], [7, 6, 11], [6, 8, 11], [8, 9, 11], [9, 10, 11], [10, 7, 11]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}, {&quot;vertices&quot;: [{&quot;x&quot;: 5.0, &quot;y&quot;: 0.0, &quot;z&quot;: 3.061616997868383e-16}, {&quot;x&quot;: 4.82, &quot;y&quot;: 0.018541019662496848, &quot;z&quot;: -0.05706339097770891}, {&quot;x&quot;: 4.82, &quot;y&quot;: 0.06, &quot;z&quot;: 2.9513987859451214e-16}, {&quot;x&quot;: 4.82, &quot;y&quot;: -0.04854101966249684, &quot;z&quot;: -0.03526711513754809}, {&quot;x&quot;: 4.82, &quot;y&quot;: -0.04854101966249685, &quot;z&quot;: 0.035267115137548675}, {&quot;x&quot;: 4.82, &quot;y&quot;: 0.018541019662496834, &quot;z&quot;: 0.057063390977709516}, {&quot;x&quot;: 4.82, &quot;y&quot;: 0.0, &quot;z&quot;: 2.9513987859451214e-16}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 2, 5], [2, 1, 6], [1, 3, 6], [3, 4, 6], [4, 5, 6], [5, 2, 6]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}, {&quot;vertices&quot;: [{&quot;x&quot;: 0.0, &quot;y&quot;: 4.82, &quot;z&quot;: 2.9513987859451214e-16}, {&quot;x&quot;: 0.019021130325903073, &quot;y&quot;: 4.82, &quot;z&quot;: -0.006180339887498654}, {&quot;x&quot;: 0.0, &quot;y&quot;: 4.82, &quot;z&quot;: -0.019999999999999706}, {&quot;x&quot;: 0.011755705045849465, &quot;y&quot;: 4.82, &quot;z&quot;: 0.016180339887499243}, {&quot;x&quot;: -0.011755705045849461, &quot;y&quot;: 4.82, &quot;z&quot;: 0.016180339887499243}, {&quot;x&quot;: -0.019021130325903073, &quot;y&quot;: 4.82, &quot;z&quot;: -0.00618033988749865}, {&quot;x&quot;: 0.019021130325903073, &quot;y&quot;: 3.7843667304341507e-19, &quot;z&quot;: -0.006180339887498949}, {&quot;x&quot;: 0.0, &quot;y&quot;: 1.2246467991473532e-18, &quot;z&quot;: -0.02}, {&quot;x&quot;: 0.011755705045849465, &quot;y&quot;: -9.907600726170915e-19, &quot;z&quot;: 0.016180339887498948}, {&quot;x&quot;: -0.011755705045849461, &quot;y&quot;: -9.907600726170915e-19, &quot;z&quot;: 0.016180339887498948}, {&quot;x&quot;: -0.019021130325903073, &quot;y&quot;: 3.784366730434148e-19, &quot;z&quot;: -0.006180339887498945}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 0.0}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 2, 5], [2, 1, 6, 7], [1, 3, 8, 6], [3, 4, 9, 8], [4, 5, 10, 9], [5, 2, 7, 10], [7, 6, 11], [6, 8, 11], [8, 9, 11], [9, 10, 11], [10, 7, 11]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}, {&quot;vertices&quot;: [{&quot;x&quot;: 0.0, &quot;y&quot;: 5.0, &quot;z&quot;: 3.061616997868383e-16}, {&quot;x&quot;: 0.05706339097770921, &quot;y&quot;: 4.82, &quot;z&quot;: -0.018541019662496553}, {&quot;x&quot;: 0.0, &quot;y&quot;: 4.82, &quot;z&quot;: -0.0599999999999997}, {&quot;x&quot;: 0.03526711513754839, &quot;y&quot;: 4.82, &quot;z&quot;: 0.04854101966249714}, {&quot;x&quot;: -0.03526711513754838, &quot;y&quot;: 4.82, &quot;z&quot;: 0.048541019662497145}, {&quot;x&quot;: -0.05706339097770922, &quot;y&quot;: 4.82, &quot;z&quot;: -0.01854101966249654}, {&quot;x&quot;: 0.0, &quot;y&quot;: 4.82, &quot;z&quot;: 2.9513987859451214e-16}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 2, 5], [2, 1, 6], [1, 3, 6], [3, 4, 6], [4, 5, 6], [5, 2, 6]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}, {&quot;vertices&quot;: [{&quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 4.82}, {&quot;x&quot;: 0.019021130325903073, &quot;y&quot;: 0.006180339887498949, &quot;z&quot;: 4.82}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.02, &quot;z&quot;: 4.82}, {&quot;x&quot;: 0.011755705045849465, &quot;y&quot;: -0.016180339887498948, &quot;z&quot;: 4.82}, {&quot;x&quot;: -0.011755705045849461, &quot;y&quot;: -0.016180339887498948, &quot;z&quot;: 4.82}, {&quot;x&quot;: -0.019021130325903073, &quot;y&quot;: 0.006180339887498945, &quot;z&quot;: 4.82}, {&quot;x&quot;: 0.019021130325903073, &quot;y&quot;: 0.006180339887498949, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.02, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.011755705045849465, &quot;y&quot;: -0.016180339887498948, &quot;z&quot;: 0.0}, {&quot;x&quot;: -0.011755705045849461, &quot;y&quot;: -0.016180339887498948, &quot;z&quot;: 0.0}, {&quot;x&quot;: -0.019021130325903073, &quot;y&quot;: 0.006180339887498945, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 0.0}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 2, 5], [2, 1, 6, 7], [1, 3, 8, 6], [3, 4, 9, 8], [4, 5, 10, 9], [5, 2, 7, 10], [7, 6, 11], [6, 8, 11], [8, 9, 11], [9, 10, 11], [10, 7, 11]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}, {&quot;vertices&quot;: [{&quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 5.0}, {&quot;x&quot;: 0.05706339097770921, &quot;y&quot;: 0.018541019662496848, &quot;z&quot;: 4.82}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.06, &quot;z&quot;: 4.82}, {&quot;x&quot;: 0.03526711513754839, &quot;y&quot;: -0.04854101966249684, &quot;z&quot;: 4.82}, {&quot;x&quot;: -0.03526711513754838, &quot;y&quot;: -0.04854101966249685, &quot;z&quot;: 4.82}, {&quot;x&quot;: -0.05706339097770922, &quot;y&quot;: 0.018541019662496834, &quot;z&quot;: 4.82}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 4.82}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 2, 5], [2, 1, 6], [1, 3, 6], [3, 4, 6], [4, 5, 6], [5, 2, 6]], &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}];\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"
    }
   ],
   "source": [
    "# Or refinied again:\n",
    "# 1. Define your vectors\n",
    "V1 = vector([1, 2, 3])\n",
    "V2 = vector([2, 0, 4])\n",
    "origin_point = point((0, 0, 0), color='black', size=30)\n",
    "\n",
    "# 2. Calculate vector lengths for labels\n",
    "len_v1 = V1.norm().n(digits=3)  # Get numerical approximation\n",
    "len_v2 = V2.norm().n(digits=3)\n",
    "\n",
    "# 3. Create vector plots (now with thickness and new labels)\n",
    "plot_v1 = plot(V1, color='blue', width=4, \n",
    "               legend_label=f'V1 = [1, 2, 3] (Length: {len_v1})')\n",
    "plot_v2 = plot(V2, color='red', width=4, \n",
    "               legend_label=f'V2 = [2, 0, 4] (Length: {len_v2})')\n",
    "\n",
    "# 4. Define and plot the axes\n",
    "axis_len = 5       # How long to draw the axes\n",
    "label_offset = 0.3 # How far to put the 'x, y, z' label\n",
    "\n",
    "x_axis = arrow((0, 0, 0), (axis_len, 0, 0), color='black', width=1.0)\n",
    "y_axis = arrow((0, 0, 0), (0, axis_len, 0), color='black', width=1.0)\n",
    "z_axis = arrow((0, 0, 0), (0, 0, axis_len), color='black', width=1.0)\n",
    "\n",
    "x_label = text3d(\"x\", (axis_len + label_offset, 0, 0), color='black', fontsize=14)\n",
    "y_label = text3d(\"y\", (0, axis_len + label_offset, 0), color='black', fontsize=14)\n",
    "z_label = text3d(\"z\", (0, 0, axis_len + label_offset), color='black', fontsize=14)\n",
    "\n",
    "# 5. --- NEW: Add ticks and number labels ---\n",
    "tick_offset = 0.3  # How far to put number labels from the axis\n",
    "tick_size = 15     # Size of the tick mark points\n",
    "\n",
    "# Use sum() to create a single plot object from a generator\n",
    "x_ticks = sum(point((i, 0, 0), color='black', size=tick_size) for i in range(1, axis_len + 1))\n",
    "y_ticks = sum(point((0, i, 0), color='black', size=tick_size) for i in range(1, axis_len + 1))\n",
    "z_ticks = sum(point((0, 0, i), color='black', size=tick_size) for i in range(1, axis_len + 1))\n",
    "\n",
    "x_tick_labels = sum(text3d(str(i), (i, -tick_offset, 0), color='black', fontsize=10) for i in range(1, axis_len + 1))\n",
    "y_tick_labels = sum(text3d(str(i), (tick_offset, i, 0), color='black', fontsize=10) for i in range(1, axis_len + 1))\n",
    "z_tick_labels = sum(text3d(str(i), (0, tick_offset, i), color='black', fontsize=10) for i in range(1, axis_len + 1))\n",
    "\n",
    "\n",
    "# 6. Combine ALL the plot objects\n",
    "pl = plot_v1 + plot_v2 + origin_point\n",
    "pl += x_axis + y_axis + z_axis + x_label + y_label + z_label\n",
    "pl += x_ticks + y_ticks + z_ticks\n",
    "pl += x_tick_labels + y_tick_labels + z_tick_labels\n",
    "\n",
    "# 7. Show the final plot\n",
    "pl.show(frame=False, aspect_ratio=1)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "d45abd6c-8d5f-46cc-aea3-bdf4c75465f4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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81INxZIl16MRn2IXvizfosRV6hXbsBYBUOeqzHkxKQ0xmZqY5pLCprfSRfrqbMGfyAEAwLfJZDyalIUZPeu7Tp48cPnzYnPfy+uuvy2mnnSZZWVmpuUMkTQOMfn8AAMGyyYc9mJRPJ2mQ0evbb7+VmpoaOXToECHGQnruEgAgGA74tAfj2LEDGlz05F2d1z948GAqXxopot8fAiYA+NtRH/dgwmUcjeX45TjoSAx/47eXBhid7oOdQsVenZrVwAkAiSgsFJk6tb4HU1Hhr2kkxw6A1F+Q/JKE182fP99cocK69r4eeOABGTt2bMTnl5aWyogRI054fMuWLXLBBRc4fr8AEJQeTDhOsQYi6Nq1qzz22GPSq1cv8/bzzz8v1113naxfv94EmqZs27atwQjKWWed5cr9AkBQejDhCDFABOPHj2/w9qOPPmpGZtauXRs1xHTo0EHatm3rwh0CQDB7MOHYsReIYYn6Sy+9ZPpeQ4cOjfrcAQMGSOfOnWXkyJGyevXqZl9b+2Pagwm/ACARi3y8H0xTCDFAEzZu3CitW7c2Zehp06ZJSUmJORogEg0uCxYskOLiYlm6dKnk5OSYIFNWVhb1c+Tn55sib+jKzs526KsB4GebAtSDcXR1EuAXuk3Arl275MsvvzTh5Pe//72sWbOmySATaUoqIyNDli1bFnUkJnw1n47EaJBhdRKAeHowgwfXTyNpD6agQAKBEAPEaNSoUXLuuefKs88+G9PztUezZMkSs0IpViyxBhCPo0dFcnPrp5G0B1Ne7v9ppBCmk4AYad6PZw8kXcmk00wA4JRFAezBhGN1EhDBrFmzzJ4wOrWzf/9+U+zVvWBWrFhh3p+XlyeVlZWyePFi8/acOXOke/fuZuWSTkPpCIxOQekFAE7YFNAeTDhCDBDBZ599JrfeeqtUVVWZ6Z1+/fqZAHP11Veb9+vj2pcJ0eAyY8YME2xatWplwszy5ctl3LhxafwqAPhV0PaDaQqdGMAidGIANCfoPZhwdGIAAPCQoPdgwhFiAADwCHowDRFiAADwAHowJyLEABaYN2+e2URvsO5YBQCNBPFcpFhQ7AUsQrEXQCSFhSJTp9b3YCoqgj2NFMJIDAAAFqMH0zRCDAAAlqIHEx0hBgAAC9GDaR4hBgAAC7EfTPMIMQAAWIYeTGwIMQAAWIQeTOwIMQAAWIIeTHwIMQAAWIIeTHwIMYAF2LEXAD2Y+LFjL2ARduwFgtuD0b/DhKaRtAdTUJDuu7IfIzEAAKQRPZjEEWIAAEgjejCJI8QAAJAm9GCSQ4gBACAN2A8meYQYAABcRg8mNQgxAAC4jB5MahBiAABwET2Y1CHEAADgEnowqUWIAQDABfRgUo8QA1iAYwcA/6MHk3ocOwBYhGMHAP/2YIYMqZ9GKipiGikVGIkBAMBB9GCcQ4gBAMAh9GCcRYgBAMAh9GCcRYgBAMAB7AfjPEIMAAApRg/GHYQYAABSiB6MewgxAACkED0Y9xBiAABIkc2b6cG4iRADWIAdewHvq6mhB+M2duwFLMKOvYB3TZ4s8vzz9T2Y8nKmkZzGSAwAACnowYQCDD0Y9xBiAABIsgejq5FC6MG4hxADAECC6MGkFyEGAIAE6UqkLVuO/Zn9YNxHiAEAIAH0YNKPEANEMH/+fOnXr59ZIaTX0KFD5c0334z6MWvWrJGBAwdKy5YtpWfPnlJQUODa/QJwFz0YOxBigAi6du0qjz32mKxbt85cV111lVx33XWyWX9yRbBjxw4ZN26cDBs2TNavXy+zZs2Se++9V4qLi12/dwDOogdjD/aJAWLUrl07efzxx+W222474X0zZ86UZcuWyZbQ5LiITJs2Td5//30p180iYsQ+MYD92A/GHozEAM04cuSIvPTSS1JTU2OmlSLRoDJ69OgGj40ZM8aM4hw6dKjJ166trTXBJfwCYC96MHYhxABN2Lhxo7Ru3VqysrLMqEpJSYk5GiCS6upq6dixY4PH9O3Dhw/Lvn37mvwc+fn5ZuQldGVnZ6f86wCQGvRg7EOIAZqQk5MjGzZskLVr18ovfvELyc3NlQ8//LDJ52dkZDR4OzRT2/jxcHl5eWbqKHTt3r07hV8BgFShB2OnzHTfAGCrFi1aSK9evcyfBw0aJO+995489dRT8uyzz57w3E6dOpnRmHB79uyRzMxMad++fZOfQ0d59AJgN/aDsRMjMUCMdGRFOyyRaFdm1apVDR5buXKlCT+nnHKKS3cIwAn0YOxFiAEi0CXS77zzjuzcudN0Y2bPni2lpaVyyy231E0DTZo0qe752pn55JNPZPr06WaF0nPPPScLFy6UGTNmpPGrAJAsejB2YzoJiOCzzz6TW2+9VaqqqkzhVje+W7FihVx99dXm/fr4rl276p7fo0cPeeONN+T++++XefPmSZcuXeTpp5+WiRMnpvGrAJAMejD2Y58YwCLsEwPYg/1g7Md0EgAAjdCD8QZCDAAAYejBeAchBgCA4+jBeAshBrCAloF1N+DBgwen+1aAQGM/GG+h2AtYhGIvkN4ezJQp9T2YigqmkWzHSAwAIPDowXgTIQYAEGj0YLyLEAMACDR6MN5FiAEABBb7wXgbIQYAEEj0YLyPEAMACBx6MP5AiAEABA49GH8gxAAAAoUejH8QYgAAgUEPxl8IMYAFOHYAcB49GP/h2AHAIhw7ADhn8uT6aSTtwZSXM43kdYzEAAB8jx6MPxFiAAC+Rg/GvwgxAADfogfjb4QYAIBvsR+MvxFiAAC+RA/G/wgxAADfoQcTDIQYAICv0IMJDkIMAMBX6MEEByEGsAA79gKpQQ8mWNixF7AIO/YCyfVg9O8BoWmkoiKmkfyOkRgAgOfRgwkmQgwAwPPowQQTIQYA4Gn0YIKLEAMA8Cz2gwk2QgwAwJPowYAQAwDwJHowIMQAADyHHgwUIQYA4Cn0YBBCiAEswI69QGzowSAcO/YCFmHHXiC6yZPrp5G0B1NezjRSkDESAwDwBHowaIwQAwCwHj0YREKIAQBYjR4MmkKIAQBYjf1g0BRCDADAWvRgEA0hBoggPz/fLHdu06aNdOjQQSZMmCDbtm2L+jGlpaWSkZFxwrV161bX7hvwE3owaA4hBohgzZo1ctddd8natWtl1apVcvjwYRk9erTU6OR8MzTsVFVV1V3nnXeeK/cM+Ak9GMQiM6ZnAQGzYsWKBm8XFhaaEZmKigq54ooron6sPq9t27YO3yHgb/RgEAtGYoAY6OZzql27ds0+d8CAAdK5c2cZOXKkrF69Oupza2trzQZ34RcQdPRgECtCDNAM3dR6+vTpcvnll0vfvn2bfJ4GlwULFkhxcbEsXbpUcnJyTJApKyuL2r3RHXpDV3Z2tkNfBeAN9GAQD44dAJqh3Zjly5fLu+++K127do3rY8ePH2/KvcuWLWtyJEavEB2J0SDDsQMIag9Gjw8LTSNpD6agIN13BZsxEgNEcc8995gAotNC8QYYdemll8pHH33U5PuzsrJMWAm/gKCiB4N4UewFItABSg0wJSUlZul0jx49Enqd9evXm2kmANHRg0EiCDFAE1NIRUVF8tprr5m9Yqqrq83j2ltpdfwna15enlRWVsrixYvN23PmzJHu3btLnz595ODBg7JkyRLTj9ELQNPowSBRhBgggvnz55t/Dh8+/ISl1pMnTzZ/1j1gdu3aVfc+DS4zZswwwUaDjoYZ7dKMGzfO5bsHvIP9YJAMir2ARbTYq6M9FHsRFPp3gtA0kvZgysuZRkLsKPYCANKCHgySRYgBALiOHgxSgRADAHAVPRikCiEGsMC8efOkd+/e5uRswO/YDwapQrEXsAjFXgShBzNlSn0PpqKCaSQkjpEYAIAr6MEg1QgxAADH0YOBEwgxAADH0YOBEwgxAABHsR8MnEKIAQA4hh4MnESIAQA4gh4MnEaIAQA4gh4MnEaIAQCkHD0YuIEQAwBIKXowcAshBrAAxw7AL+jBwE0cOwBYhGMH4HWTJ9dPI2kPprycaSQ4h5EYAEBK0IOB2wgxAICk0YNBOhBiAABJoQeDdCHEAACSwn4wSBdCDAAgYfRgkE6EGABAQujBIN0IMQCAuNGDgQ0IMQCAuNGDgQ0IMYAF2LEXXkIPBrZgx17AIuzYCy/0YDRrh6aRioqYRkL6MBIDAIgJPRjYhhADAIgJPRjYhhADAGgWPRjYiBADAIiK/WBgK0IMAKBJ9GBgM0IMAKBJ9GBgM0IMACAiejCwHSEGAHACejDwAkIMAKABejDwCkIMYAGOHYBN6MHAKzh2ALAIxw7Ahh7MlCn1PZiKCqaRYC9GYgAABj0YeA0hBgBADwaeRIgBANCDgScRYgAg4NgPBl5FiAEiyM/PNyuF2rRpIx06dJAJEybItm3bmv24NWvWyMCBA6Vly5bSs2dPKSgocOV+gUTRg4GXEWKAJsLIXXfdJWvXrpVVq1bJ4cOHZfTo0VKjxYEm7NixQ8aNGyfDhg2T9evXy6xZs+Tee++V4uJiV+8diBU9GHgdS6yBGOzdu9eMyGi4ueKKKyI+Z+bMmbJs2TLZEioWiMi0adPk/fffl/Ly8pg+D0us4abJk+unkbQHo/+ZMo0EL2EkBoiBhgrVrl27Jp+jQUVHa8KNGTNG1q1bJ4cOHYr4MbW1tSa4hF+AG+jBwA+cCTHfHRHZtVTkm88ceXnATTpYOX36dLn88sulb9++TT6vurpaOnbs2OAxfVunovbt29dk90ZHXkJXdnZ2yu8faIweDPzCmRDz1giRdyeKlHQS+a8rRbb/XqT2r458KsBpd999t3zwwQfy4osvNvvcjIyMBm+HZmsbPx6Sl5dnRnlC1+7du1N010Bk9GDgJ5mOvOrXYT+I95Qdu96bJtJxpEi3m0SyrxfJau/IpwZS6Z577jE9l7KyMunatWvU53bq1MmMxoTbs2ePZGZmSvv2kf97z8rKMhfgFvaDgZ84MxJzXtg4ZcjRIyLVK0X+93aRpR1F3h7DCA2spSMoOgKzdOlSefvtt6VHjx7NfszQoUPNSqZwK1eulEGDBskpp5zi4N0CsaEHA79xZnXSt/uOTSVpcGn2Dk5mhAbWufPOO6WoqEhee+01ycnJqXtceyutjv/U16mgyspKWbx4cd0Sa+3M/PznP5fbb7/dFH11dZJOQ02cODGmz8vqJDjZg9FD0kPTSEVFTCPB+5xbYq0jLTryEtfdHA8059ws0mOSyEknO3JrQHOa6rAUFhbKZF2XapanTpadO3dKaWlp3ft1Cfb9998vmzdvli5duphl1xpkYkWIgVM9GA0woWkk7cGwDyP8wLkQo1NFOnWUqN4zRS56LJV3BFiPEAMnsB8M/Mq5fWK6Tjg2spKoI7WpvBsACCR6MPAz50JMyzOPTQ0lov0lIt9/KNV3BACBwn4w8Dtnd+zVsm4iAWbEH0VanOHEHQFAILAfDILA2RAT75QSAQYBNW/ePOndu7c5ORtIBfaDQRA4fwBkrKuUvjdAZORqAgwCjWIvUtWDmTKlvgdTUcE0EvzJ+QMgY51SOviFyOH9Tt8NAPgaPRgEifMhprkppZOOb7les1Pkv4aLfP0Xx28JAPyIHgyCxvkQE22VknZgxv5JpPW5x94+8DFBBgASRA8GQeN8iGlqSilU4j2jt8ioUoIMACSB/WAQRO6EmMZTSo1XIZ3alSADAAmiB4OgcifE6JRSj9xjfz7rB5GXURNkACBu9GAQZM4vsQ757sixYNKml0hGlOykoUXDiz5XaajRcKMhB/A5llgjXpyLhCA7yb3PdLLI6edHDzCKERkAiAk9GASdeyEmHgQZAIiKHgxga4hRBBkECMcOIB70YAC3OzGJoiODAKETg+boT2ztwSxefOxtejAIMntHYkIYkQGABj2YUIChB4Ogsz/EKIIMAMimTcd25Q2hB4Og80aIUQQZAAF24AA9GMC7IUYRZAAEtAejK5G2bj32NuciAV4MMYogAyCAPZgXXjj2Z3owgJdDjCLIAAgIejCA30KMIsgA8Dl6MIBfQ4wiyADwKXowgN9DjCLIwAfYsReN0YMB/LBjb6zY2Rc+wI69CPVghgypn0YqKmIaCfDnSEwIIzIAfIAeDBDEEKMIMgA8jB4MEOQQowgyADyKHgwQ9BCjCDIAPIb9YID4+TPEKIIMAI+gBwMkxr8hRhFkAFiOHgyQOH+HGEWQAWAxejBA4vwfYhRBBoCF6MEAyQlGiFEEGQAWoQcDJC84IUYRZGApjh0IFnowQGr459iBeHBEASzFsQPBUFgoMnVqfQ+mooJpJCARwRqJCWFEBkCa0IMBUieYIUYRZAC4jB4MkFrBDTGKIAPAJfRggNQLdohRBBkALmA/GCD1CDGKIINGysrKZPz48dKlSxfJyMiQV199NerzS0tLzfMaX1tDf+1GoNGDAZxBiAkhyCBMTU2N9O/fX+bOnRvXx23btk2qqqrqrvPOO8+xe4Q30IMBnJPp4Gt7N8iEll+HggzLrwNn7Nix5opXhw4dpG3bto7cE7yHHgzgLEZiGmNEBkkYMGCAdO7cWUaOHCmrV69u9vm1tbVmb5jwC/5BDwZwFiEmEoIM4qTBZcGCBVJcXCxLly6VnJwcE2S0WxNNfn6+2dwudGVnZ7t2z3AWPRjAecHcsTdW7OwL/Z8kI0NKSkpkwoQJcX2cFoP1Y5ctWxZ1JEavEB2J0SDDjr3e78HoCRKhaSTtwRQUpPuuAP9hJCYaRmSQhEsvvVQ++uijqM/JysoyYSX8grfRgwHcQ4hpDkEGCVq/fr2ZZkKw0IMB3MPqpFiwailwDhw4INu3b697e8eOHbJhwwZp166ddOvWTfLy8qSyslIWL15s3j9nzhzp3r279OnTRw4ePChLliwx/Ri9EBz0YAB3EWJiRZAJlHXr1smIESPq3p4+fbr5Z25urixatMjsAbNr166692twmTFjhgk2rVq1MmFm+fLlMm7cuLTcP9zHfjCA+yj2xouyLxykxV5dpUSx11v0p2hubv00kvZgysuZRgKcRicmXnRkADRCDwZID0JMIggyAI6jBwOkDyEmUQQZIPDowQDpRYhJBkEGKTJv3jzp3bu3DNYd0uAJ7AcDpB/F3lSg7IsUodjrHYWFIlOn1vdgKiqYRgLcxkhMKjAiAwQKPRjADoSYVCHIAIFADwawByEmlQgygK/RgwHsQohJNYIM4FvsBwPYhRDjBIIM4Dv0YAD7EGKcQpABfIMeDGAnQoyTCDKA59GDAexFiHEaQQbwNHowgL0IMW4gyACeRA8GsBshxi0EGUTBsQP2oQcD2I9jB9zGEQWIgmMH7KA/FXNz66eRtAdTXs40EmAbRmLcxogMYD16MIA3EGLSgSADWIseDOAdhJh0IcgA1qEHA3gLISadCDKANdgPBvAeQky6EWQAK9CDAbyHEGMDggyQVvRgAG8ixNiCIAOkBT0YwLsIMTYhyACuogcDeBshxjYEmUBix970oAcDeBs79tqKnX0DiR173e3BDBlSP41UVMQ0EuA1jMTYihEZwDH0YAB/IMTYjCADpBw9GMA/CDG2I8gAKUUPBvAPQowXEGSAlGA/GMBfCDFeQZABkkIPBvAfQoyXEGSAhNCDAfyJEOM1BBkgbvRgAH8ixHgRQQaIGT0YwL8IMV5FkAGaRQ8G8DdCjJcRZHyDYwdSjx4M4H8cO+AHHFHgGxw7kDqFhSJTp9b3YCoqmEYC/IaRGD9gRAZogB4MEAyEGL8gyAAGPRggOAgxfkKQQcDRgwGChRDjNwQZBBj7wQDBQojxI4JM0srKymT8+PHSpUsXycjIkFdffbXZj1mzZo0MHDhQWrZsKT179pSCggJX7hXH0IMBgocQ41cEmaTU1NRI//79Ze7cuTE9f8eOHTJu3DgZNmyYrF+/XmbNmiX33nuvFBcXO36voAcDBBVLrP2O5ddJ05GYkpISmTBhQpPPmTlzpixbtky2bNlS99i0adPk/fffl/Ly8pg/F0us46c/wXJz66eRtAej/8qZRgL8j5EYv2NExhUaVEaPHt3gsTFjxsi6devk0KFDTX5cbW2tCS7hF+JDDwYILkJMYIPMCIJMClVXV0vHjh0bPKZvHz58WPbt29fkx+Xn55uRl9CVnZ3twt36Bz0YINgIMYENMtsJMg5MO4ULzdQ2fjxcXl6emToKXbt373b8Pv2CHgwAQkyQEGQc06lTJzMaE27Pnj2SmZkp7du3b/LjsrKyTPcl/ELz2A8GgCLEBA1BxhFDhw6VVatWNXhs5cqVMmjQIDnllFPSdl9+RQ8GgCLEBBFBplkHDhyQDRs2mCu0hFr/vGvXrrppoEmTJjVYifTJJ5/I9OnTzQql5557ThYuXCgzZsxI29fgV/RgAIQQYoKKIBOVrioaMGCAuZSGE/3zAw88YN6uqqqqCzSqR48e8sYbb0hpaalcdNFF8utf/1qefvppmThxYtq+Bj+iBwMgHPvEBN0J+8j0Ehm1mn1k0oR9YprGfjAAGmMkJugYkYFH0IMB0BghBgQZWI8eDIBICDE4hiADS9GDAdAUQgzqEWRgGfaDARANIQYNEWTSYt68edK7d28ZPHhwum/FKvRgAETD6iRExqqltGB1UsMezJAh9dNIRUVMIwFoiJEYRMaIDNKIHgyAWBBi0DSCDNKAHgyAWBFi0HyQGbmaIAPX0IMBECtCDJp3WjZBBq5gPxgA8SDEIDYEGTiMHgyAeBFiEDuCDBxCDwZAIggxiA9BBg6gBwMgEYQYxI8ggxSiBwMgUYQYJIYgk1JB3bGXHgyAZLBjL5JTs1vkrRHs7JsiQdqxV3/y5ObWTyNpD6a8nGkkALFjJAbJYUQGCaIHAyBZhBgkjyCDONGDAZAKhBikBkEGMaIHAyBVCDFIHYIMmsF+MABSiRCD1CLIIAp6MABSiRCD1CPIIAJ6MABSjRADZxBkEIYeDAAnEGLgHIIM6MEAcBAhBs4iyEjQd+ylBwPAKezYC3ews28gd+zVHsyQIfXTSEVFTCMBSB1GYuAORmQChx4MAKcRYuAegkxg0IMB4AZCDNxFkAkEejAA3ECIgfsIMr7GfjAA3EKIQXoQZHyJHgwANxFikD4EGV+hBwPAbYQYpBdBxjfowQBwGyEG6UeQ8Tx6MADSgRADOxBkPIseDIB0IcTAHgEOMl49doAeDIB04tgB2CfARxR47diBwkKRqVPrezAVFUwjAXAPIzGwT4BHZLyEHgyAdCPEwE4EGavRgwFgA0IM7EWQsRI9GAC2IMTAbgQZ67AfDABbEGJgvzQGmd/97nfSo0cPadmypQwcOFDeeeedJp9bWloqGRkZJ1xbQ0MWPkAPBoBNCDHwhjQEmZdfflnuu+8+mT17tqxfv16GDRsmY8eOlV27dkX9uG3btklVVVXddd5554kf0IMBYBuWWMNbXFx+fckll8jFF18s8+fPr3vswgsvlAkTJkh+fn7EkZgRI0bIF198IW3btvXVEmv9KZGbWz+NpD2Y8nKmkQCkFyMx8BaXRmQOHjwoFRUVMnr06AaP69v/8z//E/VjBwwYIJ07d5aRI0fK6tWroz63trbWBJfwy0b0YADYiBAD73EhyOzbt0+OHDkiHTt2bPC4vl1dXR3xYzS4LFiwQIqLi2Xp0qWSk5NjgkxZWVmTn0dHdHTkJXRlZ2eLbejBALAV00nwLgenlj799FM5++yzzajL0KFD6x5/9NFH5YUXXoi5rDt+/HhT7l22bFmTIzF6hehIjAYZW6aTtAejJyGEvlztwRQUpPuuAOAYRmLgXQ6OyJx55ply8sknnzDqsmfPnhNGZ6K59NJL5aOPPmry/VlZWSashF+2YD8YALYjxMDbHAoyLVq0MEuqV61a1eBxffuyyy6L+XV0VZNOM3kRPRgAtstM9w0AKQsyoamlUJBJcmpp+vTpcuutt8qgQYPMlJL2XXR59bRp08z78/LypLKyUhYvXmzenjNnjnTv3l369OljisFLliwx/Ri9vIYeDAAvIMTAHxwIMj/+8Y/lr3/9qzz88MNmv5e+ffvKG2+8Ieecc455vz4WvmeMBpcZM2aYYNOqVSsTZpYvXy7jxo0TL2E/GABeQbEX/uLiPjJOSPc+MewHA8BL6MTAXzhrKSn0YAB4CSEG/kOQSQg9GABeQ4iBPxFk4kIPBoAXEWLgXwSZmLAfDACvIsTA3zwSZObNmye9e/eWwbo9rsvowQDwKlYnIRg8smrJ7dVJ2oMZMqR+GqmoiGkkAN7BSAyCwSMjMm6iBwPA6wgxCA6CTB16MAD8gBCDYCHIGPRgAPgBIQbBE/Agw34wAPyCEINgCmiQoQcDwE8IMQiugAUZejAA/IYQg2ALUJChBwPAbwgxQACCDD0YAH5EiAEsCDJO7thLDwaAX7FjL2DRzr6p3rFX/+/Oza2fRtIeTHk500gA/IGRGMDHU0v0YAD4GSEG8GmQoQcDwO8IMYAPgww9GABBQIgBfBZk2A8GQFAQYgCfBRl6MACCghAD+CjI0IMBECSEGMAnQYYeDICgIcQAPggy9GAABBEhBrAgyCS7Yy89GABBxI69gEU7+yayY6/2YIYMqZ9GKipiGglAMDASA3h4aokeDIAgI8QAHg0y9GAABB0hBvBokKEHAyDoCDGAB4MM+8EAACEG8FyQoQcDAMcQYgAPBRl6MABQjxADeCjI0IMBgHqEGMAjQYYeDAA0RIgBPBBk6MEAwIkIMYAFQSbasQP0YAAgMo4dACw6oiDSsQOFhSJTpx7/0NYiFRVMIwGAYiQGsHhqiR4MADSNEANYGmTowQBAdIQYwMIgQw8GAJpHiAEsDDJ/+AP7wQBAczKbfQYAZ4JMqOx7PMi82/LX8vrbG8xTpk+v79vTgwGAyFidBFiyaumjapErH2kpVV98KyJ/E5HTTQ+moCDdNwoAdiLEAOlUs1v2vdxPzmz5pXlz/c6z5OLZe4+HmB3yf//XXs4778Sl2ACAGKeTNOfs37/f+bsBAqa2tqVc+fB38oc7RM7tIHJWGw0wqlJEbpAXXrhFZsyYkea7BID0aNOmjWRkZCQ3EhPagAsAAMAt4Rt/JhxiYh2J0bCTnZ0tu3fvjvpJ46Vbsb/33nspe70gv6YT3yMvfN22vmZtba2MHj1aNmzYIB3OOF2mXnmBPLbsf+vev2jRIrn++uvTfp9Ov55XXjPIP+O8cI9B/v544XueyPenuZGYmKaT9AXi+Q9Cn5vK/4BOPvnklL5e0F8z1d8jr3zdtr7mI488IuPHj5c9f/uqQYDp16+f3HLLLZKZmWnd127rv0s3XjOoP+O8cI9B/v545Xue6u+PJ/aJuSt833Ve0zpe+bptfc1rr71WSkpKpL/uaHfcjTfeKG+99VZKAowTX7ut/y7deE0neOFr98I9OsUrX/tdHvieW706KdLhdbAL3yO7ffjhh9KnTx8z3Nq1K6uSbMP/P3bj+xO8709KR2KysrLkwQcfNP+Enfge2e2ss84y/+T7Yyf+/7Eb35/gfX/YJwawCH+TBACfdWIAAAAaI8QAAABPIsQAAABPohMDWCS0sWRzGzwBAAgxAADAoxybTtq5c6fcdttt0qNHD2nVqpWce+65ZmnVwYMHnfqUiNOjjz4ql112mZx66qnStm3bdN8OYK2ysjKzq3KXLl3MCNmrr76a7ltCmPz8fLNFvo5gdujQQSZMmCDbtm1L923huPnz55sdyEM79Q4dOlTefPNNsTrEbN26Vb777jt59tlnZfPmzfLkk09KQUGBzJo1y6lPiThpoLzpppvkF7/4RbpvBbBaTU2N2VF57ty56b4VRLBmzRqzu+zatWtl1apVcvjwYXMmmX7fkH66cedjjz0m69atM9dVV10l1113nckGnppOevzxx00i+/Of/+zWp0QM9JDB++67T7788st03wpgPR2J0WMi9G/7sNPevXvNiIyGmyuuuCLdt4MI2rVrZzKBztgkIzUHs8RIN/DSGwcAwMnfNYrfN/Y5cuSIvPLKK2aUTKeVkuVaiPn444/lmWeekd/85jdufUoAQMDo5ML06dPl8ssvl759+6b7dnDcxo0bTWj59ttvpXXr1mY0s3fv3uJ6J+ahhx4yw6nRLp3zCvfpp5/KNddcY/oXP/vZz5K+aaT2+wMAfnH33XfLBx98IC+++GK6bwVhcnJyZMOGDaa3pD3M3Nxcc+Ct652Yffv2mSua7t27S8uWLesCzIgRI+SSSy4x3YuTTmJ/PSfF+/1RdGKA2NGJsdc999xjVo7pajJdGQt7jRo1yqxa1sU/rk4nnXnmmeaKRWVlpQkwAwcOlMLCQgKMC+L5/gCAH+jfxTXAaLgsLS0lwHjke1ZbW2tvJ0ZHYIYPHy7dunWTJ554wrTFQzp16uTUp0Ucdu3aJZ9//rn5p5atdKhP9erVy8xZAjjmwIEDsn379rq3d+zYYf5/0eKo/oxDeuny6qKiInnttdfMXjHV1dXmcT0RXvcpQ3rp1ipjx46V7OxssyP5Sy+9ZMLmihUr7F1irVMUU6ZMifg+Ngm2w+TJk+X5558/4fHVq1ebAArgGP2Bq6PKjem8vv6sQ3o1dUSHzgDozzmkly6jfuutt6SqqsoES934bubMmXL11Vcn/docOwAAADyJkgoAAPAkQgwAAPAkQgwAAPAkQgwAAPAkQgwAAPAkQgwAAPAkQgwAAPAkQgwAAPAkQgwAAPAkQgwAAPAkQgwAABAv+n+iOuZDz29oxwAAAABJRU5ErkJggg==",
      "text/plain": [
       "Graphics object consisting of 3 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Another example (thanks to Gemini):\n",
    "\n",
    "# 1. Define vectors\n",
    "v1 = vector([3, 4])\n",
    "v2 = vector([-2, 2])\n",
    "\n",
    "# 2. Create plot objects for each vector\n",
    "p_v1 = plot(v1, color='blue', legend_label='v1 = (3, 4)')\n",
    "p_v2 = plot(v2, color='orange', legend_label='v2 = (-2, 2)')\n",
    "\n",
    "# 3. Add an origin point\n",
    "origin = point((0,0), color='black', size=30)\n",
    "\n",
    "# 4. Combine and show\n",
    "# (aspect_ratio=1 ensures x and y axes are scaled equally)\n",
    "(p_v1 + p_v2 + origin).show(aspect_ratio=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "b6f1c056-d77e-44d2-9d58-63c2d3acf162",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(8.41421356237310, -9.65685424949238)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "12.8083497715031"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Solve a problem, 11.1 q9 -- enterered rounded answers in Moer, it's correct!\n",
    "# A person starts walking from home and walks:\n",
    "# 4 miles East\n",
    "# 5 miles Southeast\n",
    "# 4 miles South\n",
    "# 3 miles Southwest\n",
    "# 3 miles East\n",
    "# Have to find total miles (easy) and displacement vector and ||displacement vector||\n",
    "\n",
    "southeast = 7*pi/4\n",
    "southwest = 5*pi/4\n",
    "\n",
    "# Solve # 11.1 q 9:\n",
    "v1 = vector((4, 0))\n",
    "v2 = vector([N(5*cos(southeast)), N(5*sin(southeast))])\n",
    "v3 = vector((0, -4))\n",
    "v4 = vector([N(3*cos(southwest)), N(3*sin(southwest))])\n",
    "v5 = vector((3, 0))\n",
    "\n",
    "displacement = v1 + v2 + v3 + v4 + v5\n",
    "print(displacement)\n",
    "sqrt(displacement[0]**2 + displacement[1]**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "7ffadf9b-d82f-4416-8003-d34770c534e2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.65685424949238"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N((4*sqrt(2)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2034d444-e253-455f-b82d-797668c96b90",
   "metadata": {},
   "source": [
    "$\\vec{v} = \\langle-1,2\\rangle\\quad\\quad\\vec{u} = \\langle1,1\\rangle\\quad\\quad \\vec{u} + \\vec{v} = \\langle0, 3\\rangle$\n",
    "\n",
    "$\\vec{w} = \\langle6, -3\\rangle$\n",
    "\n",
    "If a and b are scalars, then $a\\vec{u} + b\\vec{v} = \\vec{w}$ \n",
    "\n",
    "So $a \\cdot \\langle1,1\\rangle + b\\cdot \\langle-1,2\\rangle = \\langle6, -3\\rangle$\n",
    "\n",
    "So system of equations to solve is: \n",
    "\n",
    "$$\n",
    "\\begin{alignat*}{2}\n",
    "a - b &= 6 \\\\\n",
    "a + 2b &= -3\n",
    "\\\\\n",
    "\\therefore\n",
    "\\\\\n",
    "a &= b + 6 \\\\\n",
    "b + 6 + 2b &= -3\\\\\n",
    "3b &= -9\\\\\n",
    "b&=\\boxed{-3} \\\\\n",
    "a&=\\boxed{3}\n",
    "\\end{alignat*}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "590d7a2c-983d-4876-baa8-87f127550643",
   "metadata": {},
   "source": [
    "# Magnitude, Distance, Midpoints\n",
    "\n",
    "Note that the distance formula between two points (x,y,z) and $(x_1, y_1, z_1)$\n",
    "is effectively the same as the magnitude formula, i.e., \n",
    "\n",
    "$\\sqrt{(x - x_1)^2 + (y - y_1)^2 + (z - z_1)^2}$\n",
    "\n",
    "The midpoint formula for the two points would be:\n",
    "\n",
    "$\\langle \\frac{x+x_1}{2}, \\frac{y+y_1}{2}, \\frac{z+z_1}{2}\\rangle$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "308cc0c0-4171-4cbc-b8aa-d32c7a9615e4",
   "metadata": {},
   "outputs": [],
   "source": [
    "def magnitude_list(l: list):\n",
    "    \"\"\"return the Euclidean length of a point specified as a list of scalars\"\"\"\n",
    "    square_sum = 0\n",
    "    for i in l:\n",
    "        square_sum += i**2\n",
    "    return sqrt(square_sum)\n",
    "\n",
    "# But see the vector.norm() method, which will give you the same thing!\n",
    "\n",
    "def magnitude(v: vector):\n",
    "    \"\"\"return the Euclidean length of a vector\"\"\"\n",
    "    l = v.list()\n",
    "    return magnitude_list(l)\n",
    "\n",
    "# Note we could also implement distance between two points \n",
    "# doing the subtraction in that fn and calling magnitude_list\n",
    "# from sympy import Point3D\n",
    "# p = Point3D([1,2,3])\n",
    "# print(p)\n",
    "\n",
    "V = vector((1,2,3))\n",
    "assert magnitude(V) == sqrt(14)\n",
    "assert magnitude(V) == V.norm()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bc0292bf-1a30-47ba-a25c-d0876626affd",
   "metadata": {},
   "source": [
    "## Implicit Plots of 3D Surfaces\n",
    "\n",
    "For implicit plotting, we used https://c3d.libretexts.org/CalcPlot3D/index.html, selecting \"implicit surface\" from the dropdown.  This is easier to set up than SageMath.\n",
    "\n",
    "The cell below is commented out because it throws off Jupyter's scrolling somewhat"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "5cae0671-8409-48c4-b097-75531e521639",
   "metadata": {},
   "outputs": [],
   "source": [
    "# x = var('x')\n",
    "# y = var('y')\n",
    "# z = var('z')\n",
    "\n",
    "# equation = (x-2)**2 + y**2 + z**2 == 36\n",
    "# implicit_plot3d(equation, (x,-8, 8), (y,-8,8), (z,-8, 8))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "b52a9eaa-402f-433f-9c87-303031cb8358",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-1, 11, -6)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Checking a solution gotten by hand:\n",
    "a = vector((-3, 3, -2))\n",
    "b = vector((-1, -4, 2))\n",
    "a - 2*b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "8324a329-b2f0-4443-8cee-1a59dc451c69",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(12, -18, -4)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Checking a solution gotten by hand:\n",
    "a = vector((3, -3, -1))\n",
    "b = vector((3, -5, -1))\n",
    "a + 3*b"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8e73582b-8314-4c8f-a87a-6f8a0a4882d8",
   "metadata": {},
   "source": [
    "## The Dot Product\n",
    "\n",
    "The dot product gives us information about the angle $\\theta$ between two vectors.\n",
    "\n",
    "$$\n",
    "\\begin{alignat*}{2}\n",
    "&\\vec{V1} \\cdot \\vec{V2} = 0 &\\quad\\quad \\theta = \\frac{pi}{2} \\\\\n",
    "&\\vec{V1} \\cdot \\vec{V2} \\gt 0 &\\quad\\quad \\theta \\lt \\frac{pi}{2} \\\\\n",
    "&\\vec{V1} \\cdot \\vec{V2} \\lt 0 &\\quad\\quad \\theta \\gt \\frac{pi}{2} \\\\\n",
    "\\end{alignat*}\n",
    "$$\n",
    "\n",
    "The dot product is related to the angle as follows:\n",
    "\n",
    "$$\n",
    "\\begin{align*}\n",
    "\\vec{v} \\cdot \\vec{w} &= \\|\\vec{v}\\| \\|\\vec{w}\\| \\cos \\theta \\\\ \\\\\n",
    "\\cos \\theta &= \\frac{\\vec{v} \\cdot \\vec{w}}{\\|\\vec{v}\\| \\|\\vec{w}\\|}\n",
    "\\end{align*}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "8554a0cd-b3f3-4b8d-b88b-cdcde05070f0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.89902877949431"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N(acos(-35/(10*sqrt(13))))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "12cf872f-395e-4e8e-9174-652e8e016af4",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-35\n",
      "sqrt(26)\n",
      "5*sqrt(2)\n",
      "5.09901951359278\n",
      "5.09901951359279\n",
      "-7/52*sqrt(26)*sqrt(2)\n"
     ]
    }
   ],
   "source": [
    "a = vector((4,-1,-3))\n",
    "b = vector((-5,0,5))\n",
    "print(a*b)\n",
    "print(a.norm())\n",
    "print(b.norm())\n",
    "print(N(sqrt(26)))\n",
    "print(N(sqrt(2) * sqrt(13)))\n",
    "lhs = a*b / (a.norm() * b.norm())\n",
    "print(lhs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "611332a3-c5d1-4bc7-9bc5-ce24455692dd",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-0.970725343394151\n",
      "-0.970725343394151\n"
     ]
    }
   ],
   "source": [
    "print(N(-7/52*sqrt(26)*sqrt(2) ))\n",
    "print(N(a*b/(10*sqrt(13))))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "6d22830a-5a1b-40e4-8573-dee329a8d7f9",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.89902877949431  approx =   arccos(-174.730561810947/pi)\n"
     ]
    }
   ],
   "source": [
    "from sympy import deg\n",
    "theta_radians = acos(-0.970725343394151)\n",
    "theta_degrees = (acos(deg(-0.970725343394151)))\n",
    "print(theta_radians, \" approx =  \", theta_degrees)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "78fdf928-314a-448f-ac26-943dd845face",
   "metadata": {},
   "outputs": [
    {
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       "&lt;meta name=viewport content=&quot;width=device-width, user-scalable=no, minimum-scale=1.0, maximum-scale=1.0&quot;&gt;\n",
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       "\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;: [&quot;x&quot;, &quot;y&quot;, &quot;z&quot;], &quot;axesLabelsStyle&quot;: null, &quot;decimals&quot;: 2, &quot;delay&quot;: 20, &quot;frame&quot;: true, &quot;loop&quot;: true, &quot;projection&quot;: &quot;perspective&quot;, &quot;theme&quot;: &quot;light&quot;, &quot;viewpoint&quot;: false};\n",
       "    var animate = options.animate;\n",
       "\n",
       "    if ( options.theme === &#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;:-5.0, &quot;y&quot;:-1.0, &quot;z&quot;:-3.0}, {&quot;x&quot;:4.0, &quot;y&quot;:0.0, &quot;z&quot;:5.0}]; // bounds\n",
       "\n",
       "    if ( b[0].x === b[1].x ) {\n",
       "        b[0].x -= 1;\n",
       "        b[1].x += 1;\n",
       "    }\n",
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       "        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 = [];\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 = [{&quot;point&quot;: [0.0, 0.0, 0.0], &quot;size&quot;: 30.0, &quot;color&quot;: &quot;#000000&quot;, &quot;opacity&quot;: 1.0}];\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 = [];\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 = [{&quot;vertices&quot;: [{&quot;x&quot;: 3.9293981913502534, &quot;y&quot;: -0.9823495478375633, &quot;z&quot;: -2.9470486435126904}, {&quot;x&quot;: 3.9258461453268407, &quot;y&quot;: -0.9759937250972228, &quot;z&quot;: -2.9539033124573537}, {&quot;x&quot;: 3.9331354817159347, &quot;y&quot;: -0.9732838704289837, &quot;z&quot;: -2.9450874821613087}, {&quot;x&quot;: 3.9234656158124994, &quot;y&quot;: -0.9874871107661432, &quot;z&quot;: -2.9532462232535024}, {&quot;x&quot;: 3.9292837040505075, &quot;y&quot;: -0.9918805590871075, &quot;z&quot;: -2.944024289495837}, {&quot;x&quot;: 3.9352600098454844, &quot;y&quot;: -0.9831024738083595, &quot;z&quot;: -2.9389819101954506}, {&quot;x&quot;: -0.003552046023412369, &quot;y&quot;: 0.00635582274034045, &quot;z&quot;: -0.006854668944663308}, {&quot;x&quot;: 0.003737290365681299, &quot;y&quot;: 0.009065677408579675, &quot;z&quot;: 0.00196116135138184}, {&quot;x&quot;: -0.005932575537754049, &quot;y&quot;: -0.005137562928579778, &quot;z&quot;: -0.006197579740812137}, {&quot;x&quot;: -0.0001144872997458178, &quot;y&quot;: -0.0095310112495442, &quot;z&quot;: 0.003024354016853643}, {&quot;x&quot;: 0.005861818495230935, &quot;y&quot;: -0.0007529259707961455, &quot;z&quot;: 0.008066733317239961}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 0.0}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 2, 5], [2, 1, 6, 7], [1, 3, 8, 6], [3, 4, 9, 8], [4, 5, 10, 9], [5, 2, 7, 10], [7, 6, 11], [6, 8, 11], [8, 9, 11], [9, 10, 11], [10, 7, 11]], &quot;color&quot;: &quot;#ff0000&quot;, &quot;opacity&quot;: 1.0}, {&quot;vertices&quot;: [{&quot;x&quot;: 3.9999999999999996, &quot;y&quot;: -0.9999999999999999, &quot;z&quot;: -3.0}, {&quot;x&quot;: 3.9187420532800163, &quot;y&quot;: -0.963282079616542, &quot;z&quot;: -2.9676126503466804}, {&quot;x&quot;: 3.940610062447297, &quot;y&quot;: -0.9551525156118243, &quot;z&quot;: -2.941165159458545}, {&quot;x&quot;: 3.911600464736991, &quot;y&quot;: -0.9977622366233027, &quot;z&quot;: -2.965641382735127}, {&quot;x&quot;: 3.929054729451016, &quot;y&quot;: -1.0109425815861959, &quot;z&quot;: -2.9379755814621293}, {&quot;x&quot;: 3.946983646835946, &quot;y&quot;: -0.9846083257499518, &quot;z&quot;: -2.9228484435609703}, {&quot;x&quot;: 3.9293981913502534, &quot;y&quot;: -0.9823495478375633, &quot;z&quot;: -2.9470486435126904}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 2, 5], [2, 1, 6], [1, 3, 6], [3, 4, 6], [4, 5, 6], [5, 2, 6]], &quot;color&quot;: &quot;#ff0000&quot;, &quot;opacity&quot;: 1.0}, {&quot;vertices&quot;: [{&quot;x&quot;: -4.936360389693212, &quot;y&quot;: 0.0, &quot;z&quot;: 4.936360389693211}, {&quot;x&quot;: -4.929635404573572, &quot;y&quot;: 0.0030901699437494747, &quot;z&quot;: 4.94308537481285}, {&quot;x&quot;: -4.936360389693212, &quot;y&quot;: 0.01, &quot;z&quot;: 4.936360389693211}, {&quot;x&quot;: -4.932204120315437, &quot;y&quot;: -0.008090169943749474, &quot;z&quot;: 4.940516659070985}, {&quot;x&quot;: -4.940516659070986, &quot;y&quot;: -0.008090169943749474, &quot;z&quot;: 4.932204120315436}, {&quot;x&quot;: -4.943085374812851, &quot;y&quot;: 0.0030901699437494725, &quot;z&quot;: 4.929635404573571}, {&quot;x&quot;: 0.006724985119639573, &quot;y&quot;: 0.0030901699437494747, &quot;z&quot;: 0.0067249851196395745}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.01, &quot;z&quot;: 0.0}, {&quot;x&quot;: 0.0041562693777745345, &quot;y&quot;: -0.008090169943749474, &quot;z&quot;: 0.004156269377774535}, {&quot;x&quot;: -0.004156269377774534, &quot;y&quot;: -0.008090169943749474, &quot;z&quot;: -0.0041562693777745345}, {&quot;x&quot;: -0.006724985119639573, &quot;y&quot;: 0.0030901699437494725, &quot;z&quot;: -0.0067249851196395745}, {&quot;x&quot;: 0.0, &quot;y&quot;: 0.0, &quot;z&quot;: 0.0}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 2, 5], [2, 1, 6, 7], [1, 3, 8, 6], [3, 4, 9, 8], [4, 5, 10, 9], [5, 2, 7, 10], [7, 6, 11], [6, 8, 11], [8, 9, 11], [9, 10, 11], [10, 7, 11]], &quot;color&quot;: &quot;#0000ff&quot;, &quot;opacity&quot;: 1.0}, {&quot;vertices&quot;: [{&quot;x&quot;: -5.000000000000001, &quot;y&quot;: 0.0, &quot;z&quot;: 5.0}, {&quot;x&quot;: -4.916185434334293, &quot;y&quot;: 0.009270509831248424, &quot;z&quot;: 4.95653534505213}, {&quot;x&quot;: -4.936360389693212, &quot;y&quot;: 0.03, &quot;z&quot;: 4.936360389693211}, {&quot;x&quot;: -4.923891581559888, &quot;y&quot;: -0.02427050983124842, &quot;z&quot;: 4.948829197826535}, {&quot;x&quot;: -4.9488291978265355, &quot;y&quot;: -0.024270509831248423, &quot;z&quot;: 4.923891581559887}, {&quot;x&quot;: -4.956535345052131, &quot;y&quot;: 0.009270509831248417, &quot;z&quot;: 4.916185434334292}, {&quot;x&quot;: -4.936360389693212, &quot;y&quot;: 0.0, &quot;z&quot;: 4.936360389693211}], &quot;faces&quot;: [[0, 1, 2], [0, 3, 1], [0, 4, 3], [0, 5, 4], [0, 2, 5], [2, 1, 6], [1, 3, 6], [3, 4, 6], [4, 5, 6], [5, 2, 6]], &quot;color&quot;: &quot;#0000ff&quot;, &quot;opacity&quot;: 1.0}];\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"
    }
   ],
   "source": [
    "\n",
    "origin = point((0,0), color='black', size=30)\n",
    "pl = plot(a, color=\"red\") + plot(b, color=\"blue\") + origin\n",
    "pl.show(aspect_ratio=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "a2f5d361-a333-490c-885c-83724e50cdc9",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0\n"
     ]
    }
   ],
   "source": [
    "# Checking an answer.  Are these two vectors orthogonal?\n",
    "\n",
    "a = vector((2, -2, 3))\n",
    "b = vector((4, -1, -10/3))\n",
    "\n",
    "print(a * b)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "72d4d2c4-dff4-497e-8205-e869062c063d",
   "metadata": {},
   "source": [
    "## Projection of vectors\n",
    "\n",
    "A vector projection is a kind of \"shadow that one vector casts onto another.  When we want to find the projection of $\\vec{b}$ onto $\\vec{a}$, we write it as $\\text{proj}_{\\vec{a}}{\\vec{b}}$.\n",
    "\n",
    "Here is the formula to use:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\boldsymbol{\n",
    "\\text{proj}_{\\vec{w}}\\vec{v} = \\left( \\frac{\\vec{v}\\cdot\\vec{w}}{\\|\\vec{w}\\|^2} \\right) \n",
    "\\vec{w}}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "The following problem, (11.4 Question 6), finds the projection $\\vec{b}$ onto $\\vec{a}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "962d5798-a64c-444c-9436-cfa00fb0f4f3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(72/17, 0, -18/17)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = vector((4, 0, -1))\n",
    "B = vector ((5,0,2))\n",
    "\n",
    "\n",
    "def get_projection_b_onto_a(b, a):\n",
    "    \"\"\"project b onto a\"\"\"\n",
    "    return (b*a)/(a.norm()**2) * a\n",
    "\n",
    "get_projection_b_onto_a(B, A)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "42c37abb-25b5-4326-928e-7c132731c4d6",
   "metadata": {},
   "source": [
    "11.4 question 7.\n",
    "Find the angle of intersection, in radians, of two functions that intersect at (-2,3).\n",
    "\n",
    "The functions are $f(x) = x^2 + 2x + 3$ and $g(x) = -5x - 7$\n",
    "\n",
    "First, we need to find the derivatives, to get our slopes as something we can represent as vectors.  \n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "f^{\\prime}(x) = 2x + 2 \\\\\n",
    "g^{\\prime}(x) = -5\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Find the slope of tangent line for $f(x)$:\n",
    "\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "(y - 3) = -2(x - (-2)) \\\\\n",
    "y - 3 = -2x -4 \\\\\n",
    "y = -2x - 1 \\\\\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "The slope is -2 at $f(x)$ = -2.  (For confirmation, see https://www.desmos.com/calculator/5ftj0x5ccz)\n",
    "\n",
    "$g(x)$ is linear, so the slope is -5 wherever you go.\n",
    "\n",
    "Next, per the video, we need to create two vectors, lets call them $\\vec{F}$ and $\\vec{G}$  We can read these off the graph.  \n",
    "\n",
    "For $\\vec{F}$, we can take (-2,3) to the point (0,-1).  Subtracting we have the position vector, (-2,4)\n",
    "\n",
    "For vector $\\vec{G}$, we can take the points (-2,3) to the points (0, -7).  Subtracting we get a position vector (-2, 10).  \n",
    "\n",
    "Next, as we documented above, we'll need to use the formula below to find the angle begween the vectors.\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\cos \\theta &= \\frac{\\vec{v} \\cdot \\vec{w}}{\\|\\vec{v}\\| \\|\\vec{w}\\|}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Earlier, however, we didn't stop to implement that formula in Sage.  Let's do so now, and use it to find our angle between our two vectors, $\\vec{F} = (-2,4)$ and $\\vec{G} = (-2,10)$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "1c98e265-e105-4851-8d4e-264882ebb12b",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Raw angle:   arccos(11/130*sqrt(26)*sqrt(5))\n",
      "Evaluated:   0.266252049150925\n",
      "In degree:   15.2551187030578\n"
     ]
    }
   ],
   "source": [
    "def get_angle_between_vectors(v: vector, w: vector):\n",
    "    \"\"\"returns the angle between two vectors in radians.\"\"\"\n",
    "    cos_theta = (v*w)/(v.norm() * w.norm())\n",
    "    return acos(cos_theta)\n",
    "\n",
    "F = vector((-2, 4))\n",
    "G = vector((-2, 10))\n",
    "print(\"Raw angle:  \", get_angle_between_vectors(F, G))\n",
    "print(\"Evaluated:  \", N(get_angle_between_vectors(F, G)))\n",
    "print(\"In degree:  \", N(deg((get_angle_between_vectors(F, G)))))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6cf1c48f-b123-4865-8994-0a4687477462",
   "metadata": {},
   "source": [
    "## 3D Vector Cross Products\n",
    "\n",
    "Earlier we used the ```vector.cross_product``` method in Sage to find a cross-product.  Let's review how to do it without sage.  \n",
    "\n",
    "There is an unmemorizable formula for the vector cross-product, but fortunately, we can also think about it in terms of the determinant of a 3x3 matrix based on a and b.  We use the regular multiplication symbol for the cross-product:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\vec{a} \\times \\vec{b} = \\det{\\begin{vmatrix}\n",
    "i & j & k \\\\\n",
    "a_1 & a_2 & a_3 \\\\\n",
    "b_1 & b_2 & b_3 \\\\\n",
    "\\end{vmatrix}}\n",
    "= &\\left(\\det{\\begin{vmatrix}\n",
    "a_2 & a_3 \\\\\n",
    "b_2 & b_3 \\\\\n",
    "\\end{vmatrix}} \\cdot i\\right)  - \\left(\\det{\\begin{vmatrix}\n",
    "a_1 & a_3 \\\\\n",
    "b_1 & b_3 \\\\\n",
    "\\end{vmatrix}} \\cdot j\\right) + \n",
    "\\left(\\det{\\begin{vmatrix}\n",
    "a_1 & a_2 \\\\\n",
    "b_1 & b_2 \\\\\n",
    "\\end{vmatrix}}\n",
    "\\cdot k\\right)\\\\ \n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "The first thirty-eight minutes or so of [this video](https://www.youtube.com/watch?v=qqfhgStQ-cA) provides a good treatment of the details of how we got that formula in terms of three 2x2 matrices.  "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "79e6ee24-1f0e-4467-9bbd-ba4e7cde2870",
   "metadata": {},
   "source": [
    "Let's use this method to solve 11.5, question 3, which asks us to find the cross-product $\\vec{u} \\times \\vec{b}$ where:\n",
    "\n",
    "$\\vec{u} = 6i + 9j + 6k$\n",
    "\n",
    "$\\vec{v} = \\langle4, -8, 5\\rangle$\n",
    "\n",
    "Solving for\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\det{\\begin{vmatrix}\n",
    "i & j & k \\\\\n",
    "6 & 9 & 6 \\\\\n",
    "4 & -8 & 5\\\\\n",
    "\\end{vmatrix}}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "gives us:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "& [(45) - (-48)]i - [(30)-(24)]j + [(-48) -(36)]k \\\\\n",
    "\\\\\n",
    "= &\\;\\boxed{\\langle 93, -6, -84\\rangle}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Now, since doing this manually is error-prone as heck, let's check our result using Sage. \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "c62dccaf-fc2c-442b-8e78-e562e21c2269",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(93, -6, -84)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u = vector((6, 9, 6))\n",
    "v = vector((4, -8, 5))\n",
    "u.cross_product(v)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bd7a838e-8f69-44a4-bb41-bfd46ecbfa05",
   "metadata": {},
   "source": [
    "Sure enough, I had a sign error in the middle term, since corrected."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b3672228-9047-49e4-824e-b5aa2594177a",
   "metadata": {},
   "source": [
    "## Some Properties of the Cross Product\n",
    "\n",
    "1. The cross product is a vector that is orthogonal to both of the original two vectors.  Since they are orthogonal, the dot product will be zero."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "877d64f3-4009-4e1f-8c66-6ee95223bddb",
   "metadata": {},
   "outputs": [],
   "source": [
    "u = vector((3, 6, 9))\n",
    "v = vector((4, -8, -1))\n",
    "w = u.cross_product(v)\n",
    "\n",
    "# Show w is orthogonal to both u and v\n",
    "assert w * u == 0\n",
    "assert w * v == 0"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "706bdc52-7aa1-4b10-ab60-11230fce56db",
   "metadata": {},
   "source": [
    "2. If we consider two vectors as forming two sides of a parallelogram, then the magnitude of the cross product is equal to the area of this parallelogram.  If theta is the angle between them, then we may find this area using the [parallelogram area formula](https://en.wikipedia.org/wiki/Parallelogram)\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "|\\vec{A}\\times\\vec{B}| = |\\vec{A}| \\cdot |\\vec{B}| \\cdot \\sin\\theta\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "1348f6f0-401c-4d37-b137-0ef40a5b9f10",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "9*sqrt(101)\n",
      "9*sqrt(101)\n"
     ]
    }
   ],
   "source": [
    "# Show that a cross product represents the \n",
    "# area of a parallelogram between two vectors.\n",
    "\n",
    "u = vector((3,6,9))\n",
    "v = vector((4, -8, -1))\n",
    "w = u.cross_product(v)\n",
    "\n",
    "# get_angle_between_vectors defined earlier in this notebook.\n",
    "theta = get_angle_between_vectors(u, v)\n",
    "print(w.norm())\n",
    "print((u.norm() * v.norm() * sin(theta)).simplify())\n",
    "assert (w.norm() == u.norm() * v.norm() * sin(theta))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0f338447-bce3-4710-ac68-abda3a16c295",
   "metadata": {},
   "source": [
    "3. The cross product is \"anti-commutative\".  That is, if we reverse the order of the vectors, we get the negation of the cross-product."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "fcb78df0-badc-4f29-a473-b54ac3707541",
   "metadata": {},
   "outputs": [],
   "source": [
    "u = vector((3, 2, 1))\n",
    "v = vector((4, 3, -1))\n",
    "assert (u.cross_product(v) == - v.cross_product(u))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "aba6b22d-a399-45c2-b2ba-3e2767b3e84e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.34846922834953"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u = vector(RR, (3, 2, -4))\n",
    "v = vector(RR, (-6, 6, -4))\n",
    "w = vector(RR, (0,-2,-2))\n",
    "\n",
    "z = u.cross_product(w)\n",
    "z.norm() * 1/2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "741529d2-ff4d-4715-a19b-cbeab5b9cb81",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sqrt(673)"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P = vector((3, 2, -4))\n",
    "Q = vector((-6, 6, -4))\n",
    "R = vector((0, -2, -2))\n",
    "\n",
    "PQ = Q-P\n",
    "PR = R-P\n",
    "\n",
    "(PQ.cross_product(PR)).norm() * 1/2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8f8b208a-0718-4a66-a043-4b7b29739b29",
   "metadata": {},
   "source": [
    "PQ = <(-6 -3), (6-4), (-4 -(-4)> = <-9, 4, 0>\n",
    "PR = <(0 - 3), (-2 -2), (-2) - (-4)> = <-3, -4, 2>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "3f7ff289-7a25-471f-8fb7-04bedce68c3f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "45.8693797647189"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pq = vector(RR, (-9, 2, 0))\n",
    "pr = vector(RR, (-3, -4, 2))\n",
    "\n",
    "pq.cross_product(pr).norm()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1b7c471c-b78c-4b9f-951d-3f379b7c4e3c",
   "metadata": {},
   "source": [
    "# Triple Scalar Product\n",
    "\n",
    "This is used to find the volume of a parallepiped (a lopsided cube, i.e., a paralellogram in 3D).\n",
    "\n",
    "Formally we use $V = | u \\cdot (v \\times w)|$ (That's an abslolute value -- result will be scalar)\n",
    "\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "u \\cdot (v \\times w) = \\begin{vmatrix}\n",
    "u_1 & u_2 & u3 \\\\\n",
    "v_1 & v_2 & v_3 \\\\\n",
    "w_1 & w_2 & w_3\n",
    "\\end{vmatrix}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "So you'd work it like a 3D determinant cross product.  (The order doesn't matter of the three terms)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "884579e4-c72f-4644-a79f-f338edcfd527",
   "metadata": {},
   "outputs": [],
   "source": [
    "a = vector([3, 5, -1])\n",
    "b = vector([0, 1, 4])\n",
    "c = vector([2, 3, 1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "fd7ce10a-6e1a-4e89-9340-7b6d3618b19d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a * b.cross_product(c)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "4cfe4e79-b411-4bb1-bd8b-95fe13bf1a1b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b * c.cross_product(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "c2d10098-bd05-46c7-9c4e-da7514767bbb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-9"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a * c.cross_product(b)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6470d009-25f2-472c-b1e7-ef63e5e25e6f",
   "metadata": {},
   "source": [
    "## Finding the equation of a plane given 3 points\n",
    "\n",
    "11.6 Q 4.  \"Find a plane through the points (-8,-5,3), (-4,6,1), (3,4,-2)\"\n",
    "\n",
    "We need to find a normal vector to the plane as a first step.  Let's call these vectors, a, b & c, respectively.  Then:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "ac81ed3f-8d4a-4c2b-9b3a-8498287f060c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(37.0000000000000, 2.00000000000000, 85.0000000000000)"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = vector(RR, [-8,-5,3])\n",
    "b = vector(RR, [-4,6,1])\n",
    "c = vector(RR, [3,4,-2])\n",
    "\n",
    "bc = c - b\n",
    "ba = a - b\n",
    "\n",
    "normal_vector = ba.cross_product(bc)\n",
    "normal_vector"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ef181be0-291c-4dee-959c-493c6d212cb5",
   "metadata": {},
   "source": [
    "Next, we use the Point-Normal equation to find the equation of the plane.\n",
    "\n",
    "## Equation of a Plane (Point - Normal Form)\n",
    "\n",
    "The plane containing the point $(x_1, y_1, z_1)$ and having a normal vector $\\vec{n} = \\langle a, b, c \\rangle$ has equation:\n",
    "\n",
    "$$\\boldsymbol{a(x - x_1) + b(y - y_1) + c(z - z_1) = 0}$$\n",
    "\n",
    "Or a, b, and c terms are given by the components of our normal vector.  Meanwhile, we can use any point in the plane as the terms of our $x_1, y_1, z_1$.  We'll use point b.\n",
    "\n",
    "Given that, our equation is $37(x - (-4)) + 2(y - 6) + 85(z - 1) = 0$, and simplifying we have $37(x +4) + 2(y - 6) + 85(z - 1) = 0$\n",
    "\n",
    "[Checking it graphically on Desmos](https://www.desmos.com/3d/fm6wbveeqf), it is correct.\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "73aa9901-d006-46d3-9a8c-2a875f240b42",
   "metadata": {},
   "source": [
    "## Finding the intersection of two planes:\n",
    "\n",
    "In exercise 11.6 question 5, we're asked to find the equation of a line where two planes intersect.  The planes are $-4y\\ +\\ 5z\\ =\\ -7$ and $-9x\\ -7y\\ -4z\\ =\\ -25$, and they are [graphed here](https://www.desmos.com/3d/qcyywxvipn).  The [associated video](https://moer.maricopa.edu/assessment/watchvid.php?url=http%3A%2F%2Fyoutu.be%2FPlcZDfKv2_Q) tells us to find the normal vector for each plane, and then the cross product of those two normal vectors will give us the direction vector of their intersection.  Then given a point on that vector, we can figure out the equation.\n",
    "\n",
    "The Point-Normal equation from the last cell means we can read off our normal vector components from the coefficients of each term.  So we have two normal vectors:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\vec{n_1} &= \\langle0, -4, 5\\rangle \\\\\n",
    "\\vec{n_2} &= \\langle-9, -7, -4\\rangle\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "\n",
    "Their cross-product (see below), is $\\langle51, -45, -36\\rangle$\n",
    "\n",
    "Note we're asked to find a parametric equation based on $x(t) = 51t$  Copying our point normal equation from above, we have:  \n",
    "\n",
    "$$\\boldsymbol{a(x - x_1) + b(y - y_1) + c(z - z_1) = 0}$$\n",
    "\n",
    "So we want to find three points $P(0, , )$.  For points y and z, we have a system of equations:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "4y + 5z = -7 \\\\\n",
    "-7y - 4z = -25\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "We solve it in a code cell below using Sage.  $y =3, z = 1$ \n",
    "\n",
    "Multiplying row two by $\\frac{5}{4}$ gives $-\\frac{35}{4}y - 5z = - \\frac{125}{4}$\n",
    "\n",
    "Subtracting \n",
    "\n",
    "\n",
    "\n",
    "So at this point we have a point, $P$ on the line, and $P = \\left(0, 3, 1\\right)$ and a direction vector, $\\langle51, -45, -36\\rangle$\n",
    "\n",
    "So we have\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "x &= 0 + 51t \\\\\n",
    "y &= 3 -45t \\\\\n",
    "z &= 1 - 36t\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "3df5f865-fc97-4502-a0a7-5fc038c16ad5",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(51, -45, -36) (-51, 45, 36)\n"
     ]
    }
   ],
   "source": [
    "# Get the cross product for exercise below:\n",
    "n1 = vector([0, -4, 5])\n",
    "n2 = vector([-9, -7, -4])\n",
    "print(n1.cross_product(n2), n2.cross_product(n1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "7c424439-9246-436a-b606-03de2a6a4296",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3, 1)"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Use sage to solve the system of \n",
    "M = Matrix(QQ, [[-4, 5], [-7, -4]])\n",
    "M\n",
    "V = vector(QQ, [-7, -25])\n",
    "M.solve_right(V)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "6c089635-9733-4c66-bdfb-f067ec37baff",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "87/10"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(-174/-20)"
   ]
  }
 ],
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