{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# polymake and Optimization\n",
    "\n",
    "By Sebastian Pokutta and Marc E. Pfetsch.\n",
    "\n",
    "## Introduction\n",
    "\n",
    "Polymake offers many interesting features that can help (discrete) optimizers to analyze optimization problems. For example\n",
    "\n",
    "*  linear optimization can be performed exactly and visualized in small dimensions\n",
    "\n",
    "*  the convex hull of feasible points of an integer program can be computed and analyzed\n",
    "\n",
    "*  Hilbert bases can be computed\n",
    "\n",
    "There are several other tutorials that cover similar topics:\n",
    "\n",
    "*  [ILP and Hilbert Bases](ilp_and_hilbertbases.ipynb)\n",
    "\n",
    "*  [Gomory Cuts](branchnbound.ipynb)\n",
    "\n",
    "*  [Branch and Bound](gomory.ipynb)\n",
    "\n",
    "*  [Tutorial for Lattice Polytopes](lattice_polytopes_tutorial.ipynb)\n",
    "\n",
    "This tutorial is targeted towards the optimization community, since, surprisingly, polymake does not seem to be well known in this community. In particular, the community still tends to use the quite old program `porta` to compute convex hulls or inequality descriptions. While `porta` still does a decent job here, `polymake` offers a much broader feature set. Polymake supports several convex hull algorithms which might be better suited depending on the data. Moreover it offers many visualization tools that can help to better *understand* a given polytope. We think that polymake has many advantages for discrete optimizers and hope that this tutorial will help to spread the usage of polymake.\n",
    "\n",
    "You can find files of the example LPs in the folder demo/files/optimization in your polymake directory. \n",
    "\n",
    "## Input: lp2poly\n",
    "\n",
    "The first important step is to get the desired input into `polymake`. While it is possible to define polytopes in `polymake` right away, a more convenient methods, especially for people working in discrete optimization, is the possibility to read an optimization problem from a file. This has the advantage that input that is usually generated outside `polymake` can simply be loaded. A standard file format for linear (integer) programs is the so-called CPLEX LP-format, which is well known in the optimization community and can be read into `polymake` in a very straightforward way as shown in the following example:\n",
    "\n",
    "    \n",
    "    Minimize\n",
    "     obj:  x1 + x2 + x3\n",
    "    Subject to\n",
    "     C1: x1 + x2 + x3 <= 2\n",
    "    Bounds\n",
    "     0 <= x1 <= 1\n",
    "     0 <= x2 <= 1\n",
    "     0 <= x3 <= 1\n",
    "    End\n",
    "\n",
    "\n",
    "Thus, the file describes a 0/1-cube in three dimensions. It should be easy to adapt this format to other cases (If for example `x1` does not have any bounds you can write `x1 free` instead).\n",
    "\n",
    "Now assume that this example is contained in file `c3t.lp`. We create a polytope from the file via:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "$f=lp2poly('files/optimization/c3t.lp');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "The polytope `$f` is coded via floating point numbers:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Polytope<Rational>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print $f->type->full_name;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "We convert it to a rational polytope via:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "$p = new Polytope<Rational>($f);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "Now, `$p` contains data for the linear optimization problem in the file `ct3.lp`. The corresponding polyhedron is stored directly in `$p`. The information about the linear objective is contained in `$p->LP`.\n",
    "\n",
    "\n",
    "## Linear Optimization\n",
    "\n",
    "Polymake can be used to perform several actions related to linear optimization (linear programming - LP). For instance, one can exactly solve a linear program (via lrs or cdd). Before we explain the corresponding usage, we first need to have a linear optimization problem at hand.\n",
    "\n",
    "Assuming that we are given the above example in variable `$p`, the linear objective can be accessed via\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0 1 1 1"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print $p->LP->LINEAR_OBJECTIVE;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus - as described in the file - the objective function coefficients are 1 for all three variables (and there is an offset of 0).\n",
    "\n",
    "Now, we can solve the corresponding linear program via\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/html": [
       "<details><summary><pre style=\"display:inline\"><small>Click here for additional output</small></pre></summary>\n",
       "<pre>\n",
       "polymake: used package cdd\n",
       "  cddlib\n",
       "  Implementation of the double description method of Motzkin et al.\n",
       "  Copyright by Komei Fukuda.\n",
       "  http://www-oldurls.inf.ethz.ch/personal/fukudak/cdd_home/\n",
       "\n",
       "</pre>\n",
       "</details>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print $p->LP->MAXIMAL_VALUE;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus, the maximal value that we can obtain via the above linear objective function is 2. We can also get an optimal vertex via\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1 0 1 1"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print $p->LP->MAXIMAL_VERTEX;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This vertex corresponds to setting `x1=0, x2=1, x3=1`. The optimal face can also be computed:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{0 1 3}"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/html": [
       "<details><summary><pre style=\"display:inline\"><small>Click here for additional output</small></pre></summary>\n",
       "<pre>\n",
       "polymake: used package ppl\n",
       "  The Parma Polyhedra Library ([[wiki:external_software#PPL]]): A C++ library for convex polyhedra\n",
       "  and other numerical abstractions.\n",
       "  http://www.cs.unipr.it/ppl/\n",
       "\n",
       "</pre>\n",
       "</details>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print $p->LP->MAXIMAL_FACE;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This means that the optimal face is the convex hull of three vertices (with indices 4, 5, 6).\n",
    "\n",
    "Of course, by replacing `MAXIMAL` by `MINIMAL`, we obtain the corresponding values for minimization.\n",
    "\n",
    "The directed graph obtained by directing the graph of the polytope in the direction of increasing objective function can be obtained via\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
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       "<!--\n",
       "polymake for daniel\n",
       "Thu Aug 19 09:55:16 2021\n",
       "c3t\n",
       "-->\n",
       "\n",
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       "      <style>\n",
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       "   <script>\n",
       "    requirejs.config({\n",
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       "        WEBGL: '/kernelspecs/polymake/WebGL',\n",
       "      },\n",
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       "        'three': { exports: 'THREE'},\n",
       "        'SVGRenderer': { deps: [ 'three' ], exports: 'THREE.SVGRenderer' },\n",
       "        'WEBGL': { deps: [ 'three' ], exports: 'THREE.WEBGL' },\n",
       "        'Projector': { deps: [ 'three' ], exports: 'THREE.Projector' },\n",
       "        'TrackballControls': { deps: [ 'three' ], exports: 'THREE.TrackballControls' },\n",
       "        'OrbitControls': { deps: [ 'three' ], exports: 'THREE.OrbitControls' },\n",
       "      }\n",
       "    });\n",
       "    \n",
       "    require(['three'],function(THREE){\n",
       "        window.THREE = THREE;\n",
       "      require(['TrackballControls', 'OrbitControls', 'Projector', 'SVGRenderer', 'WEBGL'],\n",
       "               function(TrackballControls, OrbitControls, Projector, SVGRenderer, WEBGL) {\n",
       "    THREE.TrackballControls = TrackballControls;\n",
       "    THREE.OrbitControls = OrbitControls;\n",
       "    THREE.Projector = Projector;\n",
       "    THREE.SVGRenderer = SVGRenderer;\n",
       "    THREE.WEBGL = WEBGL;\n",
       "\n",
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       "\n",
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       "const explodableModel = true; \n",
       "const modelContains = { points: false, pointlabels: false, lines: false, edgelabels: false, faces: false, arrowheads: false };\n",
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       "\n",
       "var three = document.getElementById(\"model77883637379\");\n",
       "var scene = new THREE.Scene();\n",
       "var renderer = new THREE.WebGLRenderer( { antialias: true } );\n",
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       "renderer.setPixelRatio( window.devicePixelRatio );\n",
       "renderer.setClearColor(0xFFFFFF, 1);\n",
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       "three.appendChild(renderer.domElement);\n",
       "\n",
       "var frustumSize = 4;\n",
       "var cameras = [new THREE.PerspectiveCamera(75, 1, 0.1, 1000), new THREE.OrthographicCamera()];\n",
       "cameras.forEach(function(cam) {\n",
       "    cam.position.set(0, 0, 5);\n",
       "    cam.lookAt(0, 0, 0);  \n",
       "    cam.up.set(0, 1, 0);         \n",
       "});\n",
       "var controls = [new THREE.TrackballControls(cameras[0], three), new THREE.OrbitControls(cameras[1], three)];\n",
       "var camera, control;\n",
       "\n",
       "controls[0].zoomSpeed = 0.2;\n",
       "controls[0].rotateSpeed = 4;\n",
       "\n",
       "\n",
       "// class to allow move points together with labels and spheres\n",
       "var PMPoint = function (x,y,z) {\n",
       "   this.vector = new THREE.Vector3(x,y,z);\n",
       "   this.sprite = null;\n",
       "   this.sphere = null;\n",
       "}\n",
       "PMPoint.prototype.addLabel = function(labelsprite) {\n",
       "   this.sprite = labelsprite;\n",
       "   this.sprite.position.copy(this.vector);\n",
       "}\n",
       "PMPoint.prototype.addSphere = function(spheremesh) {\n",
       "   this.sphere = spheremesh;\n",
       "   this.sphere.position.copy(this.vector);\n",
       "}\n",
       "PMPoint.prototype.set = function(x,y,z) {\n",
       "   this.vector.set(x,y,z);\n",
       "   if (this.sprite) {\n",
       "      this.sprite.position.copy(this.vector);\n",
       "   }\n",
       "   if (this.sphere) {\n",
       "      this.sphere.position.copy(this.vector);\n",
       "   }\n",
       "}\n",
       "PMPoint.prototype.radius = function() {\n",
       "   if (this.sphere) {\n",
       "      return this.sphere.geometry.parameters.radius;\n",
       "   } else {\n",
       "      return 0;\n",
       "   }\n",
       "};\n",
       "// select the target node\n",
       "var target = document.querySelector('#model77883637379');\n",
       "\n",
       "// create an observer instance\n",
       "var observer = new MutationObserver(function(mutations) {\n",
       "   mutations.forEach(function(mutation) {\n",
       "      if (mutation.removedNodes && mutation.removedNodes.length > 0) {\n",
       "         cancelAnimationFrame(renderId);\n",
       "         observer.disconnect();\n",
       "         console.log(\"cancelled frame \"+renderId);\n",
       "      }\n",
       "   });\n",
       "});\n",
       "\n",
       "// configuration of the observer:\n",
       "var config = { childList: true, characterData: true }\n",
       "\n",
       "// pass in the target node, as well as the observer options\n",
       "while (target) {\n",
       "   if (target.className==\"output\") {\n",
       "      observer.observe(target, config);\n",
       "      break;\n",
       "   }\n",
       "   target = target.parentNode;\n",
       "}\n",
       "\n",
       "// COMMON_CODE_BLOCK_END\n",
       "\n",
       "var obj0 = new THREE.Object3D();\n",
       "obj0.name = \"c3t\";\n",
       "obj0.userData.explodable = 1;\n",
       "obj0.userData.points = [];\n",
       "obj0.userData.points.push(new PMPoint(0, 1, 1));\n",
       "obj0.userData.points.push(new PMPoint(1, 0, 1));\n",
       "obj0.userData.points.push(new PMPoint(1, 0, 0));\n",
       "obj0.userData.points.push(new PMPoint(1, 1, 0));\n",
       "obj0.userData.points.push(new PMPoint(0, 1, 0));\n",
       "obj0.userData.points.push(new PMPoint(0, 0, 1));\n",
       "obj0.userData.points.push(new PMPoint(0, 0, 0));\n",
       "\n",
       "obj0.userData.pointradii = 0.02;\n",
       "   <!-- Vertex style -->\n",
       "obj0.userData.pointmaterial = new THREE.MeshBasicMaterial( { color: 0xFF0000, side: THREE.DoubleSide, transparent: false } );\n",
       "obj0.userData.pointlabels = [\"0\", \"1\", \"2\", \"3\", \"4\", \"5\", \"6\"];\n",
       "obj0.userData.edgeindices = [0, 1, 1, 2, 0, 3, 1, 3, 2, 3, 0, 4, 3, 4, 0, 5, 1, 5, 2, 6, 4, 6, 5, 6];\n",
       "obj0.userData.facets = [[1, 3, 0], [0, 4, 6, 5], [2, 3, 1], [1, 5, 6, 2], [0, 3, 4], [4, 3, 2, 6], [1, 0, 5]];\n",
       "   <!-- Facet style -->\n",
       "obj0.userData.facetmaterial = new THREE.MeshBasicMaterial( { color: 0x77EC9E, depthFunc: THREE.LessDepth, opacity: 1, polygonOffset: true, polygonOffsetFactor: 1, polygonOffsetUnits: 0.5, side: THREE.DoubleSide, transparent: true } );\n",
       "init_object(obj0);\n",
       "scene.add(obj0);\n",
       "\n",
       "var obj1 = new THREE.Object3D();\n",
       "obj1.name = \"GRAPH directed with LP unnamed#0\";\n",
       "obj1.userData.explodable = 1;\n",
       "obj1.userData.points = [];\n",
       "obj1.userData.points.push(new PMPoint(0, 1, 1));\n",
       "obj1.userData.points.push(new PMPoint(1, 0, 1));\n",
       "obj1.userData.points.push(new PMPoint(1, 0, 0));\n",
       "obj1.userData.points.push(new PMPoint(1, 1, 0));\n",
       "obj1.userData.points.push(new PMPoint(0, 1, 0));\n",
       "obj1.userData.points.push(new PMPoint(0, 0, 1));\n",
       "obj1.userData.points.push(new PMPoint(0, 0, 0));\n",
       "\n",
       "obj1.userData.edgeindices = [2, 1, 2, 3, 4, 0, 4, 3, 5, 0, 5, 1, 6, 2, 6, 4, 6, 5];\n",
       "   <!-- Edge style -->\n",
       "obj1.userData.edgematerial = new THREE.LineBasicMaterial( { color: 0x000000, depthTest: true, linewidth: 1.5, transparent: false } );\n",
       "obj1.userData.arrowstyle = 1;\n",
       "init_object(obj1);\n",
       "scene.add(obj1);\n",
       "\n",
       "// COMMON_CODE_BLOCK_BEGIN\n",
       "function textSpriteMaterial(message, parameters) {\n",
       "    if ( parameters === undefined ) parameters = {};\n",
       "    var fontface = \"Helvetica\";\n",
       "    var fontsize = parameters.hasOwnProperty(\"fontsize\") ? parameters[\"fontsize\"] : 15;\n",
       "    fontsize = fontsize*10;\n",
       "    var lines = message.split('\\\\n');\n",
       "    var size = 512;\n",
       "    for(var i = 0; i<lines.length; i++){\n",
       "        var tmp = lines[i].length;\n",
       "        while(tmp*fontsize > size){\n",
       "           fontsize--;\n",
       "        }\n",
       "    }\n",
       "    \n",
       "    var canvas = document.createElement('canvas');\n",
       "    canvas.width = size;\n",
       "    canvas.height = size;\n",
       "    var context = canvas.getContext('2d');\n",
       "    context.fillStyle = \"rgba(255, 255, 255, 0)\";\n",
       "    context.fill();\n",
       "    context.font = fontsize + \"px \" + fontface;\n",
       "    \n",
       "    // text color\n",
       "    context.fillStyle = \"rgba(0, 0, 0, 1.0)\";\n",
       "     for(var i = 0; i<lines.length; i++){\n",
       "        context.fillText(lines[i], size/2, size/2+i*fontsize);\n",
       "     }\n",
       "    \n",
       "    // canvas contents will be used for a texture\n",
       "    var texture = new THREE.Texture(canvas);\n",
       "    texture.needsUpdate = true;\n",
       "    \n",
       "    var spriteMaterial = new THREE.SpriteMaterial({map: texture, depthTest: true, depthWrite: false, polygonOffset: true, polygonOffsetFactor: -1, polygonOffsetUnits: 1 });\n",
       "    return spriteMaterial;\n",
       "}\n",
       "\n",
       "\n",
       "// ---------------------- INITIALIZING OBJECTS--------------------------------------\n",
       "// ---------------------------------------------------------------------------------\n",
       "\n",
       "function init_object(obj) {\n",
       "    if (obj.userData.hasOwnProperty(\"pointmaterial\")) {\n",
       "        init_points(obj);\n",
       "        modelContains.points = true;\n",
       "    }\n",
       "    if (obj.userData.hasOwnProperty(\"pointlabels\")) {\n",
       "        init_pointlabels(obj);\n",
       "        modelContains.pointlabels = true;\n",
       "    }\n",
       "    if (obj.userData.hasOwnProperty(\"edgematerial\")) {\n",
       "        init_lines(obj);\n",
       "        modelContains.lines = true;\n",
       "    }\n",
       "    if (obj.userData.hasOwnProperty(\"edgelabels\")) {\n",
       "        init_edgelabels(obj);\n",
       "        modelContains.edgelabels = true;\n",
       "    }\n",
       "    if (obj.userData.hasOwnProperty(\"arrowstyle\")) {\n",
       "        init_arrowheads(obj);\n",
       "        modelContains.arrowheads = true;\n",
       "    }\n",
       "    if (obj.userData.hasOwnProperty(\"facetmaterial\")) {\n",
       "        init_faces(obj);\n",
       "        modelContains.faces = true;\n",
       "    }\n",
       "}\n",
       "\n",
       "function init_points(obj) {\n",
       "    var pointgroup = new THREE.Group();\n",
       "    pointgroup.name = \"points\";\n",
       "    var points = obj.userData.points;\n",
       "    var radii = obj.userData.pointradii;\n",
       "    var materials = obj.userData.pointmaterial;\n",
       "    var geometry,material;\n",
       "    if (!Array.isArray(radii)) {\n",
       "        geometry = new THREE.SphereBufferGeometry(radii);  \n",
       "    }\n",
       "    if (!Array.isArray(materials)) {\n",
       "        material = materials;\n",
       "    }\n",
       "    for (var i=0; i<points.length; i++) {\n",
       "        var point = points[i];\n",
       "        if (Array.isArray(radii)) {\n",
       "            if (radii[i] == 0) {\n",
       "                continue;\n",
       "            }\n",
       "            geometry = new THREE.SphereBufferGeometry(radii[i]);  \n",
       "        } \n",
       "        if (Array.isArray(materials)) {\n",
       "            material = materials[i];     \n",
       "        } \n",
       "        var sphere = new THREE.Mesh(geometry, material);\n",
       "        point.addSphere(sphere);\n",
       "        pointgroup.add(sphere);\n",
       "    }\n",
       "    obj.add(pointgroup);\n",
       "}\n",
       "\n",
       "function init_pointlabels(obj) {\n",
       "    var points = obj.userData.points;\n",
       "    var labels = obj.userData.pointlabels;\n",
       "    var pointlabels = new THREE.Group();\n",
       "    pointlabels.name = \"pointlabels\";\n",
       "    if (Array.isArray(labels)) {\n",
       "        for (var i=0; i<points.length; i++) {\n",
       "            var point = points[i];\n",
       "            var spriteMaterial = textSpriteMaterial( labels[i] );\n",
       "\t        var sprite = new THREE.Sprite(spriteMaterial);\n",
       "            point.addLabel(sprite);\n",
       "            pointlabels.add(sprite);\n",
       "        }\n",
       "    } else {\n",
       "        var spriteMaterial = textSpriteMaterial( labels );\n",
       "        for (var i=0; i<points.length; i++) {\n",
       "            var point = points[i];\n",
       "\t        var sprite = new THREE.Sprite(spriteMaterial);\n",
       "            point.addLabel(sprite);\n",
       "            pointlabels.add(sprite);\n",
       "        }\n",
       "    }\n",
       "    obj.add(pointlabels);\n",
       "}\n",
       "\n",
       "function init_lines(obj) {\n",
       "    var edgeindices = obj.userData.edgeindices;\n",
       "    var points = obj.userData.points;\n",
       "    var materials = obj.userData.edgematerial;\n",
       "    var geometry = new THREE.BufferGeometry();\n",
       "    var bufarr = new Float32Array( obj.userData.edgeindices.length * 3 );\n",
       "    var bufattr = new THREE.Float32BufferAttribute( bufarr, 3 );\n",
       "    var geometry = new THREE.BufferGeometry();\n",
       "    geometry.setAttribute('position', bufattr);\n",
       "    if (Array.isArray(materials)) {     \n",
       "        for (var i=0; i<materials.length; i++) {\n",
       "            geometry.addGroup(2*i,2,i);\n",
       "        }\n",
       "    }\n",
       "    var lines = new THREE.LineSegments(geometry, materials);\n",
       "    lines.name = \"lines\";\n",
       "    obj.add(lines);\n",
       "    updateEdgesPosition(obj);\n",
       "}\n",
       "\n",
       "function init_edgelabels(obj) {\n",
       "    var points = obj.userData.points;\n",
       "    var edgeindices = obj.userData.edgeindices;\n",
       "    var labels = obj.userData.edgelabels;\n",
       "    var edgelabels = new THREE.Group();\n",
       "    edgelabels.name = \"edgelabels\";\n",
       "    if (Array.isArray(labels)) {\n",
       "        for (var i=0; i<edgeindices.length/2; i++) {\n",
       "            var spriteMaterial = textSpriteMaterial( labels[i] );\n",
       "            var sprite = new THREE.Sprite(spriteMaterial);\n",
       "            sprite.position.copy(new THREE.Vector3().addVectors(points[edgeindices[2*i]].vector,points[edgeindices[2*i+1]].vector).multiplyScalar(0.5));\n",
       "            edgelabels.add(sprite);\n",
       "        }\n",
       "    } else {\n",
       "        var spriteMaterial = textSpriteMaterial( labels );\n",
       "        for (var i=0; i<edgeindices.length/2; i++) {\n",
       "            var sprite = new THREE.Sprite(spriteMaterial);\n",
       "            sprite.position.copy(new THREE.Vector3().addVectors(points[edgeindices[2*i]].vector,points[edgeindices[2*i+1]].vector).multiplyScalar(0.5));\n",
       "            edgelabels.add(sprite);\n",
       "        }\n",
       "    }\n",
       "    obj.add(edgelabels);\n",
       "}\n",
       "\n",
       "function init_arrowheads(obj) {\n",
       "    var arrowheads = new THREE.Group();\n",
       "    arrowheads.name = \"arrowheads\";\n",
       "    var arrowstyle = obj.userData.arrowstyle;\n",
       "    var edgeindices = obj.userData.edgeindices;\n",
       "    var edgematerials = obj.userData.edgematerial;\n",
       "    var points = obj.userData.points;\n",
       "    var material;\n",
       "    if (!Array.isArray(edgematerials)) {\n",
       "        material = new THREE.MeshBasicMaterial( {color: edgematerials.color} );\n",
       "    }\n",
       "\n",
       "    for (var i=0; i<edgeindices.length; i=i+2) {\n",
       "        var start = points[edgeindices[i]];\n",
       "        var end = points[edgeindices[i+1]];\n",
       "        var dist = start.vector.distanceTo( end.vector ) - start.radius() - end.radius();\n",
       "        if (dist <= 0) {\n",
       "            continue;\n",
       "        }\n",
       "        var dir = new THREE.Vector3().subVectors(end.vector,start.vector);\n",
       "        dir.normalize();\n",
       "        var axis = new THREE.Vector3().set(dir.z,0,-dir.x);\n",
       "        axis.normalize();\n",
       "        var radians = Math.acos( dir.y );\n",
       "        var radius = dist/25;\n",
       "        var height = dist/5;\n",
       "        var geometry = new THREE.ConeBufferGeometry(radius,height);\n",
       "        var position = new THREE.Vector3().addVectors(start.vector,dir.clone().multiplyScalar(start.radius()+dist-height/2));\n",
       "        if (Array.isArray(edgematerials)) {\n",
       "            material = new THREE.MeshBasicMaterial( {color: edgematerials[i].color} );\n",
       "        }\n",
       "        var cone = new THREE.Mesh( geometry, material );\n",
       "        cone.quaternion.setFromAxisAngle(axis,radians);;\n",
       "        cone.position.copy(position);;\n",
       "        arrowheads.add(cone);\n",
       "    }\n",
       "    obj.add(arrowheads);\n",
       "}\n",
       "\n",
       "function init_faces(obj) {\n",
       "    var points = obj.userData.points;\n",
       "    var facets = obj.userData.facets;\n",
       "    obj.userData.triangleindices = [];\n",
       "    for (var i=0; i<facets.length; i++) {\n",
       "        facet = facets[i];\n",
       "        for (var t=0; t<facet.length-2; t++) {\n",
       "            obj.userData.triangleindices.push(facet[0],facet[t+1],facet[t+2]);  \n",
       "        }\n",
       "    }\n",
       "    var bufarr = new Float32Array( obj.userData.triangleindices.length * 3 );\n",
       "    var bufattr = new THREE.Float32BufferAttribute(bufarr,3);\n",
       "    \n",
       "    var materials = obj.userData.facetmaterial;\n",
       "    var geometry = new THREE.BufferGeometry();\n",
       "    geometry.setAttribute('position',bufattr);\n",
       "    if (Array.isArray(materials)) {\n",
       "        var tricount = 0;\n",
       "        var facet;\n",
       "        for (var i=0; i<facets.length; i++) {\n",
       "            facet = facets[i];\n",
       "            geometry.addGroup(tricount,(facet.length-2)*3,i);\n",
       "            tricount += (facet.length-2)*3;\n",
       "        }\n",
       "    }\n",
       "    var mesh = new THREE.Mesh(geometry, materials);\n",
       "    mesh.name = \"faces\";\n",
       "    obj.add(mesh); \n",
       "    updateFacesPosition(obj);\n",
       "}\n",
       "// //INITIALIZING\n",
       "\n",
       "\n",
       "function updateFacesPosition(obj) {\n",
       "    var points = obj.userData.points;\n",
       "    var indices = obj.userData.triangleindices;\n",
       "    var faces = obj.getObjectByName(\"faces\");\n",
       "    var ba = faces.geometry.getAttribute(\"position\");\n",
       "    for (var i=0; i<indices.length; i++) {\n",
       "        ba.setXYZ(i, points[indices[i]].vector.x, points[indices[i]].vector.y ,points[indices[i]].vector.z); \n",
       "    }\n",
       "    faces.geometry.attributes.position.needsUpdate = true;\n",
       "    \n",
       "}\n",
       "\n",
       "function updateEdgesPosition(obj) {\n",
       "    var points = obj.userData.points;\n",
       "    var indices = obj.userData.edgeindices;\n",
       "    var lines = obj.getObjectByName(\"lines\");\n",
       "    var ba = lines.geometry.getAttribute(\"position\"); \n",
       "    for (var i=0; i<indices.length; i++) {\n",
       "        ba.setXYZ(i, points[indices[i]].vector.x, points[indices[i]].vector.y ,points[indices[i]].vector.z); \n",
       "    }\n",
       "    lines.geometry.attributes.position.needsUpdate = true;\n",
       "}\n",
       "\n",
       "function onWindowResize() {\n",
       "    renderer.setSize( three.clientWidth, three.clientHeight );\n",
       "    svgRenderer.setSize( three.clientWidth, three.clientHeight );\n",
       "    updateCamera();\n",
       "}\n",
       "\n",
       "function updateCamera() {\n",
       "    var width = three.clientWidth;\n",
       "    var height = three.clientHeight;\n",
       "    var aspect = width / height;\n",
       "    if (camera.type == \"OrthographicCamera\") {\n",
       "        camera.left = frustumSize * aspect / - 2;\n",
       "        camera.right = frustumSize * aspect / 2;\n",
       "        camera.top = frustumSize / 2;\n",
       "        camera.bottom = - frustumSize / 2;\n",
       "    } else if (camera.type == \"PerspectiveCamera\") {\n",
       "        camera.aspect = aspect;\n",
       "    }\n",
       "    camera.updateProjectionMatrix();\n",
       "}\n",
       "\n",
       "function changeCamera(event) {\n",
       "    var selindex = event.currentTarget.selectedIndex;\n",
       "    camera = cameras[selindex];\n",
       "    control = controls[selindex];\n",
       "    control.enabled = true; \n",
       "    for (var i=0; i<controls.length; i++) {\n",
       "        if (i!=selindex) {\n",
       "            controls[i].enabled = false;\n",
       "        }\n",
       "    }\n",
       "    updateCamera();\n",
       "}\n",
       "\n",
       "var camtypenode = document.getElementById('cameraType_0');\n",
       "camtypenode.onchange = changeCamera;\n",
       "camtypenode.dispatchEvent(new Event('change'));\n",
       "\n",
       "onWindowResize();\n",
       "window.addEventListener('resize', onWindowResize);\t\n",
       "\n",
       "\n",
       "var xRotationEnabled = false;\n",
       "var yRotationEnabled = false;\n",
       "var zRotationEnabled = false;\n",
       "var rotationSpeedFactor = 1;\n",
       "var settingsShown = false;\n",
       "var labelsShown = true;\n",
       "var intervals = [];\n",
       "var timeouts = [];\n",
       "var explodingSpeed = 0.05;\n",
       "var explodeScale = 0;\n",
       "var XMLS = new XMLSerializer();\n",
       "var svgElement;\n",
       "var renderId;\n",
       "\n",
       "var render = function () {\n",
       "\n",
       "\trenderId = requestAnimationFrame(render);\n",
       "\n",
       "//\tcomment in for automatic explosion\n",
       "//\texplode(updateFactor());\n",
       "\n",
       "    var phi = 0.02 * rotationSpeedFactor;\n",
       "\n",
       "    if (xRotationEnabled) {\n",
       "        scene.rotation.x += phi;\n",
       "    }\n",
       "    if (yRotationEnabled) {\n",
       "        scene.rotation.y += phi;\n",
       "    }\n",
       "    if (zRotationEnabled) {\n",
       "        scene.rotation.z += phi;\n",
       "    }\n",
       "\n",
       "    control.update();\n",
       "    renderer.render(scene, camera);\n",
       "};\n",
       "\n",
       "if ( THREE.WEBGL.isWebGLAvailable() ) {\n",
       "\trender();\n",
       "} else {\n",
       "\tvar warning = WEBGL.getWebGLErrorMessage();\n",
       "\tthree.appendChild( warning );\n",
       "}\n",
       "    \n",
       "function changeTransparency() {\n",
       "    var opacity = 1-Number(event.currentTarget.value);\n",
       "    for (var i=0; i<scene.children.length; i++) {\n",
       "        child = scene.children[i];\n",
       "        if ( child.userData.hasOwnProperty(\"facetmaterial\") ) {\n",
       "            if (Array.isArray(child.userData.facetmaterial)) {\n",
       "                for (var j=0; j<child.userData.facetmaterial.length; j++) {\n",
       "                    child.userData.facetmaterial[j].opacity = opacity;\n",
       "                }\n",
       "            } else {\n",
       "                child.userData.facetmaterial.opacity = opacity;\n",
       "            }    \n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "function changeRotationX(event){\n",
       "    xRotationEnabled = event.currentTarget.checked;\n",
       "}\t\n",
       "\n",
       "function changeRotationY(event){\n",
       "    yRotationEnabled = event.currentTarget.checked;\n",
       "}\t\n",
       "\n",
       "function changeRotationZ(event){\n",
       "    zRotationEnabled = event.currentTarget.checked;\n",
       "}\t\n",
       "\n",
       "\n",
       "function changeRotationSpeedFactor(event){\n",
       "    rotationSpeedFactor = Number(event.currentTarget.value);\n",
       "}\n",
       "\n",
       "function resetScene(){\n",
       "    scene.rotation.set(0,0,0);\n",
       "    camera.position.set(0,0,5);\n",
       "    camera.up.set(0,1,0);\n",
       "}\n",
       "\n",
       "function showSettings(event){\n",
       "    document.getElementById('settings_0').style.visibility = 'visible';\n",
       "    document.getElementById('showSettingsButton_0').style.visibility = 'hidden';\n",
       "    document.getElementById('hideSettingsButton_0').style.visibility = 'visible';\n",
       "    settingsShown = true;\n",
       "}\n",
       "\n",
       "function hideSettings(event){\n",
       "    document.getElementById('settings_0').style.visibility = 'hidden';\n",
       "    document.getElementById('showSettingsButton_0').style.visibility = 'visible';\n",
       "    document.getElementById('hideSettingsButton_0').style.visibility = 'hidden';\n",
       "    settingsShown = false;\n",
       "}\n",
       "\n",
       "\n",
       "\n",
       "var pos = 150* Math.PI;\n",
       "\n",
       "function updateFactor() {\n",
       "    pos++;\n",
       "    return Math.sin(.01*pos)+1;\n",
       "}\n",
       "\n",
       "// ------------------------ FOLDING ------------------------------------------------\n",
       "// ---------------------------------------------------------------------------------\n",
       "// rotate point p around axis defined by points p1 and p2 by given angle\n",
       "function rotate(p, p1, p2, angle ){   \n",
       "    angle = -angle;\n",
       "    var x = p.x, y = p.y, z = p.z, \n",
       "    a = p1.x, b = p1.y, c = p1.z, \n",
       "    u = p2.x-p1.x, v = p2.y-p1.y, w = p2.z-p1.z;\n",
       "    var result = [];\n",
       "    var L = u*u + v*v + w*w;\n",
       "    var sqrt = Math.sqrt;\n",
       "    var cos = Math.cos;\n",
       "    var sin = Math.sin;\n",
       "\n",
       "    result[0] = ((a*(v*v+w*w)-u*(b*v+c*w-u*x-v*y-w*z))*(1-cos(angle))+L*x*cos(angle)+sqrt(L)*(-c*v+b*w-w*y+v*z)*sin(angle))/L;\n",
       "    result[1] = ((b*(u*u+w*w)-v*(a*u+c*w-u*x-v*y-w*z))*(1-cos(angle))+L*y*cos(angle)+sqrt(L)*(c*u-a*w+w*x-u*z)*sin(angle))/L;\n",
       "    result[2] = ((c*(u*u+v*v)-w*(a*u+b*v-u*x-v*y-w*z))*(1-cos(angle))+L*z*cos(angle)+sqrt(L)*(-b*u+a*v-v*x+u*y)*sin(angle))/L;\n",
       "\n",
       "    return result;\n",
       "}\n",
       "\n",
       "var fold = function(event){\n",
       "    var obj = foldables[Number(event.currentTarget.name)];\n",
       "    var foldvalue = Number(event.currentTarget.value);\n",
       "    var scale = foldvalue - obj.userData.oldscale;\n",
       "\n",
       "    for (var j=0; j<obj.userData.axes.length; j++) {\n",
       "        rotateVertices(obj, j, scale);\n",
       "    }\n",
       "    update(obj);\n",
       "    obj.userData.oldscale += scale;\n",
       "    lookAtBarycenter(obj);\n",
       "}\n",
       "\n",
       "function lookAtBarycenter(obj){\n",
       "    control.target = barycenter(obj);\n",
       "}\n",
       "\n",
       "function barycenter(obj) {\n",
       "    var center = new THREE.Vector3(0,0,0);\n",
       "    var points = obj.userData.points;\n",
       "    for (var i=0; i<points.length; i++){\n",
       "        center.add(points[i].vector);\n",
       "    }\n",
       "    center.divideScalar(points.length);\n",
       "    return center;\n",
       "}\n",
       "\n",
       "function rotateVertices(obj, edge, scale) {\n",
       "    var axes = obj.userData.axes;\n",
       "    var subtrees = obj.userData.subtrees;\n",
       "    var points = obj.userData.points;\n",
       "    var angles = obj.userData.angles;\n",
       "    if (edge < axes.length){\n",
       "        for (var j=0; j<subtrees[edge].length; j++){\n",
       "            var rotP = rotate(points[subtrees[edge][j]].vector, points[axes[edge][0]].vector,points[axes[edge][1]].vector, scale * (Math.PI - angles[edge]));\n",
       "            points[subtrees[edge][j]].set(rotP[0],rotP[1],rotP[2]);\n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "function update(obj) {\n",
       "   updateFacesPosition(obj);\n",
       "   updateEdgesPosition(obj);\n",
       "}\n",
       "\n",
       "if (foldables.length) {\n",
       "    var settings = document.getElementById('settings_0');\n",
       "    var foldDiv = document.createElement('div');\n",
       "    foldDiv.id = 'fold_0';\n",
       "    var title = document.createElement('strong');\n",
       "    title.innerHTML = 'Fold';\n",
       "    foldDiv.appendChild(title);\n",
       "    foldDiv.className = 'group';\n",
       "    for (var i=0; i<foldables.length; i++) {\n",
       "        var range = document.createElement('input');\n",
       "        range.type = 'range';\n",
       "        range.min = 0;\n",
       "        range.max = 1;\n",
       "        range.value = 0;\n",
       "        range.step = 0.001;\n",
       "        range.name = String(i);\n",
       "        range.oninput = fold;\n",
       "        foldDiv.appendChild(range);\n",
       "    }\n",
       "    lookAtBarycenter(foldables[0]);\n",
       "    settings.insertBefore(foldDiv,settings.childNodes[0]);\n",
       "}\n",
       "\n",
       "    \n",
       "// ---------------------- EXPLOSION ------------------------------------------------\n",
       "// ---------------------------------------------------------------------------------\n",
       "\n",
       "if (explodableModel) {\n",
       "    for (var i=0; i<scene.children.length; i++) {\n",
       "        obj = scene.children[i];\n",
       "        if ( obj.userData.explodable ) {\n",
       "            computeCentroid(obj);\n",
       "        }\n",
       "    }\n",
       "    document.getElementById('explodeRange_0').oninput = triggerExplode;\n",
       "    document.getElementById('explodeCheckbox_0').onchange = triggerAutomaticExplode;\n",
       "    document.getElementById('explodingSpeedRange_0').oninput = setExplodingSpeed;\n",
       "}\n",
       "\n",
       "function computeCentroid(obj) {\n",
       "    centroid = new THREE.Vector3();\n",
       "    obj.userData.points.forEach(function(pmpoint) {\n",
       "        centroid.add(pmpoint.vector);\t\t\t\n",
       "    });\n",
       "    centroid.divideScalar(obj.userData.points.length);\n",
       "    obj.userData.centroid = centroid;\n",
       "}\n",
       "\n",
       "function explode(factor) {\n",
       "    for (var i=0; i<scene.children.length; i++) {\n",
       "        var obj = scene.children[i];\n",
       "        if (obj.userData.hasOwnProperty(\"centroid\")) { \n",
       "            var c = obj.userData.centroid;\n",
       "            obj.position.set(c.x*factor, c.y*factor, c.z*factor);\n",
       "        }\n",
       "    }\t\n",
       "}\n",
       "\n",
       "function triggerExplode(event){\n",
       "    explodeScale = Number(event.currentTarget.value);\n",
       "    explode(explodeScale);\n",
       "}\n",
       "\n",
       "function setExplodingSpeed(event){\n",
       "    explodingSpeed = Number(event.currentTarget.value);\n",
       "}\n",
       "\n",
       "function triggerAutomaticExplode(event){\n",
       "    if (event.currentTarget.checked){\n",
       "        startExploding();\n",
       "    } else {\n",
       "        clearIntervals();\n",
       "    }\t\n",
       "}\n",
       "\n",
       "function startExploding(){\n",
       "    intervals.push(setInterval(explodingInterval, 25));\n",
       "}\n",
       "\n",
       "\n",
       "function explodingInterval(){\n",
       "    explodeScale += explodingSpeed;\n",
       "    if (explodeScale <= 6){ \n",
       "        explode(explodeScale);\n",
       "    }\n",
       "    else{\n",
       "        explode(6);\n",
       "        explodeScale = 6;\n",
       "        clearIntervals();\n",
       "        timeouts.push(setTimeout(startUnexploding, 3000));\n",
       "    }\n",
       "    document.getElementById('explodeRange_0').value = explodeScale;\n",
       "}\n",
       "\n",
       "\n",
       "function startUnexploding(){\n",
       "    intervals.push(setInterval(unexplodingInterval, 25));\n",
       "}\n",
       "\n",
       "function unexplodingInterval(){\n",
       "    explodeScale -= explodingSpeed;\n",
       "    if (explodeScale >= 0){\t\n",
       "        explode(explodeScale);\n",
       "    }\n",
       "    else {\n",
       "        explode(0);\n",
       "        explodeScale = 0;\n",
       "        clearIntervals();\n",
       "        timeouts.push(setTimeout(startExploding, 3000));\n",
       "    }\n",
       "    document.getElementById('explodeRange_0').value = explodeScale;\n",
       "}\n",
       "\n",
       "function clearIntervals(){\n",
       "    intervals.forEach(function(interval){\n",
       "        clearInterval(interval);\n",
       "    });\n",
       "    intervals = [];\n",
       "    timeouts.forEach(function(timeout){\n",
       "        clearTimeout(timeout);\n",
       "    });\n",
       "    timeouts = [];\n",
       "}\n",
       "\n",
       "// ---------------------- DISPLAY --------------------------------------------------\n",
       "// ---------------------------------------------------------------------------------\n",
       "\n",
       "const objectTypeInnerHTMLs = { points: \"Points\", pointlabels: \"Point labels\", lines: \"Edges\", edgelabels: \"Edge labels\", faces: \"Faces\", arrowheads: \"Arrow heads\" };\n",
       "const objectTypeVisible = {};\n",
       "Object.assign(objectTypeVisible,modelContains);\n",
       "const sortedObjectTypeKeys = Object.keys(objectTypeInnerHTMLs).sort();\n",
       "const shownObjectTypesList = document.getElementById('shownObjectTypesList_0');\n",
       "\n",
       "function setVisibility(bool,objname) {\n",
       "    for (var i=0; i<scene.children.length; i++){\n",
       "        var obj = scene.children[i].getObjectByName(objname);\n",
       "        if (obj) {\n",
       "            obj.visible = bool;\n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "function toggleObjectTypeVisibility(event){\n",
       "    var name = event.currentTarget.name;\n",
       "    var checked = event.currentTarget.checked;\n",
       "    objectTypeVisible[name] = checked;\n",
       "    setVisibility(checked,name);\n",
       "}\n",
       "\n",
       "for (var i=0; i<sortedObjectTypeKeys.length; i++){\n",
       "    var key = sortedObjectTypeKeys[i];\n",
       "    if (modelContains[key]) {\n",
       "        var objTypeNode = document.createElement('span');\n",
       "        objTypeNode.innerHTML = objectTypeInnerHTMLs[key] + '<br>';\n",
       "        var checkbox = document.createElement('input');\n",
       "        checkbox.type = 'checkbox';\n",
       "        checkbox.checked = true;\n",
       "        checkbox.name = key;\n",
       "        checkbox.onchange = toggleObjectTypeVisibility;\n",
       "        shownObjectTypesList.appendChild(checkbox);\n",
       "        shownObjectTypesList.appendChild(objTypeNode);\n",
       "    }\n",
       "}\n",
       "\n",
       "// ------------------------------------------------------\n",
       "\n",
       "function toggleObjectVisibility(event){\n",
       "    var nr = Number(event.currentTarget.name);\n",
       "    scene.children[nr].visible = event.currentTarget.checked;\n",
       "}\n",
       "\n",
       "// append checkboxes for displaying or hiding objects\n",
       "var shownObjectsList = document.getElementById('shownObjectsList_0');\n",
       "for (var i=0; i<scene.children.length; i++){\n",
       "    obj = scene.children[i];\n",
       "    var objNode = document.createElement('span');\n",
       "    objNode.innerHTML = obj.name + '<br>';\n",
       "    var checkbox = document.createElement('input');\n",
       "    checkbox.type = 'checkbox';\n",
       "    checkbox.checked = true;\n",
       "    checkbox.name = String(i);\n",
       "    checkbox.onchange = toggleObjectVisibility;\n",
       "    shownObjectsList.appendChild(checkbox);\n",
       "    shownObjectsList.appendChild(objNode);\n",
       "}\n",
       "\n",
       "// ---------------------- SVG ------------------------------------------------------\n",
       "// ---------------------------------------------------------------------------------\n",
       "\n",
       "function takeSvgScreenshot() {\n",
       "    if (objectTypeVisible[\"pointlabels\"]) {\n",
       "        setVisibility(false,\"pointlabels\");\n",
       "    }\n",
       "    if (objectTypeVisible[\"edgelabels\"]) {\n",
       "        setVisibility(false,\"edgelabels\");\n",
       "    }\n",
       "    svgRenderer.render(scene,camera);\n",
       "    svgElement = XMLS.serializeToString(svgRenderer.domElement);\n",
       "    \n",
       "    if (objectTypeVisible[\"pointlabels\"]) {\n",
       "        setVisibility(true,\"pointlabels\");\n",
       "    }\n",
       "    if (objectTypeVisible[\"edgelabels\"]) {\n",
       "        setVisibility(true,\"edgelabels\");\n",
       "    }\n",
       "\n",
       "    if (document.getElementById('tab_0').checked){\n",
       "        //show in new tab\n",
       "        var myWindow = window.open(\"\",\"\");\n",
       "        myWindow.document.body.innerHTML = svgElement;\n",
       "    } else{\n",
       "        // download svg file \n",
       "        download(\"screenshot.svg\", svgElement);\n",
       "    }\n",
       "}\n",
       "\n",
       "function download(filename, text) {\n",
       "    var element = document.createElement('a');\n",
       "    element.setAttribute('href', 'data:text/plain;charset=utf-8,' + encodeURIComponent(text));\n",
       "    element.setAttribute('download', filename);\n",
       "\n",
       "    element.style.display = 'none';\n",
       "    document.body.appendChild(element);\n",
       "\n",
       "    element.click();\n",
       "\n",
       "    document.body.removeChild(element);\n",
       "}\n",
       "\n",
       "\n",
       "document.getElementById('transparencyRange_0').oninput = changeTransparency;\n",
       "document.getElementById('changeRotationX_0').onchange = changeRotationX;\n",
       "document.getElementById('changeRotationY_0').onchange = changeRotationY;\n",
       "document.getElementById('changeRotationZ_0').onchange = changeRotationZ;\n",
       "document.getElementById('resetButton_0').onclick = resetScene;\n",
       "document.getElementById('rotationSpeedRange_0').oninput = changeRotationSpeedFactor;\n",
       "document.getElementById('takeScreenshot_0').onclick = takeSvgScreenshot;\n",
       "document.getElementById('showSettingsButton_0').onclick = showSettings;\n",
       "document.getElementById('hideSettingsButton_0').onclick = hideSettings;\n",
       "\n",
       "\n",
       "// ------------------ SHORTCUTS --------------------------------------------\n",
       "// -------------------------------------------------------------------------\n",
       "\n",
       "/**\n",
       " * http://www.openjs.com/scripts/events/keyboard_shortcuts/\n",
       " * Version : 2.01.B\n",
       " * By Binny V A\n",
       " * License : BSD\n",
       " */\n",
       "shortcut = {\n",
       "\t'all_shortcuts':{},//All the shortcuts are stored in this array\n",
       "\t'add': function(shortcut_combination,callback,opt) {\n",
       "\t\t//Provide a set of default options\n",
       "\t\tvar default_options = {\n",
       "\t\t\t'type':'keydown',\n",
       "\t\t\t'propagate':false,\n",
       "\t\t\t'disable_in_input':false,\n",
       "\t\t\t'target':document,\n",
       "\t\t\t'keycode':false\n",
       "\t\t}\n",
       "\t\tif(!opt) opt = default_options;\n",
       "\t\telse {\n",
       "\t\t\tfor(var dfo in default_options) {\n",
       "\t\t\t\tif(typeof opt[dfo] == 'undefined') opt[dfo] = default_options[dfo];\n",
       "\t\t\t}\n",
       "\t\t}\n",
       "\n",
       "\t\tvar ele = opt.target;\n",
       "\t\tif(typeof opt.target == 'string') ele = document.getElementById(opt.target);\n",
       "\t\tvar ths = this;\n",
       "\t\tshortcut_combination = shortcut_combination.toLowerCase();\n",
       "\n",
       "\t\t//The function to be called at keypress\n",
       "\t\tvar func = function(e) {\n",
       "\t\t\te = e || window.event;\n",
       "\t\t\t\n",
       "\t\t\tif(opt['disable_in_input']) { //Don't enable shortcut keys in Input, Textarea fields\n",
       "\t\t\t\tvar element;\n",
       "\t\t\t\tif(e.target) element=e.target;\n",
       "\t\t\t\telse if(e.srcElement) element=e.srcElement;\n",
       "\t\t\t\tif(element.nodeType==3) element=element.parentNode;\n",
       "\n",
       "\t\t\t\tif(element.tagName == 'INPUT' || element.tagName == 'TEXTAREA') return;\n",
       "\t\t\t}\n",
       "\t\n",
       "\t\t\t//Find Which key is pressed\n",
       "\t\t\tif (e.keyCode) code = e.keyCode;\n",
       "\t\t\telse if (e.which) code = e.which;\n",
       "\t\t\tvar character = String.fromCharCode(code).toLowerCase();\n",
       "\t\t\t\n",
       "\t\t\tif(code == 188) character=\",\"; //If the user presses , when the type is onkeydown\n",
       "\t\t\tif(code == 190) character=\".\"; //If the user presses , when the type is onkeydown\n",
       "\n",
       "\t\t\tvar keys = shortcut_combination.split(\"+\");\n",
       "\t\t\t//Key Pressed - counts the number of valid keypresses - if it is same as the number of keys, the shortcut function is invoked\n",
       "\t\t\tvar kp = 0;\n",
       "\t\t\t\n",
       "\t\t\t//Work around for stupid Shift key bug created by using lowercase - as a result the shift+num combination was broken\n",
       "\t\t\tvar shift_nums = {\n",
       "\t\t\t\t\"`\":\"~\",\n",
       "\t\t\t\t\"1\":\"!\",\n",
       "\t\t\t\t\"2\":\"@\",\n",
       "\t\t\t\t\"3\":\"#\",\n",
       "\t\t\t\t\"4\":\"$\",\n",
       "\t\t\t\t\"5\":\"%\",\n",
       "\t\t\t\t\"6\":\"^\",\n",
       "\t\t\t\t\"7\":\"&\",\n",
       "\t\t\t\t\"8\":\"*\",\n",
       "\t\t\t\t\"9\":\"(\",\n",
       "\t\t\t\t\"0\":\")\",\n",
       "\t\t\t\t\"-\":\"_\",\n",
       "\t\t\t\t\"=\":\"+\",\n",
       "\t\t\t\t\";\":\":\",\n",
       "\t\t\t\t\"'\":\"\\\"\",\n",
       "\t\t\t\t\",\":\"<\",\n",
       "\t\t\t\t\".\":\">\",\n",
       "\t\t\t\t\"/\":\"?\",\n",
       "\t\t\t\t\"\\\\\":\"|\"\n",
       "\t\t\t}\n",
       "\t\t\t//Special Keys - and their codes\n",
       "\t\t\tvar special_keys = {\n",
       "\t\t\t\t'esc':27,\n",
       "\t\t\t\t'escape':27,\n",
       "\t\t\t\t'tab':9,\n",
       "\t\t\t\t'space':32,\n",
       "\t\t\t\t'return':13,\n",
       "\t\t\t\t'enter':13,\n",
       "\t\t\t\t'backspace':8,\n",
       "\t\n",
       "\t\t\t\t'scrolllock':145,\n",
       "\t\t\t\t'scroll_lock':145,\n",
       "\t\t\t\t'scroll':145,\n",
       "\t\t\t\t'capslock':20,\n",
       "\t\t\t\t'caps_lock':20,\n",
       "\t\t\t\t'caps':20,\n",
       "\t\t\t\t'numlock':144,\n",
       "\t\t\t\t'num_lock':144,\n",
       "\t\t\t\t'num':144,\n",
       "\t\t\t\t\n",
       "\t\t\t\t'pause':19,\n",
       "\t\t\t\t'break':19,\n",
       "\t\t\t\t\n",
       "\t\t\t\t'insert':45,\n",
       "\t\t\t\t'home':36,\n",
       "\t\t\t\t'delete':46,\n",
       "\t\t\t\t'end':35,\n",
       "\t\t\t\t\n",
       "\t\t\t\t'pageup':33,\n",
       "\t\t\t\t'page_up':33,\n",
       "\t\t\t\t'pu':33,\n",
       "\t\n",
       "\t\t\t\t'pagedown':34,\n",
       "\t\t\t\t'page_down':34,\n",
       "\t\t\t\t'pd':34,\n",
       "\t\n",
       "\t\t\t\t'left':37,\n",
       "\t\t\t\t'up':38,\n",
       "\t\t\t\t'right':39,\n",
       "\t\t\t\t'down':40,\n",
       "\t\n",
       "\t\t\t\t'f1':112,\n",
       "\t\t\t\t'f2':113,\n",
       "\t\t\t\t'f3':114,\n",
       "\t\t\t\t'f4':115,\n",
       "\t\t\t\t'f5':116,\n",
       "\t\t\t\t'f6':117,\n",
       "\t\t\t\t'f7':118,\n",
       "\t\t\t\t'f8':119,\n",
       "\t\t\t\t'f9':120,\n",
       "\t\t\t\t'f10':121,\n",
       "\t\t\t\t'f11':122,\n",
       "\t\t\t\t'f12':123\n",
       "\t\t\t}\n",
       "\t\n",
       "\t\t\tvar modifiers = { \n",
       "\t\t\t\tshift: { wanted:false, pressed:false},\n",
       "\t\t\t\tctrl : { wanted:false, pressed:false},\n",
       "\t\t\t\talt  : { wanted:false, pressed:false},\n",
       "\t\t\t\tmeta : { wanted:false, pressed:false}\t//Meta is Mac specific\n",
       "\t\t\t};\n",
       "                        \n",
       "\t\t\tif(e.ctrlKey)\tmodifiers.ctrl.pressed = true;\n",
       "\t\t\tif(e.shiftKey)\tmodifiers.shift.pressed = true;\n",
       "\t\t\tif(e.altKey)\tmodifiers.alt.pressed = true;\n",
       "\t\t\tif(e.metaKey)   modifiers.meta.pressed = true;\n",
       "                        \n",
       "\t\t\tfor(var i=0; k=keys[i],i<keys.length; i++) {\n",
       "\t\t\t\t//Modifiers\n",
       "\t\t\t\tif(k == 'ctrl' || k == 'control') {\n",
       "\t\t\t\t\tkp++;\n",
       "\t\t\t\t\tmodifiers.ctrl.wanted = true;\n",
       "\n",
       "\t\t\t\t} else if(k == 'shift') {\n",
       "\t\t\t\t\tkp++;\n",
       "\t\t\t\t\tmodifiers.shift.wanted = true;\n",
       "\n",
       "\t\t\t\t} else if(k == 'alt') {\n",
       "\t\t\t\t\tkp++;\n",
       "\t\t\t\t\tmodifiers.alt.wanted = true;\n",
       "\t\t\t\t} else if(k == 'meta') {\n",
       "\t\t\t\t\tkp++;\n",
       "\t\t\t\t\tmodifiers.meta.wanted = true;\n",
       "\t\t\t\t} else if(k.length > 1) { //If it is a special key\n",
       "\t\t\t\t\tif(special_keys[k] == code) kp++;\n",
       "\t\t\t\t\t\n",
       "\t\t\t\t} else if(opt['keycode']) {\n",
       "\t\t\t\t\tif(opt['keycode'] == code) kp++;\n",
       "\n",
       "\t\t\t\t} else { //The special keys did not match\n",
       "\t\t\t\t\tif(character == k) kp++;\n",
       "\t\t\t\t\telse {\n",
       "\t\t\t\t\t\tif(shift_nums[character] && e.shiftKey) { //Stupid Shift key bug created by using lowercase\n",
       "\t\t\t\t\t\t\tcharacter = shift_nums[character]; \n",
       "\t\t\t\t\t\t\tif(character == k) kp++;\n",
       "\t\t\t\t\t\t}\n",
       "\t\t\t\t\t}\n",
       "\t\t\t\t}\n",
       "\t\t\t}\n",
       "\t\t\t\n",
       "\t\t\tif(kp == keys.length && \n",
       "\t\t\t\t\t\tmodifiers.ctrl.pressed == modifiers.ctrl.wanted &&\n",
       "\t\t\t\t\t\tmodifiers.shift.pressed == modifiers.shift.wanted &&\n",
       "\t\t\t\t\t\tmodifiers.alt.pressed == modifiers.alt.wanted &&\n",
       "\t\t\t\t\t\tmodifiers.meta.pressed == modifiers.meta.wanted) {\n",
       "\t\t\t\tcallback(e);\n",
       "\t\n",
       "\t\t\t\tif(!opt['propagate']) { //Stop the event\n",
       "\t\t\t\t\t//e.cancelBubble is supported by IE - this will kill the bubbling process.\n",
       "\t\t\t\t\te.cancelBubble = true;\n",
       "\t\t\t\t\te.returnValue = false;\n",
       "\t\n",
       "\t\t\t\t\t//e.stopPropagation works in Firefox.\n",
       "\t\t\t\t\tif (e.stopPropagation) {\n",
       "\t\t\t\t\t\te.stopPropagation();\n",
       "\t\t\t\t\t\te.preventDefault();\n",
       "\t\t\t\t\t}\n",
       "\t\t\t\t\treturn false;\n",
       "\t\t\t\t}\n",
       "\t\t\t}\n",
       "\t\t}\n",
       "\t\tthis.all_shortcuts[shortcut_combination] = {\n",
       "\t\t\t'callback':func, \n",
       "\t\t\t'target':ele, \n",
       "\t\t\t'event': opt['type']\n",
       "\t\t};\n",
       "\t\t//Attach the function with the event\n",
       "\t\tif(ele.addEventListener) ele.addEventListener(opt['type'], func, false);\n",
       "\t\telse if(ele.attachEvent) ele.attachEvent('on'+opt['type'], func);\n",
       "\t\telse ele['on'+opt['type']] = func;\n",
       "\t},\n",
       "\n",
       "\t//Remove the shortcut - just specify the shortcut and I will remove the binding\n",
       "\t'remove':function(shortcut_combination) {\n",
       "\t\tshortcut_combination = shortcut_combination.toLowerCase();\n",
       "\t\tvar binding = this.all_shortcuts[shortcut_combination];\n",
       "\t\tdelete(this.all_shortcuts[shortcut_combination])\n",
       "\t\tif(!binding) return;\n",
       "\t\tvar type = binding['event'];\n",
       "\t\tvar ele = binding['target'];\n",
       "\t\tvar callback = binding['callback'];\n",
       "\n",
       "\t\tif(ele.detachEvent) ele.detachEvent('on'+type, callback);\n",
       "\t\telse if(ele.removeEventListener) ele.removeEventListener(type, callback, false);\n",
       "\t\telse ele['on'+type] = false;\n",
       "\t}\n",
       "}\n",
       "\n",
       "shortcut.add(\"Alt+Left\",function() {\n",
       "\tvar event = new Event('click');\n",
       "\tif (settingsShown){\n",
       "\t\tdocument.getElementById('hideSettingsButton_0').dispatchEvent(event);\n",
       "\t} else {\n",
       "\t\tdocument.getElementById('showSettingsButton_0').dispatchEvent(event);\n",
       "\t}\n",
       "});\n",
       "\n",
       "\n",
       "// COMMON_CODE_BLOCK_END\n",
       "\n",
       "});});\n",
       "      </script>\n",
       "   </body>\n",
       "</html>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<details><summary><pre style=\"display:inline\"><small>Click here for additional output</small></pre></summary>\n",
       "<pre>\n",
       "polymake: used package threejs\n",
       "   Three.js is a lightweight cross-browser JavaScript library/API used to create and display animated 3D computer graphics on a Web browser.\n",
       "   See http://github.com/mrdoob for the source code.\n",
       "\n",
       "</pre>\n",
       "</details>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "$p->VISUAL->DIRECTED_GRAPH;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The minimal and maximal faces can be visualized via"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<!--\n",
       "polymake for daniel\n",
       "Thu Aug 19 09:55:41 2021\n",
       "c3t\n",
       "-->\n",
       "\n",
       "\n",
       "<html>\n",
       "   <head>\n",
       "      <meta charset=utf-8>\n",
       "      <title>c3t</title>\n",
       "      <style>\n",
       "/*\n",
       "// COMMON_CODE_BLOCK_BEGIN\n",
       "*/\n",
       "         html {overflow: scroll;}\n",
       "         strong{font-size: 18px;}\n",
       "         canvas { z-index: 8; }\n",
       "         input[type='radio'] {margin-left:0;}\n",
       "         input[type='checkbox'] {margin-right:7px; margin-left: 0px; padding-left:0px;}\n",
       "         .group{padding-bottom: 15px;}\n",
       "         .settings * {z-index: 11; }\n",
       "         .settings{z-index: 10; font-family: Arial, Helvetica, sans-serif; margin-left: 30px; visibility: hidden; width: 14em; height: 96%; border: solid 1px silver; padding: 2px; overflow-y: scroll; box-sizing: border-box; background-color: white; position: absolute;}\n",
       "         .indented{margin-left: 20px; margin-top: 10px; padding-bottom: 0px;} \n",
       "         .shownObjectsList{overflow: auto; max-width: 150px; max-height: 150px;}\n",
       "         .showSettingsButton{visibility: visible; z-index: 12; position: absolute }\n",
       "         .hideSettingsButton{visibility: hidden; z-index: 12; position: absolute; opacity: 0.5}\n",
       "         button{margin-left: 0; margin-top: 10px}\n",
       "         img{cursor: pointer;}\n",
       "         .suboption{padding-top: 15px;}\n",
       "         #model32789193516 { width: 100%; height: 100%; }\n",
       "         .threejs_container { width: 100%; height: 75vh;}\n",
       "         .settings{max-height: 74vh} \n",
       "         input[type=range] {\n",
       "           -webkit-appearance: none;\n",
       "           padding:0; \n",
       "           width:90%; \n",
       "           margin-left: auto;\n",
       "           margin-right: auto;\n",
       "           margin-top: 15px;\n",
       "           margin-bottom: 15px;\n",
       "           display: block;\t\n",
       "         }\n",
       "         input[type=range]:focus {\n",
       "           outline: none;\n",
       "         }\n",
       "         input[type=range]::-webkit-slider-runnable-track {\n",
       "           height: 4px;\n",
       "           cursor: pointer;\n",
       "           animate: 0.2s;\n",
       "           box-shadow: 0px 0px 0px #000000;\n",
       "           background: #E3E3E3;\n",
       "           border-radius: 0px;\n",
       "           border: 0px solid #000000;\n",
       "         }\n",
       "         input[type=range]::-webkit-slider-thumb {\n",
       "           box-shadow: 1px 1px 2px #B8B8B8;\n",
       "           border: 1px solid #ABABAB;\n",
       "           height: 13px;\n",
       "           width: 25px;\n",
       "           border-radius: 20px;\n",
       "           background: #E0E0E0;\n",
       "           cursor: pointer;\n",
       "           -webkit-appearance: none;\n",
       "           margin-top: -5px;\n",
       "         }\n",
       "         input[type=range]:focus::-webkit-slider-runnable-track {\n",
       "           background: #E3E3E3;\n",
       "         }\n",
       "         input[type=range]::-moz-range-track {\n",
       "           height: 4px;\n",
       "           cursor: pointer;\n",
       "           animate: 0.2s;\n",
       "           box-shadow: 0px 0px 0px #000000;\n",
       "           background: #E3E3E3;\n",
       "           border-radius: 0px;\n",
       "           border: 0px solid #000000;\n",
       "         }\n",
       "         input[type=range]::-moz-range-thumb {\n",
       "           box-shadow: 1px 1px 2px #B8B8B8;\n",
       "           border: 1px solid #ABABAB;\n",
       "           height: 13px;\n",
       "           width: 25px;\n",
       "           border-radius: 20px;\n",
       "           background: #E0E0E0;\n",
       "           cursor: pointer;\n",
       "         }\n",
       "         input[type=range]::-ms-track {\n",
       "           height: 4px;\n",
       "           cursor: pointer;\n",
       "           animate: 0.2s;\n",
       "           background: transparent;\n",
       "           border-color: transparent;\n",
       "           color: transparent;\n",
       "         }\n",
       "         input[type=range]::-ms-fill-lower {\n",
       "           background: #E3E3E3;\n",
       "           border: 0px solid #000000;\n",
       "           border-radius: 0px;\n",
       "           box-shadow: 0px 0px 0px #000000;\n",
       "         }\n",
       "         input[type=range]::-ms-fill-upper {\n",
       "           background: #E3E3E3;\n",
       "           border: 0px solid #000000;\n",
       "           border-radius: 0px;\n",
       "           box-shadow: 0px 0px 0px #000000;\n",
       "         }\n",
       "         input[type=range]::-ms-thumb {\n",
       "           box-shadow: 1px 1px 2px #B8B8B8;\n",
       "           border: 1px solid #ABABAB;\n",
       "           height: 13px;\n",
       "           width: 25px;\n",
       "           border-radius: 20px;\n",
       "           background: #E0E0E0;\n",
       "           cursor: pointer;\n",
       "         }\n",
       "         input[type=range]:focus::-ms-fill-lower {\n",
       "           background: #E3E3E3;\n",
       "         }\n",
       "         input[type=range]:focus::-ms-fill-upper {\n",
       "           background: #E3E3E3;\n",
       "         }\n",
       "/*\n",
       "// COMMON_CODE_BLOCK_END\n",
       "*/\n",
       "\t\t</style>\n",
       "   </head>\n",
       "<body>\n",
       "   <div class='threejs_container'>\n",
       "\t\t<div id='settings_1' class='settings'>\n",
       "\t\t\t<div class=group id='transparency_1' class='transparency'>\n",
       "\t\t\t\t<strong>Transparency</strong>\n",
       "\t\t\t\t<input id='transparencyRange_1' type='range' min=0 max=1 step=0.01 value=0>\n",
       "\t\t\t</div>\n",
       "\t\t\t\n",
       "\t\t\t<div class=group id='rotation_1'>\n",
       "\t\t\t\t<strong>Rotation</strong>\n",
       "\t\t\t\t<div class=indented>\n",
       "\t\t\t\t\t<div><input type='checkbox' id='changeRotationX_1'> x-axis</div>\n",
       "\t\t\t\t\t<div><input type='checkbox' id='changeRotationY_1'> y-axis</div>\n",
       "\t\t\t\t\t<div><input type='checkbox' id='changeRotationZ_1'> z-axis</div>\n",
       "\t\t\t\t\t<button id='resetButton_1'>Reset</button>\n",
       "\t\t\t\t</div>\n",
       "\n",
       "\t\t\t\t<div class=suboption>Rotation speed</div>\n",
       "\t\t\t\t<input id='rotationSpeedRange_1' type='range' min=0 max=5 step=0.01 value=2>\n",
       "\t\t\t</div>\n",
       "\n",
       "\n",
       "\t\t\t<div class=group id='display_1'>\n",
       "\t\t\t\t<strong>Display</strong>\n",
       "\t\t\t\t<div class=indented>\n",
       "\t\t\t\t\t<div id='shownObjectTypesList_1' class='shownObjectsList'></div>\n",
       "\t\t\t\t</div>\n",
       "\t\t\t\t<div class=suboption>Objects</div>\n",
       "\t\t\t\t<div class=indented>\n",
       "\t\t\t\t   <div id='shownObjectsList_1' class='shownObjectsList'></div>\n",
       "\t\t\t\t</div>\n",
       "\t\t\t</div>\n",
       "         \n",
       "         <div class=group id='camera_1'>\n",
       "            <strong>Camera</strong>\n",
       "            <div class=indented>\n",
       "               <form>\n",
       "                  <select id=\"cameraType_1\">\n",
       "                     <option value='perspective' selected> Perspective<br></option>\n",
       "                     <option value='orthographic' > Orthographic<br></option>\n",
       "                  </select>\n",
       "               </form>\n",
       "            </div>\n",
       "         </div>\n",
       "\n",
       "\t\t\t<div class=group id='svg_1'>\n",
       "\t\t\t\t<strong>SVG</strong>\n",
       "\t\t\t\t<div class=indented>\n",
       "\t\t\t\t\t<form>\n",
       "\t\t\t\t\t\t<input type=\"radio\" name='screenshotMode' value='download' id='download_1' checked> Download<br>\n",
       "\t\t\t\t\t\t<input type=\"radio\" name='screenshotMode' value='tab' id='tab_1' > New tab<br>\n",
       "\t\t\t\t\t</form>\n",
       "\t\t\t\t\t<button id='takeScreenshot_1'>Screenshot</button>\n",
       "\t\t\t\t</div>\n",
       "\t\t\t</div>\n",
       "\n",
       "\t\t</div>\t<!-- end of settings -->\n",
       "\t\t<img id='hideSettingsButton_1' class='hideSettingsButton' src='/kernelspecs/polymake/close.svg' width=20px\">\n",
       "\t\t<img id='showSettingsButton_1' class='showSettingsButton' src='/kernelspecs/polymake/menu.svg' width=20px\">\n",
       "<div id=\"model32789193516\"></div>\n",
       "</div>\n",
       "   <script>\n",
       "    requirejs.config({\n",
       "      paths: {\n",
       "        three: '/kernelspecs/polymake/three',\n",
       "        TrackballControls: '/kernelspecs/polymake/TrackballControls',\n",
       "        OrbitControls: '/kernelspecs/polymake/OrbitControls',\n",
       "        Projector: '/kernelspecs/polymake/Projector',\n",
       "        SVGRenderer: '/kernelspecs/polymake/SVGRenderer',\n",
       "        WEBGL: '/kernelspecs/polymake/WebGL',\n",
       "      },\n",
       "      shim: {\n",
       "        'three': { exports: 'THREE'},\n",
       "        'SVGRenderer': { deps: [ 'three' ], exports: 'THREE.SVGRenderer' },\n",
       "        'WEBGL': { deps: [ 'three' ], exports: 'THREE.WEBGL' },\n",
       "        'Projector': { deps: [ 'three' ], exports: 'THREE.Projector' },\n",
       "        'TrackballControls': { deps: [ 'three' ], exports: 'THREE.TrackballControls' },\n",
       "        'OrbitControls': { deps: [ 'three' ], exports: 'THREE.OrbitControls' },\n",
       "      }\n",
       "    });\n",
       "    \n",
       "    require(['three'],function(THREE){\n",
       "        window.THREE = THREE;\n",
       "      require(['TrackballControls', 'OrbitControls', 'Projector', 'SVGRenderer', 'WEBGL'],\n",
       "               function(TrackballControls, OrbitControls, Projector, SVGRenderer, WEBGL) {\n",
       "    THREE.TrackballControls = TrackballControls;\n",
       "    THREE.OrbitControls = OrbitControls;\n",
       "    THREE.Projector = Projector;\n",
       "    THREE.SVGRenderer = SVGRenderer;\n",
       "    THREE.WEBGL = WEBGL;\n",
       "\n",
       "// COMMON_CODE_BLOCK_BEGIN\n",
       "\n",
       "const intervalLength = 25; // for automatic animations\n",
       "const explodableModel = false; \n",
       "const modelContains = { points: false, pointlabels: false, lines: false, edgelabels: false, faces: false, arrowheads: false };\n",
       "const foldables = [];\n",
       "\n",
       "var three = document.getElementById(\"model32789193516\");\n",
       "var scene = new THREE.Scene();\n",
       "var renderer = new THREE.WebGLRenderer( { antialias: true } );\n",
       "var svgRenderer = new THREE.SVGRenderer( { antialias: true } );\n",
       "renderer.setPixelRatio( window.devicePixelRatio );\n",
       "renderer.setClearColor(0xFFFFFF, 1);\n",
       "svgRenderer.setClearColor(0xFFFFFF, 1);\n",
       "three.appendChild(renderer.domElement);\n",
       "\n",
       "var frustumSize = 4;\n",
       "var cameras = [new THREE.PerspectiveCamera(75, 1, 0.1, 1000), new THREE.OrthographicCamera()];\n",
       "cameras.forEach(function(cam) {\n",
       "    cam.position.set(0, 0, 5);\n",
       "    cam.lookAt(0, 0, 0);  \n",
       "    cam.up.set(0, 1, 0);         \n",
       "});\n",
       "var controls = [new THREE.TrackballControls(cameras[0], three), new THREE.OrbitControls(cameras[1], three)];\n",
       "var camera, control;\n",
       "\n",
       "controls[0].zoomSpeed = 0.2;\n",
       "controls[0].rotateSpeed = 4;\n",
       "\n",
       "\n",
       "// class to allow move points together with labels and spheres\n",
       "var PMPoint = function (x,y,z) {\n",
       "   this.vector = new THREE.Vector3(x,y,z);\n",
       "   this.sprite = null;\n",
       "   this.sphere = null;\n",
       "}\n",
       "PMPoint.prototype.addLabel = function(labelsprite) {\n",
       "   this.sprite = labelsprite;\n",
       "   this.sprite.position.copy(this.vector);\n",
       "}\n",
       "PMPoint.prototype.addSphere = function(spheremesh) {\n",
       "   this.sphere = spheremesh;\n",
       "   this.sphere.position.copy(this.vector);\n",
       "}\n",
       "PMPoint.prototype.set = function(x,y,z) {\n",
       "   this.vector.set(x,y,z);\n",
       "   if (this.sprite) {\n",
       "      this.sprite.position.copy(this.vector);\n",
       "   }\n",
       "   if (this.sphere) {\n",
       "      this.sphere.position.copy(this.vector);\n",
       "   }\n",
       "}\n",
       "PMPoint.prototype.radius = function() {\n",
       "   if (this.sphere) {\n",
       "      return this.sphere.geometry.parameters.radius;\n",
       "   } else {\n",
       "      return 0;\n",
       "   }\n",
       "};\n",
       "// select the target node\n",
       "var target = document.querySelector('#model32789193516');\n",
       "\n",
       "// create an observer instance\n",
       "var observer = new MutationObserver(function(mutations) {\n",
       "   mutations.forEach(function(mutation) {\n",
       "      if (mutation.removedNodes && mutation.removedNodes.length > 0) {\n",
       "         cancelAnimationFrame(renderId);\n",
       "         observer.disconnect();\n",
       "         console.log(\"cancelled frame \"+renderId);\n",
       "      }\n",
       "   });\n",
       "});\n",
       "\n",
       "// configuration of the observer:\n",
       "var config = { childList: true, characterData: true }\n",
       "\n",
       "// pass in the target node, as well as the observer options\n",
       "while (target) {\n",
       "   if (target.className==\"output\") {\n",
       "      observer.observe(target, config);\n",
       "      break;\n",
       "   }\n",
       "   target = target.parentNode;\n",
       "}\n",
       "\n",
       "// COMMON_CODE_BLOCK_END\n",
       "\n",
       "var obj0 = new THREE.Object3D();\n",
       "obj0.name = \"c3t\";\n",
       "obj0.userData.explodable = 1;\n",
       "obj0.userData.points = [];\n",
       "obj0.userData.points.push(new PMPoint(0, 1, 1));\n",
       "obj0.userData.points.push(new PMPoint(1, 0, 1));\n",
       "obj0.userData.points.push(new PMPoint(1, 0, 0));\n",
       "obj0.userData.points.push(new PMPoint(1, 1, 0));\n",
       "obj0.userData.points.push(new PMPoint(0, 1, 0));\n",
       "obj0.userData.points.push(new PMPoint(0, 0, 1));\n",
       "obj0.userData.points.push(new PMPoint(0, 0, 0));\n",
       "\n",
       "obj0.userData.pointradii = 0.02;\n",
       "   <!-- Vertex style -->\n",
       "obj0.userData.pointmaterial = [new THREE.MeshBasicMaterial( { color: 0xFF0000, side: THREE.DoubleSide, transparent: false } ),\n",
       "new THREE.MeshBasicMaterial( { color: 0xFF0000, side: THREE.DoubleSide, transparent: false } ),\n",
       "new THREE.MeshBasicMaterial( { color: 0xFF0000, side: THREE.DoubleSide, transparent: false } ),\n",
       "new THREE.MeshBasicMaterial( { color: 0xFF0000, side: THREE.DoubleSide, transparent: false } ),\n",
       "new THREE.MeshBasicMaterial( { color: 0xFF0000, side: THREE.DoubleSide, transparent: false } ),\n",
       "new THREE.MeshBasicMaterial( { color: 0xFF0000, side: THREE.DoubleSide, transparent: false } ),\n",
       "new THREE.MeshBasicMaterial( { color: 0xFFFF00, side: THREE.DoubleSide, transparent: false } )];\n",
       "obj0.userData.pointlabels = [\"0\", \"1\", \"2\", \"3\", \"4\", \"5\", \"6\"];\n",
       "obj0.userData.edgeindices = [0, 1, 1, 2, 0, 3, 1, 3, 2, 3, 0, 4, 3, 4, 0, 5, 1, 5, 2, 6, 4, 6, 5, 6];\n",
       "   <!-- Edge style -->\n",
       "obj0.userData.edgematerial = new THREE.LineBasicMaterial( { color: 0x000000, depthTest: true, linewidth: 1.5, transparent: false } );\n",
       "obj0.userData.facets = [[1, 3, 0], [0, 4, 6, 5], [2, 3, 1], [1, 5, 6, 2], [0, 3, 4], [4, 3, 2, 6], [1, 0, 5]];\n",
       "   <!-- Facet style -->\n",
       "obj0.userData.facetmaterial = [new THREE.MeshBasicMaterial( { color: 0xFF0000, depthFunc: THREE.LessDepth, opacity: 1, polygonOffset: true, polygonOffsetFactor: 1, polygonOffsetUnits: 0.5, side: THREE.DoubleSide, transparent: true } ),\n",
       "new THREE.MeshBasicMaterial( { color: 0x77EC9E, depthFunc: THREE.LessDepth, opacity: 1, polygonOffset: true, polygonOffsetFactor: 1, polygonOffsetUnits: 0.5, side: THREE.DoubleSide, transparent: true } ),\n",
       "new THREE.MeshBasicMaterial( { color: 0x77EC9E, depthFunc: THREE.LessDepth, opacity: 1, polygonOffset: true, polygonOffsetFactor: 1, polygonOffsetUnits: 0.5, side: THREE.DoubleSide, transparent: true } ),\n",
       "new THREE.MeshBasicMaterial( { color: 0x77EC9E, depthFunc: THREE.LessDepth, opacity: 1, polygonOffset: true, polygonOffsetFactor: 1, polygonOffsetUnits: 0.5, side: THREE.DoubleSide, transparent: true } ),\n",
       "new THREE.MeshBasicMaterial( { color: 0x77EC9E, depthFunc: THREE.LessDepth, opacity: 1, polygonOffset: true, polygonOffsetFactor: 1, polygonOffsetUnits: 0.5, side: THREE.DoubleSide, transparent: true } ),\n",
       "new THREE.MeshBasicMaterial( { color: 0x77EC9E, depthFunc: THREE.LessDepth, opacity: 1, polygonOffset: true, polygonOffsetFactor: 1, polygonOffsetUnits: 0.5, side: THREE.DoubleSide, transparent: true } ),\n",
       "new THREE.MeshBasicMaterial( { color: 0x77EC9E, depthFunc: THREE.LessDepth, opacity: 1, polygonOffset: true, polygonOffsetFactor: 1, polygonOffsetUnits: 0.5, side: THREE.DoubleSide, transparent: true } )];\n",
       "init_object(obj0);\n",
       "scene.add(obj0);\n",
       "\n",
       "// COMMON_CODE_BLOCK_BEGIN\n",
       "function textSpriteMaterial(message, parameters) {\n",
       "    if ( parameters === undefined ) parameters = {};\n",
       "    var fontface = \"Helvetica\";\n",
       "    var fontsize = parameters.hasOwnProperty(\"fontsize\") ? parameters[\"fontsize\"] : 15;\n",
       "    fontsize = fontsize*10;\n",
       "    var lines = message.split('\\\\n');\n",
       "    var size = 512;\n",
       "    for(var i = 0; i<lines.length; i++){\n",
       "        var tmp = lines[i].length;\n",
       "        while(tmp*fontsize > size){\n",
       "           fontsize--;\n",
       "        }\n",
       "    }\n",
       "    \n",
       "    var canvas = document.createElement('canvas');\n",
       "    canvas.width = size;\n",
       "    canvas.height = size;\n",
       "    var context = canvas.getContext('2d');\n",
       "    context.fillStyle = \"rgba(255, 255, 255, 0)\";\n",
       "    context.fill();\n",
       "    context.font = fontsize + \"px \" + fontface;\n",
       "    \n",
       "    // text color\n",
       "    context.fillStyle = \"rgba(0, 0, 0, 1.0)\";\n",
       "     for(var i = 0; i<lines.length; i++){\n",
       "        context.fillText(lines[i], size/2, size/2+i*fontsize);\n",
       "     }\n",
       "    \n",
       "    // canvas contents will be used for a texture\n",
       "    var texture = new THREE.Texture(canvas);\n",
       "    texture.needsUpdate = true;\n",
       "    \n",
       "    var spriteMaterial = new THREE.SpriteMaterial({map: texture, depthTest: true, depthWrite: false, polygonOffset: true, polygonOffsetFactor: -1, polygonOffsetUnits: 1 });\n",
       "    return spriteMaterial;\n",
       "}\n",
       "\n",
       "\n",
       "// ---------------------- INITIALIZING OBJECTS--------------------------------------\n",
       "// ---------------------------------------------------------------------------------\n",
       "\n",
       "function init_object(obj) {\n",
       "    if (obj.userData.hasOwnProperty(\"pointmaterial\")) {\n",
       "        init_points(obj);\n",
       "        modelContains.points = true;\n",
       "    }\n",
       "    if (obj.userData.hasOwnProperty(\"pointlabels\")) {\n",
       "        init_pointlabels(obj);\n",
       "        modelContains.pointlabels = true;\n",
       "    }\n",
       "    if (obj.userData.hasOwnProperty(\"edgematerial\")) {\n",
       "        init_lines(obj);\n",
       "        modelContains.lines = true;\n",
       "    }\n",
       "    if (obj.userData.hasOwnProperty(\"edgelabels\")) {\n",
       "        init_edgelabels(obj);\n",
       "        modelContains.edgelabels = true;\n",
       "    }\n",
       "    if (obj.userData.hasOwnProperty(\"arrowstyle\")) {\n",
       "        init_arrowheads(obj);\n",
       "        modelContains.arrowheads = true;\n",
       "    }\n",
       "    if (obj.userData.hasOwnProperty(\"facetmaterial\")) {\n",
       "        init_faces(obj);\n",
       "        modelContains.faces = true;\n",
       "    }\n",
       "}\n",
       "\n",
       "function init_points(obj) {\n",
       "    var pointgroup = new THREE.Group();\n",
       "    pointgroup.name = \"points\";\n",
       "    var points = obj.userData.points;\n",
       "    var radii = obj.userData.pointradii;\n",
       "    var materials = obj.userData.pointmaterial;\n",
       "    var geometry,material;\n",
       "    if (!Array.isArray(radii)) {\n",
       "        geometry = new THREE.SphereBufferGeometry(radii);  \n",
       "    }\n",
       "    if (!Array.isArray(materials)) {\n",
       "        material = materials;\n",
       "    }\n",
       "    for (var i=0; i<points.length; i++) {\n",
       "        var point = points[i];\n",
       "        if (Array.isArray(radii)) {\n",
       "            if (radii[i] == 0) {\n",
       "                continue;\n",
       "            }\n",
       "            geometry = new THREE.SphereBufferGeometry(radii[i]);  \n",
       "        } \n",
       "        if (Array.isArray(materials)) {\n",
       "            material = materials[i];     \n",
       "        } \n",
       "        var sphere = new THREE.Mesh(geometry, material);\n",
       "        point.addSphere(sphere);\n",
       "        pointgroup.add(sphere);\n",
       "    }\n",
       "    obj.add(pointgroup);\n",
       "}\n",
       "\n",
       "function init_pointlabels(obj) {\n",
       "    var points = obj.userData.points;\n",
       "    var labels = obj.userData.pointlabels;\n",
       "    var pointlabels = new THREE.Group();\n",
       "    pointlabels.name = \"pointlabels\";\n",
       "    if (Array.isArray(labels)) {\n",
       "        for (var i=0; i<points.length; i++) {\n",
       "            var point = points[i];\n",
       "            var spriteMaterial = textSpriteMaterial( labels[i] );\n",
       "\t        var sprite = new THREE.Sprite(spriteMaterial);\n",
       "            point.addLabel(sprite);\n",
       "            pointlabels.add(sprite);\n",
       "        }\n",
       "    } else {\n",
       "        var spriteMaterial = textSpriteMaterial( labels );\n",
       "        for (var i=0; i<points.length; i++) {\n",
       "            var point = points[i];\n",
       "\t        var sprite = new THREE.Sprite(spriteMaterial);\n",
       "            point.addLabel(sprite);\n",
       "            pointlabels.add(sprite);\n",
       "        }\n",
       "    }\n",
       "    obj.add(pointlabels);\n",
       "}\n",
       "\n",
       "function init_lines(obj) {\n",
       "    var edgeindices = obj.userData.edgeindices;\n",
       "    var points = obj.userData.points;\n",
       "    var materials = obj.userData.edgematerial;\n",
       "    var geometry = new THREE.BufferGeometry();\n",
       "    var bufarr = new Float32Array( obj.userData.edgeindices.length * 3 );\n",
       "    var bufattr = new THREE.Float32BufferAttribute( bufarr, 3 );\n",
       "    var geometry = new THREE.BufferGeometry();\n",
       "    geometry.setAttribute('position', bufattr);\n",
       "    if (Array.isArray(materials)) {     \n",
       "        for (var i=0; i<materials.length; i++) {\n",
       "            geometry.addGroup(2*i,2,i);\n",
       "        }\n",
       "    }\n",
       "    var lines = new THREE.LineSegments(geometry, materials);\n",
       "    lines.name = \"lines\";\n",
       "    obj.add(lines);\n",
       "    updateEdgesPosition(obj);\n",
       "}\n",
       "\n",
       "function init_edgelabels(obj) {\n",
       "    var points = obj.userData.points;\n",
       "    var edgeindices = obj.userData.edgeindices;\n",
       "    var labels = obj.userData.edgelabels;\n",
       "    var edgelabels = new THREE.Group();\n",
       "    edgelabels.name = \"edgelabels\";\n",
       "    if (Array.isArray(labels)) {\n",
       "        for (var i=0; i<edgeindices.length/2; i++) {\n",
       "            var spriteMaterial = textSpriteMaterial( labels[i] );\n",
       "            var sprite = new THREE.Sprite(spriteMaterial);\n",
       "            sprite.position.copy(new THREE.Vector3().addVectors(points[edgeindices[2*i]].vector,points[edgeindices[2*i+1]].vector).multiplyScalar(0.5));\n",
       "            edgelabels.add(sprite);\n",
       "        }\n",
       "    } else {\n",
       "        var spriteMaterial = textSpriteMaterial( labels );\n",
       "        for (var i=0; i<edgeindices.length/2; i++) {\n",
       "            var sprite = new THREE.Sprite(spriteMaterial);\n",
       "            sprite.position.copy(new THREE.Vector3().addVectors(points[edgeindices[2*i]].vector,points[edgeindices[2*i+1]].vector).multiplyScalar(0.5));\n",
       "            edgelabels.add(sprite);\n",
       "        }\n",
       "    }\n",
       "    obj.add(edgelabels);\n",
       "}\n",
       "\n",
       "function init_arrowheads(obj) {\n",
       "    var arrowheads = new THREE.Group();\n",
       "    arrowheads.name = \"arrowheads\";\n",
       "    var arrowstyle = obj.userData.arrowstyle;\n",
       "    var edgeindices = obj.userData.edgeindices;\n",
       "    var edgematerials = obj.userData.edgematerial;\n",
       "    var points = obj.userData.points;\n",
       "    var material;\n",
       "    if (!Array.isArray(edgematerials)) {\n",
       "        material = new THREE.MeshBasicMaterial( {color: edgematerials.color} );\n",
       "    }\n",
       "\n",
       "    for (var i=0; i<edgeindices.length; i=i+2) {\n",
       "        var start = points[edgeindices[i]];\n",
       "        var end = points[edgeindices[i+1]];\n",
       "        var dist = start.vector.distanceTo( end.vector ) - start.radius() - end.radius();\n",
       "        if (dist <= 0) {\n",
       "            continue;\n",
       "        }\n",
       "        var dir = new THREE.Vector3().subVectors(end.vector,start.vector);\n",
       "        dir.normalize();\n",
       "        var axis = new THREE.Vector3().set(dir.z,0,-dir.x);\n",
       "        axis.normalize();\n",
       "        var radians = Math.acos( dir.y );\n",
       "        var radius = dist/25;\n",
       "        var height = dist/5;\n",
       "        var geometry = new THREE.ConeBufferGeometry(radius,height);\n",
       "        var position = new THREE.Vector3().addVectors(start.vector,dir.clone().multiplyScalar(start.radius()+dist-height/2));\n",
       "        if (Array.isArray(edgematerials)) {\n",
       "            material = new THREE.MeshBasicMaterial( {color: edgematerials[i].color} );\n",
       "        }\n",
       "        var cone = new THREE.Mesh( geometry, material );\n",
       "        cone.quaternion.setFromAxisAngle(axis,radians);;\n",
       "        cone.position.copy(position);;\n",
       "        arrowheads.add(cone);\n",
       "    }\n",
       "    obj.add(arrowheads);\n",
       "}\n",
       "\n",
       "function init_faces(obj) {\n",
       "    var points = obj.userData.points;\n",
       "    var facets = obj.userData.facets;\n",
       "    obj.userData.triangleindices = [];\n",
       "    for (var i=0; i<facets.length; i++) {\n",
       "        facet = facets[i];\n",
       "        for (var t=0; t<facet.length-2; t++) {\n",
       "            obj.userData.triangleindices.push(facet[0],facet[t+1],facet[t+2]);  \n",
       "        }\n",
       "    }\n",
       "    var bufarr = new Float32Array( obj.userData.triangleindices.length * 3 );\n",
       "    var bufattr = new THREE.Float32BufferAttribute(bufarr,3);\n",
       "    \n",
       "    var materials = obj.userData.facetmaterial;\n",
       "    var geometry = new THREE.BufferGeometry();\n",
       "    geometry.setAttribute('position',bufattr);\n",
       "    if (Array.isArray(materials)) {\n",
       "        var tricount = 0;\n",
       "        var facet;\n",
       "        for (var i=0; i<facets.length; i++) {\n",
       "            facet = facets[i];\n",
       "            geometry.addGroup(tricount,(facet.length-2)*3,i);\n",
       "            tricount += (facet.length-2)*3;\n",
       "        }\n",
       "    }\n",
       "    var mesh = new THREE.Mesh(geometry, materials);\n",
       "    mesh.name = \"faces\";\n",
       "    obj.add(mesh); \n",
       "    updateFacesPosition(obj);\n",
       "}\n",
       "// //INITIALIZING\n",
       "\n",
       "\n",
       "function updateFacesPosition(obj) {\n",
       "    var points = obj.userData.points;\n",
       "    var indices = obj.userData.triangleindices;\n",
       "    var faces = obj.getObjectByName(\"faces\");\n",
       "    var ba = faces.geometry.getAttribute(\"position\");\n",
       "    for (var i=0; i<indices.length; i++) {\n",
       "        ba.setXYZ(i, points[indices[i]].vector.x, points[indices[i]].vector.y ,points[indices[i]].vector.z); \n",
       "    }\n",
       "    faces.geometry.attributes.position.needsUpdate = true;\n",
       "    \n",
       "}\n",
       "\n",
       "function updateEdgesPosition(obj) {\n",
       "    var points = obj.userData.points;\n",
       "    var indices = obj.userData.edgeindices;\n",
       "    var lines = obj.getObjectByName(\"lines\");\n",
       "    var ba = lines.geometry.getAttribute(\"position\"); \n",
       "    for (var i=0; i<indices.length; i++) {\n",
       "        ba.setXYZ(i, points[indices[i]].vector.x, points[indices[i]].vector.y ,points[indices[i]].vector.z); \n",
       "    }\n",
       "    lines.geometry.attributes.position.needsUpdate = true;\n",
       "}\n",
       "\n",
       "function onWindowResize() {\n",
       "    renderer.setSize( three.clientWidth, three.clientHeight );\n",
       "    svgRenderer.setSize( three.clientWidth, three.clientHeight );\n",
       "    updateCamera();\n",
       "}\n",
       "\n",
       "function updateCamera() {\n",
       "    var width = three.clientWidth;\n",
       "    var height = three.clientHeight;\n",
       "    var aspect = width / height;\n",
       "    if (camera.type == \"OrthographicCamera\") {\n",
       "        camera.left = frustumSize * aspect / - 2;\n",
       "        camera.right = frustumSize * aspect / 2;\n",
       "        camera.top = frustumSize / 2;\n",
       "        camera.bottom = - frustumSize / 2;\n",
       "    } else if (camera.type == \"PerspectiveCamera\") {\n",
       "        camera.aspect = aspect;\n",
       "    }\n",
       "    camera.updateProjectionMatrix();\n",
       "}\n",
       "\n",
       "function changeCamera(event) {\n",
       "    var selindex = event.currentTarget.selectedIndex;\n",
       "    camera = cameras[selindex];\n",
       "    control = controls[selindex];\n",
       "    control.enabled = true; \n",
       "    for (var i=0; i<controls.length; i++) {\n",
       "        if (i!=selindex) {\n",
       "            controls[i].enabled = false;\n",
       "        }\n",
       "    }\n",
       "    updateCamera();\n",
       "}\n",
       "\n",
       "var camtypenode = document.getElementById('cameraType_1');\n",
       "camtypenode.onchange = changeCamera;\n",
       "camtypenode.dispatchEvent(new Event('change'));\n",
       "\n",
       "onWindowResize();\n",
       "window.addEventListener('resize', onWindowResize);\t\n",
       "\n",
       "\n",
       "var xRotationEnabled = false;\n",
       "var yRotationEnabled = false;\n",
       "var zRotationEnabled = false;\n",
       "var rotationSpeedFactor = 1;\n",
       "var settingsShown = false;\n",
       "var labelsShown = true;\n",
       "var intervals = [];\n",
       "var timeouts = [];\n",
       "var explodingSpeed = 0.05;\n",
       "var explodeScale = 0;\n",
       "var XMLS = new XMLSerializer();\n",
       "var svgElement;\n",
       "var renderId;\n",
       "\n",
       "var render = function () {\n",
       "\n",
       "\trenderId = requestAnimationFrame(render);\n",
       "\n",
       "//\tcomment in for automatic explosion\n",
       "//\texplode(updateFactor());\n",
       "\n",
       "    var phi = 0.02 * rotationSpeedFactor;\n",
       "\n",
       "    if (xRotationEnabled) {\n",
       "        scene.rotation.x += phi;\n",
       "    }\n",
       "    if (yRotationEnabled) {\n",
       "        scene.rotation.y += phi;\n",
       "    }\n",
       "    if (zRotationEnabled) {\n",
       "        scene.rotation.z += phi;\n",
       "    }\n",
       "\n",
       "    control.update();\n",
       "    renderer.render(scene, camera);\n",
       "};\n",
       "\n",
       "if ( THREE.WEBGL.isWebGLAvailable() ) {\n",
       "\trender();\n",
       "} else {\n",
       "\tvar warning = WEBGL.getWebGLErrorMessage();\n",
       "\tthree.appendChild( warning );\n",
       "}\n",
       "    \n",
       "function changeTransparency() {\n",
       "    var opacity = 1-Number(event.currentTarget.value);\n",
       "    for (var i=0; i<scene.children.length; i++) {\n",
       "        child = scene.children[i];\n",
       "        if ( child.userData.hasOwnProperty(\"facetmaterial\") ) {\n",
       "            if (Array.isArray(child.userData.facetmaterial)) {\n",
       "                for (var j=0; j<child.userData.facetmaterial.length; j++) {\n",
       "                    child.userData.facetmaterial[j].opacity = opacity;\n",
       "                }\n",
       "            } else {\n",
       "                child.userData.facetmaterial.opacity = opacity;\n",
       "            }    \n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "function changeRotationX(event){\n",
       "    xRotationEnabled = event.currentTarget.checked;\n",
       "}\t\n",
       "\n",
       "function changeRotationY(event){\n",
       "    yRotationEnabled = event.currentTarget.checked;\n",
       "}\t\n",
       "\n",
       "function changeRotationZ(event){\n",
       "    zRotationEnabled = event.currentTarget.checked;\n",
       "}\t\n",
       "\n",
       "\n",
       "function changeRotationSpeedFactor(event){\n",
       "    rotationSpeedFactor = Number(event.currentTarget.value);\n",
       "}\n",
       "\n",
       "function resetScene(){\n",
       "    scene.rotation.set(0,0,0);\n",
       "    camera.position.set(0,0,5);\n",
       "    camera.up.set(0,1,0);\n",
       "}\n",
       "\n",
       "function showSettings(event){\n",
       "    document.getElementById('settings_1').style.visibility = 'visible';\n",
       "    document.getElementById('showSettingsButton_1').style.visibility = 'hidden';\n",
       "    document.getElementById('hideSettingsButton_1').style.visibility = 'visible';\n",
       "    settingsShown = true;\n",
       "}\n",
       "\n",
       "function hideSettings(event){\n",
       "    document.getElementById('settings_1').style.visibility = 'hidden';\n",
       "    document.getElementById('showSettingsButton_1').style.visibility = 'visible';\n",
       "    document.getElementById('hideSettingsButton_1').style.visibility = 'hidden';\n",
       "    settingsShown = false;\n",
       "}\n",
       "\n",
       "\n",
       "\n",
       "var pos = 150* Math.PI;\n",
       "\n",
       "function updateFactor() {\n",
       "    pos++;\n",
       "    return Math.sin(.01*pos)+1;\n",
       "}\n",
       "\n",
       "// ------------------------ FOLDING ------------------------------------------------\n",
       "// ---------------------------------------------------------------------------------\n",
       "// rotate point p around axis defined by points p1 and p2 by given angle\n",
       "function rotate(p, p1, p2, angle ){   \n",
       "    angle = -angle;\n",
       "    var x = p.x, y = p.y, z = p.z, \n",
       "    a = p1.x, b = p1.y, c = p1.z, \n",
       "    u = p2.x-p1.x, v = p2.y-p1.y, w = p2.z-p1.z;\n",
       "    var result = [];\n",
       "    var L = u*u + v*v + w*w;\n",
       "    var sqrt = Math.sqrt;\n",
       "    var cos = Math.cos;\n",
       "    var sin = Math.sin;\n",
       "\n",
       "    result[0] = ((a*(v*v+w*w)-u*(b*v+c*w-u*x-v*y-w*z))*(1-cos(angle))+L*x*cos(angle)+sqrt(L)*(-c*v+b*w-w*y+v*z)*sin(angle))/L;\n",
       "    result[1] = ((b*(u*u+w*w)-v*(a*u+c*w-u*x-v*y-w*z))*(1-cos(angle))+L*y*cos(angle)+sqrt(L)*(c*u-a*w+w*x-u*z)*sin(angle))/L;\n",
       "    result[2] = ((c*(u*u+v*v)-w*(a*u+b*v-u*x-v*y-w*z))*(1-cos(angle))+L*z*cos(angle)+sqrt(L)*(-b*u+a*v-v*x+u*y)*sin(angle))/L;\n",
       "\n",
       "    return result;\n",
       "}\n",
       "\n",
       "var fold = function(event){\n",
       "    var obj = foldables[Number(event.currentTarget.name)];\n",
       "    var foldvalue = Number(event.currentTarget.value);\n",
       "    var scale = foldvalue - obj.userData.oldscale;\n",
       "\n",
       "    for (var j=0; j<obj.userData.axes.length; j++) {\n",
       "        rotateVertices(obj, j, scale);\n",
       "    }\n",
       "    update(obj);\n",
       "    obj.userData.oldscale += scale;\n",
       "    lookAtBarycenter(obj);\n",
       "}\n",
       "\n",
       "function lookAtBarycenter(obj){\n",
       "    control.target = barycenter(obj);\n",
       "}\n",
       "\n",
       "function barycenter(obj) {\n",
       "    var center = new THREE.Vector3(0,0,0);\n",
       "    var points = obj.userData.points;\n",
       "    for (var i=0; i<points.length; i++){\n",
       "        center.add(points[i].vector);\n",
       "    }\n",
       "    center.divideScalar(points.length);\n",
       "    return center;\n",
       "}\n",
       "\n",
       "function rotateVertices(obj, edge, scale) {\n",
       "    var axes = obj.userData.axes;\n",
       "    var subtrees = obj.userData.subtrees;\n",
       "    var points = obj.userData.points;\n",
       "    var angles = obj.userData.angles;\n",
       "    if (edge < axes.length){\n",
       "        for (var j=0; j<subtrees[edge].length; j++){\n",
       "            var rotP = rotate(points[subtrees[edge][j]].vector, points[axes[edge][0]].vector,points[axes[edge][1]].vector, scale * (Math.PI - angles[edge]));\n",
       "            points[subtrees[edge][j]].set(rotP[0],rotP[1],rotP[2]);\n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "function update(obj) {\n",
       "   updateFacesPosition(obj);\n",
       "   updateEdgesPosition(obj);\n",
       "}\n",
       "\n",
       "if (foldables.length) {\n",
       "    var settings = document.getElementById('settings_1');\n",
       "    var foldDiv = document.createElement('div');\n",
       "    foldDiv.id = 'fold_1';\n",
       "    var title = document.createElement('strong');\n",
       "    title.innerHTML = 'Fold';\n",
       "    foldDiv.appendChild(title);\n",
       "    foldDiv.className = 'group';\n",
       "    for (var i=0; i<foldables.length; i++) {\n",
       "        var range = document.createElement('input');\n",
       "        range.type = 'range';\n",
       "        range.min = 0;\n",
       "        range.max = 1;\n",
       "        range.value = 0;\n",
       "        range.step = 0.001;\n",
       "        range.name = String(i);\n",
       "        range.oninput = fold;\n",
       "        foldDiv.appendChild(range);\n",
       "    }\n",
       "    lookAtBarycenter(foldables[0]);\n",
       "    settings.insertBefore(foldDiv,settings.childNodes[0]);\n",
       "}\n",
       "\n",
       "    \n",
       "// ---------------------- EXPLOSION ------------------------------------------------\n",
       "// ---------------------------------------------------------------------------------\n",
       "\n",
       "if (explodableModel) {\n",
       "    for (var i=0; i<scene.children.length; i++) {\n",
       "        obj = scene.children[i];\n",
       "        if ( obj.userData.explodable ) {\n",
       "            computeCentroid(obj);\n",
       "        }\n",
       "    }\n",
       "    document.getElementById('explodeRange_1').oninput = triggerExplode;\n",
       "    document.getElementById('explodeCheckbox_1').onchange = triggerAutomaticExplode;\n",
       "    document.getElementById('explodingSpeedRange_1').oninput = setExplodingSpeed;\n",
       "}\n",
       "\n",
       "function computeCentroid(obj) {\n",
       "    centroid = new THREE.Vector3();\n",
       "    obj.userData.points.forEach(function(pmpoint) {\n",
       "        centroid.add(pmpoint.vector);\t\t\t\n",
       "    });\n",
       "    centroid.divideScalar(obj.userData.points.length);\n",
       "    obj.userData.centroid = centroid;\n",
       "}\n",
       "\n",
       "function explode(factor) {\n",
       "    for (var i=0; i<scene.children.length; i++) {\n",
       "        var obj = scene.children[i];\n",
       "        if (obj.userData.hasOwnProperty(\"centroid\")) { \n",
       "            var c = obj.userData.centroid;\n",
       "            obj.position.set(c.x*factor, c.y*factor, c.z*factor);\n",
       "        }\n",
       "    }\t\n",
       "}\n",
       "\n",
       "function triggerExplode(event){\n",
       "    explodeScale = Number(event.currentTarget.value);\n",
       "    explode(explodeScale);\n",
       "}\n",
       "\n",
       "function setExplodingSpeed(event){\n",
       "    explodingSpeed = Number(event.currentTarget.value);\n",
       "}\n",
       "\n",
       "function triggerAutomaticExplode(event){\n",
       "    if (event.currentTarget.checked){\n",
       "        startExploding();\n",
       "    } else {\n",
       "        clearIntervals();\n",
       "    }\t\n",
       "}\n",
       "\n",
       "function startExploding(){\n",
       "    intervals.push(setInterval(explodingInterval, 25));\n",
       "}\n",
       "\n",
       "\n",
       "function explodingInterval(){\n",
       "    explodeScale += explodingSpeed;\n",
       "    if (explodeScale <= 6){ \n",
       "        explode(explodeScale);\n",
       "    }\n",
       "    else{\n",
       "        explode(6);\n",
       "        explodeScale = 6;\n",
       "        clearIntervals();\n",
       "        timeouts.push(setTimeout(startUnexploding, 3000));\n",
       "    }\n",
       "    document.getElementById('explodeRange_1').value = explodeScale;\n",
       "}\n",
       "\n",
       "\n",
       "function startUnexploding(){\n",
       "    intervals.push(setInterval(unexplodingInterval, 25));\n",
       "}\n",
       "\n",
       "function unexplodingInterval(){\n",
       "    explodeScale -= explodingSpeed;\n",
       "    if (explodeScale >= 0){\t\n",
       "        explode(explodeScale);\n",
       "    }\n",
       "    else {\n",
       "        explode(0);\n",
       "        explodeScale = 0;\n",
       "        clearIntervals();\n",
       "        timeouts.push(setTimeout(startExploding, 3000));\n",
       "    }\n",
       "    document.getElementById('explodeRange_1').value = explodeScale;\n",
       "}\n",
       "\n",
       "function clearIntervals(){\n",
       "    intervals.forEach(function(interval){\n",
       "        clearInterval(interval);\n",
       "    });\n",
       "    intervals = [];\n",
       "    timeouts.forEach(function(timeout){\n",
       "        clearTimeout(timeout);\n",
       "    });\n",
       "    timeouts = [];\n",
       "}\n",
       "\n",
       "// ---------------------- DISPLAY --------------------------------------------------\n",
       "// ---------------------------------------------------------------------------------\n",
       "\n",
       "const objectTypeInnerHTMLs = { points: \"Points\", pointlabels: \"Point labels\", lines: \"Edges\", edgelabels: \"Edge labels\", faces: \"Faces\", arrowheads: \"Arrow heads\" };\n",
       "const objectTypeVisible = {};\n",
       "Object.assign(objectTypeVisible,modelContains);\n",
       "const sortedObjectTypeKeys = Object.keys(objectTypeInnerHTMLs).sort();\n",
       "const shownObjectTypesList = document.getElementById('shownObjectTypesList_1');\n",
       "\n",
       "function setVisibility(bool,objname) {\n",
       "    for (var i=0; i<scene.children.length; i++){\n",
       "        var obj = scene.children[i].getObjectByName(objname);\n",
       "        if (obj) {\n",
       "            obj.visible = bool;\n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "function toggleObjectTypeVisibility(event){\n",
       "    var name = event.currentTarget.name;\n",
       "    var checked = event.currentTarget.checked;\n",
       "    objectTypeVisible[name] = checked;\n",
       "    setVisibility(checked,name);\n",
       "}\n",
       "\n",
       "for (var i=0; i<sortedObjectTypeKeys.length; i++){\n",
       "    var key = sortedObjectTypeKeys[i];\n",
       "    if (modelContains[key]) {\n",
       "        var objTypeNode = document.createElement('span');\n",
       "        objTypeNode.innerHTML = objectTypeInnerHTMLs[key] + '<br>';\n",
       "        var checkbox = document.createElement('input');\n",
       "        checkbox.type = 'checkbox';\n",
       "        checkbox.checked = true;\n",
       "        checkbox.name = key;\n",
       "        checkbox.onchange = toggleObjectTypeVisibility;\n",
       "        shownObjectTypesList.appendChild(checkbox);\n",
       "        shownObjectTypesList.appendChild(objTypeNode);\n",
       "    }\n",
       "}\n",
       "\n",
       "// ------------------------------------------------------\n",
       "\n",
       "function toggleObjectVisibility(event){\n",
       "    var nr = Number(event.currentTarget.name);\n",
       "    scene.children[nr].visible = event.currentTarget.checked;\n",
       "}\n",
       "\n",
       "// append checkboxes for displaying or hiding objects\n",
       "var shownObjectsList = document.getElementById('shownObjectsList_1');\n",
       "for (var i=0; i<scene.children.length; i++){\n",
       "    obj = scene.children[i];\n",
       "    var objNode = document.createElement('span');\n",
       "    objNode.innerHTML = obj.name + '<br>';\n",
       "    var checkbox = document.createElement('input');\n",
       "    checkbox.type = 'checkbox';\n",
       "    checkbox.checked = true;\n",
       "    checkbox.name = String(i);\n",
       "    checkbox.onchange = toggleObjectVisibility;\n",
       "    shownObjectsList.appendChild(checkbox);\n",
       "    shownObjectsList.appendChild(objNode);\n",
       "}\n",
       "\n",
       "// ---------------------- SVG ------------------------------------------------------\n",
       "// ---------------------------------------------------------------------------------\n",
       "\n",
       "function takeSvgScreenshot() {\n",
       "    if (objectTypeVisible[\"pointlabels\"]) {\n",
       "        setVisibility(false,\"pointlabels\");\n",
       "    }\n",
       "    if (objectTypeVisible[\"edgelabels\"]) {\n",
       "        setVisibility(false,\"edgelabels\");\n",
       "    }\n",
       "    svgRenderer.render(scene,camera);\n",
       "    svgElement = XMLS.serializeToString(svgRenderer.domElement);\n",
       "    \n",
       "    if (objectTypeVisible[\"pointlabels\"]) {\n",
       "        setVisibility(true,\"pointlabels\");\n",
       "    }\n",
       "    if (objectTypeVisible[\"edgelabels\"]) {\n",
       "        setVisibility(true,\"edgelabels\");\n",
       "    }\n",
       "\n",
       "    if (document.getElementById('tab_1').checked){\n",
       "        //show in new tab\n",
       "        var myWindow = window.open(\"\",\"\");\n",
       "        myWindow.document.body.innerHTML = svgElement;\n",
       "    } else{\n",
       "        // download svg file \n",
       "        download(\"screenshot.svg\", svgElement);\n",
       "    }\n",
       "}\n",
       "\n",
       "function download(filename, text) {\n",
       "    var element = document.createElement('a');\n",
       "    element.setAttribute('href', 'data:text/plain;charset=utf-8,' + encodeURIComponent(text));\n",
       "    element.setAttribute('download', filename);\n",
       "\n",
       "    element.style.display = 'none';\n",
       "    document.body.appendChild(element);\n",
       "\n",
       "    element.click();\n",
       "\n",
       "    document.body.removeChild(element);\n",
       "}\n",
       "\n",
       "\n",
       "document.getElementById('transparencyRange_1').oninput = changeTransparency;\n",
       "document.getElementById('changeRotationX_1').onchange = changeRotationX;\n",
       "document.getElementById('changeRotationY_1').onchange = changeRotationY;\n",
       "document.getElementById('changeRotationZ_1').onchange = changeRotationZ;\n",
       "document.getElementById('resetButton_1').onclick = resetScene;\n",
       "document.getElementById('rotationSpeedRange_1').oninput = changeRotationSpeedFactor;\n",
       "document.getElementById('takeScreenshot_1').onclick = takeSvgScreenshot;\n",
       "document.getElementById('showSettingsButton_1').onclick = showSettings;\n",
       "document.getElementById('hideSettingsButton_1').onclick = hideSettings;\n",
       "\n",
       "\n",
       "// ------------------ SHORTCUTS --------------------------------------------\n",
       "// -------------------------------------------------------------------------\n",
       "\n",
       "/**\n",
       " * http://www.openjs.com/scripts/events/keyboard_shortcuts/\n",
       " * Version : 2.01.B\n",
       " * By Binny V A\n",
       " * License : BSD\n",
       " */\n",
       "shortcut = {\n",
       "\t'all_shortcuts':{},//All the shortcuts are stored in this array\n",
       "\t'add': function(shortcut_combination,callback,opt) {\n",
       "\t\t//Provide a set of default options\n",
       "\t\tvar default_options = {\n",
       "\t\t\t'type':'keydown',\n",
       "\t\t\t'propagate':false,\n",
       "\t\t\t'disable_in_input':false,\n",
       "\t\t\t'target':document,\n",
       "\t\t\t'keycode':false\n",
       "\t\t}\n",
       "\t\tif(!opt) opt = default_options;\n",
       "\t\telse {\n",
       "\t\t\tfor(var dfo in default_options) {\n",
       "\t\t\t\tif(typeof opt[dfo] == 'undefined') opt[dfo] = default_options[dfo];\n",
       "\t\t\t}\n",
       "\t\t}\n",
       "\n",
       "\t\tvar ele = opt.target;\n",
       "\t\tif(typeof opt.target == 'string') ele = document.getElementById(opt.target);\n",
       "\t\tvar ths = this;\n",
       "\t\tshortcut_combination = shortcut_combination.toLowerCase();\n",
       "\n",
       "\t\t//The function to be called at keypress\n",
       "\t\tvar func = function(e) {\n",
       "\t\t\te = e || window.event;\n",
       "\t\t\t\n",
       "\t\t\tif(opt['disable_in_input']) { //Don't enable shortcut keys in Input, Textarea fields\n",
       "\t\t\t\tvar element;\n",
       "\t\t\t\tif(e.target) element=e.target;\n",
       "\t\t\t\telse if(e.srcElement) element=e.srcElement;\n",
       "\t\t\t\tif(element.nodeType==3) element=element.parentNode;\n",
       "\n",
       "\t\t\t\tif(element.tagName == 'INPUT' || element.tagName == 'TEXTAREA') return;\n",
       "\t\t\t}\n",
       "\t\n",
       "\t\t\t//Find Which key is pressed\n",
       "\t\t\tif (e.keyCode) code = e.keyCode;\n",
       "\t\t\telse if (e.which) code = e.which;\n",
       "\t\t\tvar character = String.fromCharCode(code).toLowerCase();\n",
       "\t\t\t\n",
       "\t\t\tif(code == 188) character=\",\"; //If the user presses , when the type is onkeydown\n",
       "\t\t\tif(code == 190) character=\".\"; //If the user presses , when the type is onkeydown\n",
       "\n",
       "\t\t\tvar keys = shortcut_combination.split(\"+\");\n",
       "\t\t\t//Key Pressed - counts the number of valid keypresses - if it is same as the number of keys, the shortcut function is invoked\n",
       "\t\t\tvar kp = 0;\n",
       "\t\t\t\n",
       "\t\t\t//Work around for stupid Shift key bug created by using lowercase - as a result the shift+num combination was broken\n",
       "\t\t\tvar shift_nums = {\n",
       "\t\t\t\t\"`\":\"~\",\n",
       "\t\t\t\t\"1\":\"!\",\n",
       "\t\t\t\t\"2\":\"@\",\n",
       "\t\t\t\t\"3\":\"#\",\n",
       "\t\t\t\t\"4\":\"$\",\n",
       "\t\t\t\t\"5\":\"%\",\n",
       "\t\t\t\t\"6\":\"^\",\n",
       "\t\t\t\t\"7\":\"&\",\n",
       "\t\t\t\t\"8\":\"*\",\n",
       "\t\t\t\t\"9\":\"(\",\n",
       "\t\t\t\t\"0\":\")\",\n",
       "\t\t\t\t\"-\":\"_\",\n",
       "\t\t\t\t\"=\":\"+\",\n",
       "\t\t\t\t\";\":\":\",\n",
       "\t\t\t\t\"'\":\"\\\"\",\n",
       "\t\t\t\t\",\":\"<\",\n",
       "\t\t\t\t\".\":\">\",\n",
       "\t\t\t\t\"/\":\"?\",\n",
       "\t\t\t\t\"\\\\\":\"|\"\n",
       "\t\t\t}\n",
       "\t\t\t//Special Keys - and their codes\n",
       "\t\t\tvar special_keys = {\n",
       "\t\t\t\t'esc':27,\n",
       "\t\t\t\t'escape':27,\n",
       "\t\t\t\t'tab':9,\n",
       "\t\t\t\t'space':32,\n",
       "\t\t\t\t'return':13,\n",
       "\t\t\t\t'enter':13,\n",
       "\t\t\t\t'backspace':8,\n",
       "\t\n",
       "\t\t\t\t'scrolllock':145,\n",
       "\t\t\t\t'scroll_lock':145,\n",
       "\t\t\t\t'scroll':145,\n",
       "\t\t\t\t'capslock':20,\n",
       "\t\t\t\t'caps_lock':20,\n",
       "\t\t\t\t'caps':20,\n",
       "\t\t\t\t'numlock':144,\n",
       "\t\t\t\t'num_lock':144,\n",
       "\t\t\t\t'num':144,\n",
       "\t\t\t\t\n",
       "\t\t\t\t'pause':19,\n",
       "\t\t\t\t'break':19,\n",
       "\t\t\t\t\n",
       "\t\t\t\t'insert':45,\n",
       "\t\t\t\t'home':36,\n",
       "\t\t\t\t'delete':46,\n",
       "\t\t\t\t'end':35,\n",
       "\t\t\t\t\n",
       "\t\t\t\t'pageup':33,\n",
       "\t\t\t\t'page_up':33,\n",
       "\t\t\t\t'pu':33,\n",
       "\t\n",
       "\t\t\t\t'pagedown':34,\n",
       "\t\t\t\t'page_down':34,\n",
       "\t\t\t\t'pd':34,\n",
       "\t\n",
       "\t\t\t\t'left':37,\n",
       "\t\t\t\t'up':38,\n",
       "\t\t\t\t'right':39,\n",
       "\t\t\t\t'down':40,\n",
       "\t\n",
       "\t\t\t\t'f1':112,\n",
       "\t\t\t\t'f2':113,\n",
       "\t\t\t\t'f3':114,\n",
       "\t\t\t\t'f4':115,\n",
       "\t\t\t\t'f5':116,\n",
       "\t\t\t\t'f6':117,\n",
       "\t\t\t\t'f7':118,\n",
       "\t\t\t\t'f8':119,\n",
       "\t\t\t\t'f9':120,\n",
       "\t\t\t\t'f10':121,\n",
       "\t\t\t\t'f11':122,\n",
       "\t\t\t\t'f12':123\n",
       "\t\t\t}\n",
       "\t\n",
       "\t\t\tvar modifiers = { \n",
       "\t\t\t\tshift: { wanted:false, pressed:false},\n",
       "\t\t\t\tctrl : { wanted:false, pressed:false},\n",
       "\t\t\t\talt  : { wanted:false, pressed:false},\n",
       "\t\t\t\tmeta : { wanted:false, pressed:false}\t//Meta is Mac specific\n",
       "\t\t\t};\n",
       "                        \n",
       "\t\t\tif(e.ctrlKey)\tmodifiers.ctrl.pressed = true;\n",
       "\t\t\tif(e.shiftKey)\tmodifiers.shift.pressed = true;\n",
       "\t\t\tif(e.altKey)\tmodifiers.alt.pressed = true;\n",
       "\t\t\tif(e.metaKey)   modifiers.meta.pressed = true;\n",
       "                        \n",
       "\t\t\tfor(var i=0; k=keys[i],i<keys.length; i++) {\n",
       "\t\t\t\t//Modifiers\n",
       "\t\t\t\tif(k == 'ctrl' || k == 'control') {\n",
       "\t\t\t\t\tkp++;\n",
       "\t\t\t\t\tmodifiers.ctrl.wanted = true;\n",
       "\n",
       "\t\t\t\t} else if(k == 'shift') {\n",
       "\t\t\t\t\tkp++;\n",
       "\t\t\t\t\tmodifiers.shift.wanted = true;\n",
       "\n",
       "\t\t\t\t} else if(k == 'alt') {\n",
       "\t\t\t\t\tkp++;\n",
       "\t\t\t\t\tmodifiers.alt.wanted = true;\n",
       "\t\t\t\t} else if(k == 'meta') {\n",
       "\t\t\t\t\tkp++;\n",
       "\t\t\t\t\tmodifiers.meta.wanted = true;\n",
       "\t\t\t\t} else if(k.length > 1) { //If it is a special key\n",
       "\t\t\t\t\tif(special_keys[k] == code) kp++;\n",
       "\t\t\t\t\t\n",
       "\t\t\t\t} else if(opt['keycode']) {\n",
       "\t\t\t\t\tif(opt['keycode'] == code) kp++;\n",
       "\n",
       "\t\t\t\t} else { //The special keys did not match\n",
       "\t\t\t\t\tif(character == k) kp++;\n",
       "\t\t\t\t\telse {\n",
       "\t\t\t\t\t\tif(shift_nums[character] && e.shiftKey) { //Stupid Shift key bug created by using lowercase\n",
       "\t\t\t\t\t\t\tcharacter = shift_nums[character]; \n",
       "\t\t\t\t\t\t\tif(character == k) kp++;\n",
       "\t\t\t\t\t\t}\n",
       "\t\t\t\t\t}\n",
       "\t\t\t\t}\n",
       "\t\t\t}\n",
       "\t\t\t\n",
       "\t\t\tif(kp == keys.length && \n",
       "\t\t\t\t\t\tmodifiers.ctrl.pressed == modifiers.ctrl.wanted &&\n",
       "\t\t\t\t\t\tmodifiers.shift.pressed == modifiers.shift.wanted &&\n",
       "\t\t\t\t\t\tmodifiers.alt.pressed == modifiers.alt.wanted &&\n",
       "\t\t\t\t\t\tmodifiers.meta.pressed == modifiers.meta.wanted) {\n",
       "\t\t\t\tcallback(e);\n",
       "\t\n",
       "\t\t\t\tif(!opt['propagate']) { //Stop the event\n",
       "\t\t\t\t\t//e.cancelBubble is supported by IE - this will kill the bubbling process.\n",
       "\t\t\t\t\te.cancelBubble = true;\n",
       "\t\t\t\t\te.returnValue = false;\n",
       "\t\n",
       "\t\t\t\t\t//e.stopPropagation works in Firefox.\n",
       "\t\t\t\t\tif (e.stopPropagation) {\n",
       "\t\t\t\t\t\te.stopPropagation();\n",
       "\t\t\t\t\t\te.preventDefault();\n",
       "\t\t\t\t\t}\n",
       "\t\t\t\t\treturn false;\n",
       "\t\t\t\t}\n",
       "\t\t\t}\n",
       "\t\t}\n",
       "\t\tthis.all_shortcuts[shortcut_combination] = {\n",
       "\t\t\t'callback':func, \n",
       "\t\t\t'target':ele, \n",
       "\t\t\t'event': opt['type']\n",
       "\t\t};\n",
       "\t\t//Attach the function with the event\n",
       "\t\tif(ele.addEventListener) ele.addEventListener(opt['type'], func, false);\n",
       "\t\telse if(ele.attachEvent) ele.attachEvent('on'+opt['type'], func);\n",
       "\t\telse ele['on'+opt['type']] = func;\n",
       "\t},\n",
       "\n",
       "\t//Remove the shortcut - just specify the shortcut and I will remove the binding\n",
       "\t'remove':function(shortcut_combination) {\n",
       "\t\tshortcut_combination = shortcut_combination.toLowerCase();\n",
       "\t\tvar binding = this.all_shortcuts[shortcut_combination];\n",
       "\t\tdelete(this.all_shortcuts[shortcut_combination])\n",
       "\t\tif(!binding) return;\n",
       "\t\tvar type = binding['event'];\n",
       "\t\tvar ele = binding['target'];\n",
       "\t\tvar callback = binding['callback'];\n",
       "\n",
       "\t\tif(ele.detachEvent) ele.detachEvent('on'+type, callback);\n",
       "\t\telse if(ele.removeEventListener) ele.removeEventListener(type, callback, false);\n",
       "\t\telse ele['on'+type] = false;\n",
       "\t}\n",
       "}\n",
       "\n",
       "shortcut.add(\"Alt+Left\",function() {\n",
       "\tvar event = new Event('click');\n",
       "\tif (settingsShown){\n",
       "\t\tdocument.getElementById('hideSettingsButton_1').dispatchEvent(event);\n",
       "\t} else {\n",
       "\t\tdocument.getElementById('showSettingsButton_1').dispatchEvent(event);\n",
       "\t}\n",
       "});\n",
       "\n",
       "\n",
       "// COMMON_CODE_BLOCK_END\n",
       "\n",
       "});});\n",
       "      </script>\n",
       "   </body>\n",
       "</html>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "$p->VISUAL->MIN_MAX_FACE;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Computing Facets\n",
    "\n",
    "An important action that is often needed to come up with new facet describing inequalities for combinatorial optimization problems is the computation of convex hulls for small examples.\n",
    "\n",
    "### Pure Integer Case\n",
    "\n",
    "We begin with the case in which all variables are required to be integral, i.e., the *pure integer case*. Moreover, the approach depends on whether the polyhedron is bounded or not.\n",
    "\n",
    "#### Bounded Polyhedra\n",
    "\n",
    "Let us illustrate the approach via the example of the *stable set problem*: Here one is given an (undirected) Graph G = (V,E) with node set V and edges E. The goal is to find a largest subset of nodes V' such that any two nodes in V' are not connected by an edge.\n",
    "\n",
    "For our example consider the 5-cycle, i.e., the graph C<sub>5</sub> with five nodes {1, 2, 3, 4, 5} and edges {1,2}, {2,3}, {3,4}, {4,5}, {5,1}. A formulation of the stable set problem for this graph looks as follows:\n",
    "\n",
    "    \n",
    "    Maximize\n",
    "     obj: x#1 + x#2 + x#3 + x#4 + x#5\n",
    "    Subject to\n",
    "     edge_1: x#2 + x#1 <= 1\n",
    "     edge_2: x#3 + x#2 <= 1\n",
    "     edge_3: x#4 + x#3 <= 1\n",
    "     edge_4: x#5 + x#4 <= 1\n",
    "     edge_5: x#1 + x#5 <= 1\n",
    "    Bounds\n",
    "     0 <= x#1 <= 1\n",
    "     0 <= x#2 <= 1\n",
    "     0 <= x#3 <= 1\n",
    "     0 <= x#4 <= 1\n",
    "     0 <= x#5 <= 1\n",
    "    General\n",
    "     x#1 x#2 x#3 x#4 x#5\n",
    "    End\n",
    "\n",
    "Here, `General` encodes that the following variables should be restricted to obtain integer values. Thus, all variables `x#1, x#2, x#3, x#4, x#5` are restricted to binary values (0 or 1). The value 1 encodes that the corresponding node will be in an optimal stable set (and 0 otherwise). The constraints `edge_*` encode that a most one of the two nodes covered by an edge is selected. Clearly, the corresponding polyhedron is bounded.\n",
    "\n",
    "We assume that the above information is contained in the file `stab.lp`. We now read it into polymake and convert it to rational form, as explained above:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "$f=lp2poly('files/optimization/stab.lp');\n",
    "$p = new Polytope<Rational>($f);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "We are now interested in all feasible solutions to the above problem, i.e., all assignments of 0 or 1 to the variables such that the above inequalities are satisfied. These feasible points can be computed via:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "$p->LATTICE_POINTS;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "To understand these points and make computational use of this information, we are interested in the convex hull of all feasible solutions. To this end, we construct a new polytope which is specified via points for which the convex hull is taken. This can be done as follows:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "$s=new Polytope(POINTS=>$p->LATTICE_POINTS, COORDINATE_LABELS=>$p->COORDINATE_LABELS);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "Here, the coordinate labels, i.e., the variable names, are copied to the new polytope.\n",
    "\n",
    "Now, the facets of the new polytope can be computed and listed via:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Facets:\n",
       "0: -x#1 - x#5 >= -1\n",
       "1: x#5 >= 0\n",
       "2: x#2 >= 0\n",
       "3: x#4 >= 0\n",
       "4: -x#3 - x#4 >= -1\n",
       "5: x#1 >= 0\n",
       "6: -x#4 - x#5 >= -1\n",
       "7: x#3 >= 0\n",
       "8: -x#1 - x#2 >= -1\n",
       "9: -x#2 - x#3 >= -1\n",
       "10: -x#1 - x#2 - x#3 - x#4 - x#5 >= -2\n",
       "\n"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print_constraints($s);"
   ]
  },
  {
   "attachments": {
    "ip-unb.gif": {
     "image/gif": "[encoded data, 10620 characters]"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The facet defining inequalities can be interpreted as follows:\n",
    "\n",
    "*  There are five trivial inequalities `x#? >= 0`.\n",
    "\n",
    "*  The five original 'edge' inequalities `x#i + x#j <= 1` define facets.\n",
    "\n",
    "*  We have the so-called *odd-cycle inequality* `x#1 +x#2 + x#3 + x#4 + x#5 <= 2`, stating that at most two nodes in an (odd) cycle of length 5 can be selected. This inequality can be generalized by taking the sum of all variables in an odd cycle and restricting the sum to be less or equal to the size of the cycle minus 1 divided by 2.\n",
    "\n",
    "Of course, one can also use the usual polymake output, e.g., `print $s->FACETS`.\n",
    "\n",
    "This example showed one of the routine actions often performed by discrete optimizers. Of course, this action can also be performed by a script, which makes the computation a one-line command.\n",
    "\n",
    "Note that the size of instances that can be handled will probably be small. Usually, things become difficult from dimension 15 on, but it depends on the particular structure of your instances, i.e., on the number of facets and lattice points.\n",
    "\n",
    "\n",
    "#### Unbounded Polyhedra\n",
    "\n",
    "If the underlying polyhedron is unbounded, the approach above does not work anymore, since there are infinitely many lattice points. Arguably,  this case occurs less often than the bounded case, but it is a excellent show-case for polymake's potential.\n",
    "\n",
    "The following mathematical insights are important to treat the unbounded case. First, we have to assume that the data, i.e., the inequality description of the polyhedron `P`, is rational; otherwise, we cannot expect a finite description of the convex hull. Second, we write `P` as the sum of a finite part `Q` and the recession cone `C`. If the data is rational, the recession cone of `P` and of the integer hull coincide. Third, it suffices to generate the integer points in `Q + R`, where `R` is the parallelotope generated by the rays of `C`. Thus, `R` is generated by the Minkowski sum of the interval `[0,1]` and the generating rays.\n",
    "\n",
    "To illustrate the construction, consider the following example:\n",
    "\n",
    "    \n",
    "    Minimize\n",
    "     obj:  x1 + x2\n",
    "    Subject to\n",
    "    C1: x1 + x2 >= 0.5\n",
    "    C2: x1 - 2 x2 <= 1.5\n",
    "    C3: x2 - 2 x1 <= 1.5\n",
    "    General\n",
    "     x1 x2\n",
    "    End\n",
    "    \n",
    "<img src=\"attachment:ip-unb.gif\" class=\"center-block\"><img>\n",
    "\n",
    "We now assume that the example is contained in the file `unbounded.lp` and proceed as above\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "$f = lp2poly('files/optimization/unbounded.lp');\n",
    "$pin = new Polytope<Rational>($f);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "The visualization in the picture can be generated with `$pin->VISUAL`. The lattice points can be shown with `$pin->VISUAL->LATTICE_COLORED`.\n",
    "\n",
    "We now extract the rays of the recession cone\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "$rays = $pin->VERTICES->minor($pin->FAR_FACE, All);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "This command first computes all vertices of the polyhedron (this includes unbounded vertices); note that is involves a convex hull computation. The set `FAR_FACE` contains the indices of all vertices that are unbounded. The result is:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0 1 1/2\n",
       "0 1 2\n"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print $rays;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus, there are two rays that are generators of the recession cone.\n",
    "\n",
    "We now have to construct the Minkowski hull of all intervals `[0,r]` for each ray `r` (scaled to be integral). This can be done with the following code (possibly easier):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "$zero = unit_vector<Rational>($pin->DIM + 1, 0);\n",
    "$B = new Polytope<Rational>(POINTS=>$zero);\n",
    "foreach my $r (@$rays)\n",
    "{\n",
    "    $M = new Matrix<Rational>(primitive($r));\n",
    "    $M->[0]->[0] = 1;\n",
    "    $M = $M / $zero;\n",
    "    $ptemp = new Polytope<Rational>(POINTS=>$M);\n",
    "    $B = minkowski_sum($B, $ptemp);\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The code first generates a polytope `B` consisting of 0 only. It then takes each ray `r` in turn and creates `[0,r]`. It then takes the Minkowski sum of this new polytope with `B` and stores the result in `B`.\n",
    "\n",
    "The next step is to obtain the bounded part `Q` of `P`, by first extracting the bounded vertices and the creating a new polytope:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "$Qpoints = $pin->VERTICES->minor($pin->BOUNDED_VERTICES, All);\n",
    "$Q = new Polytope<Rational>(POINTS=>$Qpoints);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "The two polytopes are now combined:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "$p = minkowski_sum($Q, $B);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "We now generate the lattice points (as in the bounded part) and add the rays from above:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "$latticemat = new Matrix<Rational>($p->LATTICE_POINTS);\n",
    "$newpoints = new Matrix<Rational>($latticemat / $rays);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "Here, `newpoints` is a matrix that contains all lattice points in `Q` and the rays from above.\n",
    "\n",
    "Finally, the polytope we are interested in is:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "$q = new Polytope(POINTS=>$newpoints, COORDINATE_LABELS=>$pin->COORDINATE_LABELS);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "The facets can be viewed as usual:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Facets:\n",
       "0: -x1 + 2 x2 >= -1\n",
       "1: x1 + x2 >= 1\n",
       "2: 2 x1 - x2 >= -1\n",
       "3: 0 >= -1\n",
       "\n"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print_constraints($q);"
   ]
  },
  {
   "attachments": {
    "ip-unb-integerhull.gif": {
     "image/gif": "[encoded data, 8296 characters]"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"attachment:ip-unb-integerhull.gif\" class=\"center-block\"><img>\n",
    "\n",
    "Note that the upper right part (including the red vertices) arises from truncation of the polyhedron for visualization.\n",
    "\n",
    "\n",
    "### Mixed-Integer Case\n",
    "\n",
    "Let us now briefly discuss how to proceed if there are variables that are allowed to be integral. In this case there are several different types of information that one might be interested in. Let us first consider the question of how to compute the convex hull of all feasible integral variables, i.e., we consider the projection to the integral variables and then consider the convex hull of all feasible solutions. We only consider the bounded case, i.e., the original polyhedron is bounded.\n",
    "\n",
    "Consider the following example:\n",
    "\n",
    "    \n",
    "    Minimize\n",
    "     obj:  x1 + x2\n",
    "    Subject to\n",
    "    C1: s1 - 10 x1 <= 0\n",
    "    C2: s2 - 10 x2 <= 0\n",
    "    C3: s1 + s2 <= 1.5\n",
    "    C4: s1 + s2 >= 0.5\n",
    "    Bounds\n",
    "     0 <= s1\n",
    "     0 <= s2\n",
    "     0 <= x1 <= 1\n",
    "     0 <= x2 <= 1\n",
    "    General\n",
    "     x1 x2\n",
    "    End\n",
    "\n",
    "In this example there are two integral variables `x1` and `x2`, while `s1` and `s2` are continuous variables. Assuming the data is contained in the file `mip.lp`, we proceed as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "$m=lp2poly('files/optimization/mip.lp');\n",
    "$p = new Polytope<Rational>($m);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "We project the polyhedron in `$p` to the third and fourth variables as follows:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "$q=projection($p, [3,4]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "We now construct the convex hull of all feasible points as above:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Facets:\n",
       "0: x1 + x2 >= 1\n",
       "1: -x2 >= -1\n",
       "2: -x1 >= -1\n",
       "\n"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "$s=new Polytope(POINTS=>$q->LATTICE_POINTS);\n",
    "print_constraints($s);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus, as expected, the convex hull equals the triangle with vertices `{(0,1),(1,0),(1,1)}`.\n",
    "\n",
    "## Integral Polytopes and Total Unimodularity\n",
    "\n",
    "As explained in the previous example, the integral points in a polytope are of particular interest in discrete optimization. These points are called *lattice points* in polymake and the corresponding convex hull *lattice polytope*. The handling of such polytopes is explained in more detail in the [Tutorial for Lattice Polytopes](lattice_polytopes_tutorial).\n",
    "\n",
    "Of particular interest for discrete optimization are properties of the original inequality system to define a lattice polytope, i.e., a polytope such that all of its vertices are integral (this can be tested by checking the property `LATTICE`). One particularly interesting case occurs if the matrix defining the polytope is *totally unimodular* and the right hand side is integral.\n",
    "\n",
    "Using the polymake extension [Unimodularity](https://github.com/xammy/unimodularity-test/wiki/Polymake-Extension) by Matthias Walter, this can be checked as illustrated in the following examples.\n",
    "\n",
    "### Example: Explicit Matrix\n",
    "\n",
    "In a first example, we directly create an integral matrix\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "$M=new Matrix<Integer>([[1,1,0,0],[1,0,1,0],[1,0,0,1]]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "The total unimodularity of this matrix can be checked as follows:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print totally_unimodular($M);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus, the given matrix is totally unimodular.\n",
    "\n",
    "### Example: Matrix from Input\n",
    "\n",
    "In the second example, we reuse the file `c3t` from the example above. We read it into polymake:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "$f=lp2poly('files/optimization/c3t.lp');\n",
    "$p = new Polytope<Rational>($f);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "We now want to check whether the constraint matrix defined by the inequalities is totally unimodular (note that there are no equations in this example). Thus we first extract the inequality matrix without the first column (as an integer matrix) and then perform the test:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "$A = new Matrix<Integer>($p->INEQUALITIES->minor(All, ~[0]));\n",
    "print totally_unimodular($A);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus, this matrix is totally unimodular as well.\n",
    "\n",
    "## Total dual integrality\n",
    "\n",
    "Computations with respect to total dual integrality (TDI) can also be performed in polymake. Currently (August 2013), you need the perpetual beta version of polymake to access this functionality.\n",
    "\n",
    "The main functions are:\n",
    "\n",
    "*  The function `totally_dual_integral` takes an inequality system (as a matrix) and checks whether it is totally dual integral.\n",
    "\n",
    "*  The function `make_totally_dual_integral` takes a polytope and returns a new polytope with inequalities that are TDI.\n",
    "\n",
    "Note that the input has to be full-dimensional in order to use these functions.\n",
    "\n",
    "To demonstrate the behavior of these functions, consider the 5-cycle example from above again:\n",
    "\n",
    "    \n",
    "    Maximize\n",
    "     obj: x#1 + x#2 + x#3 + x#4 + x#5\n",
    "    Subject to\n",
    "     edge_1: x#2 + x#1 <= 1\n",
    "     edge_2: x#3 + x#2 <= 1\n",
    "     edge_3: x#4 + x#3 <= 1\n",
    "     edge_4: x#5 + x#4 <= 1\n",
    "     edge_5: x#1 + x#5 <= 1\n",
    "    Bounds\n",
    "     0 <= x#1 <= 1\n",
    "     0 <= x#2 <= 1\n",
    "     0 <= x#3 <= 1\n",
    "     0 <= x#4 <= 1\n",
    "     0 <= x#5 <= 1\n",
    "    General\n",
    "     x#1 x#2 x#3 x#4 x#5\n",
    "    End\n",
    "\n",
    "\n",
    "Let us test whether the inequality system of this example is TDI. Thus, we first load the data as usual:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "$f = lp2poly('files/optimization/stab.lp');\n",
    "$p = new Polytope<Rational>($f);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "We now extract the corresponding inequality system and check it for TDIness:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "false"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/html": [
       "<details><summary><pre style=\"display:inline\"><small>Click here for additional output</small></pre></summary>\n",
       "<pre>\n",
       "polymake: used package libnormaliz\n",
       "  [[wiki:external_software#Normaliz]] is a tool for computations in affine monoids, vector configurations, lattice polytopes, and rational cones.\n",
       "  Copyright by Winfried Bruns, Bogdan Ichim, Christof Soeger.\n",
       "  http://www.math.uos.de/normaliz/\n",
       "\n",
       "</pre>\n",
       "</details>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "$M = new Matrix<Rational>($p->INEQUALITIES);\n",
    "print totally_dual_integral($M);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The system is not TDI, which we expected from general theory, since we know that the polytope is not integral, but the system has integral coefficients. Consequently, let us construct a TDI-system for this polytope:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Inequalities:\n",
       "0: x5 >= 0\n",
       "1: x4 >= 0\n",
       "2: x3 >= 0\n",
       "3: x2 >= 0\n",
       "4: x1 >= 0\n",
       "5: -x1 - x2 >= -1\n",
       "6: -x1 - x5 >= -1\n",
       "7: -x2 - x3 >= -1\n",
       "8: -x3 - x4 >= -1\n",
       "9: -x4 - x5 >= -1\n",
       "10: -x1 - x2 - x3 - x4 - x5 >= -5/2\n",
       "11: 0 >= -1\n",
       "\n"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "$q = make_totally_dual_integral($p);\n",
    "print_constraints($q);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As expected, the right hand side is non integral (otherwise, we know from general theory that the polytope would be integral as well). The result is now TDI:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "$N = new Matrix<Rational>($q->INEQUALITIES);\n",
    "print totally_dual_integral($N);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Note* that we need to take the inequalities instead of facets here, since facets are irredundant and thus might not be TDI, although the complete set of inequalities is TDI.\n",
    "\n",
    "\n",
    "## Chvátal-Gomory Closure\n",
    "\n",
    "In the following we want to briefly show how closures of polytopes with respect to certain cutting-plane operators can be computed. We consider the two well-known cutting-plane operators here. The first one is the Chvátal-Gomory generator and the second one is the Lift-and-project operator as defined by Balas. For simplicity we will assume that the considered polytope is full-dimensional. \n",
    "\n",
    "### Chvátal-Gomory Closure - Example 1\n",
    "\n",
    "We first consider the polytope from the stable set problem from above:\n",
    "\n",
    "    \n",
    "    Maximize\n",
    "     obj: x#1 + x#2 + x#3 + x#4 + x#5\n",
    "    Subject to\n",
    "     edge_1: x#2 + x#1 <= 1\n",
    "     edge_2: x#3 + x#2 <= 1\n",
    "     edge_3: x#4 + x#3 <= 1\n",
    "     edge_4: x#5 + x#4 <= 1\n",
    "     edge_5: x#1 + x#5 <= 1\n",
    "    Bounds\n",
    "     0 <= x#1 <= 1\n",
    "     0 <= x#2 <= 1\n",
    "     0 <= x#3 <= 1\n",
    "     0 <= x#4 <= 1\n",
    "     0 <= x#5 <= 1\n",
    "    General\n",
    "     x#1 x#2 x#3 x#4 x#5\n",
    "    End\n",
    "\n",
    "As before we read in the file using `lp2poly`:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "$f = lp2poly('files/optimization/stab.lp');\n",
    "$p = new Polytope<Rational>($f);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "The Chvátal-Gomory closure of a polytope can be computed with the function `gc_closure`. The function takes a full-dimensional polytope and returns a new polytope. This contains the system of inequalities defining the closure in the property `INEQUALITIES`. For our example, we obtain:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Inequalities:\n",
       "0: x5 >= 0\n",
       "1: x4 >= 0\n",
       "2: x3 >= 0\n",
       "3: x2 >= 0\n",
       "4: x1 >= 0\n",
       "5: -x1 - x2 >= -1\n",
       "6: -x1 - x5 >= -1\n",
       "7: -x2 - x3 >= -1\n",
       "8: -x3 - x4 >= -1\n",
       "9: -x4 - x5 >= -1\n",
       "10: -x1 - x2 - x3 - x4 - x5 >= -2\n",
       "11: 0 >= -1\n",
       "\n"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "$g = gc_closure($p);\n",
    "print print_constraints($g);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us check whether the resulting polytope is integral:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print $g->LATTICE;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus, in this case, we have obtained the integer hull by one step of the Chvatal-Gomory-closure.\n",
    "\n",
    "### Chvátal-Gomory Closure - Example 2\n",
    "\n",
    "Let us now consider the classical example of a polytope with the vertices of simplex in d dimensions and the point 1/2 times (1, ..., 1). It can be shown that such a polytope has rank at least log(d) - 1, see [Pokutta, 2011](http://www.box.net/shared/at1y8i3pq434bxt6m9xm)]. In our example, we use d = 4:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Facets:\n",
       "0: x4 >= 0\n",
       "1: x3 >= 0\n",
       "2: x2 >= 0\n",
       "3: x1 >= 0\n",
       "4: -x1 - x2 - x3 >= -1\n",
       "5: -x1 - x2 - x4 >= -1\n",
       "6: -x1 - x3 - x4 >= -1\n",
       "7: -x2 - x3 - x4 >= -1\n",
       "\n"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "$M = new Matrix<Rational>([[1,0,0,0,0],[1,1,0,0,0],[1,0,1,0,0],[1,0,0,1,0],[1,0,0,0,1],[1,1/2,1/2,1/2,1/2]]);\n",
    "$t = new Polytope<Rational>(POINTS => $M);\n",
    "$t1 = gc_closure($t);\n",
    "$t1->FACETS;\n",
    "print_constraints($t1);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "false"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print $t1->LATTICE;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus, one round was not enough to produce an integral polytope. Indeed, the vertices are\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1 0 0 1 0\n",
       "1 0 0 0 1\n",
       "1 0 1 0 0\n",
       "1 1 0 0 0\n",
       "1 1/3 1/3 1/3 1/3\n",
       "1 0 0 0 0\n"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "$t1->VERTICES;\n",
    "print $t1->VERTICES;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "However, one more round is enough:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Facets:\n",
       "0: x4 >= 0\n",
       "1: x3 >= 0\n",
       "2: x2 >= 0\n",
       "3: x1 >= 0\n",
       "4: -x1 - x2 - x3 - x4 >= -1\n",
       "\n"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "$t2 = gc_closure($t1);\n",
    "$t2->FACETS;\n",
    "print_constraints($t2);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print $t2->LATTICE;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "## Lift-and-project closure\n",
    "\n",
    "The lift-and-project closure of a 0/1-polytope P can be generated as follows: for each coordinate compute the intersection of P with the pair of opposite cube faces and compute the convex hull. Then intersect the result with P. The following script defines a subroutine performing this operation - the code is somewhat complicated throught the fact that we need to check whether parts are empty."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```perl\n",
    "use application \"polytope\";\n",
    "\n",
    "sub lpclosure\n",
    "{\n",
    "    my $p = shift;\n",
    "    my $d = $p->AMBIENT_DIM;\n",
    "    my $q = new Polytope<Rational>($p);\n",
    "    for (my $k = 0; $k < $d; $k = $k+1)\n",
    "    {\n",
    "        if ( $q->DIM == -1 )         # can stop as soon as $q is empty\n",
    "        {\n",
    "             return $q;\n",
    "        }\n",
    "    \n",
    "        # create reversed opposite inequalities of 0/1-cube and corresponding polyhedra\n",
    "        my $v1 = new Vector<Rational>(0 | -unit_vector($d, $k));\n",
    "        my $v2 = new Vector<Rational>(-1 | unit_vector($d, $k));\n",
    "    \n",
    "        # create intersection of corresponding polyhedra with iterated polyhedron $q\n",
    "        my $b1 = new Polytope<Rational>(INEQUALITIES => $v1 / $q->FACETS);\n",
    "        my $b2 = new Polytope<Rational>(INEQUALITIES => $v2 / $q->FACETS);\n",
    "    \n",
    "        if ( ($b1->DIM > -1) && ($b2->DIM > -1) )\n",
    "        {\n",
    "            my $c = conv($b1, $b2);\n",
    "            $q = intersection($q, $c);\n",
    "        }\n",
    "        elsif ( ($b1->DIM > -1) && ($b2->DIM == -1) )\n",
    "        {\n",
    "            $q = intersection($q, $b1);\n",
    "        }\n",
    "        elsif ( ($b1->DIM == -1) && ($b2->DIM > -1) )\n",
    "        {\n",
    "            $q = intersection($q, $b2);\n",
    "        }\n",
    "    }\n",
    "    return $q;\n",
    "}\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "script(\"files/optimization/lpclosure.pl\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Lift-and-Project Closure - Example 1\n",
    "\n",
    "For our well known stable set example, we get the following:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Facets:\n",
       "0: -x1 - x5 >= -1\n",
       "1: -x3 - x4 >= -1\n",
       "2: -x1 - x2 >= -1\n",
       "3: -x1 - x2 - x3 - x4 - x5 >= -2\n",
       "4: x4 >= 0\n",
       "5: x5 >= 0\n",
       "6: x2 >= 0\n",
       "7: x1 >= 0\n",
       "8: -x4 - x5 >= -1\n",
       "9: x3 >= 0\n",
       "10: -x2 - x3 >= -1\n",
       "\n"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "$q = lpclosure($p);\n",
    "$q->FACETS;\n",
    "print_constraints($q);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus, the lift-and-project closure in this case gives the integral hull (as we have seen above).\n",
    "\n",
    "### Lift-and-Project Closure - Example 2\n",
    "\n",
    "Let us now consider the same example as for CG-closures:\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Facets:\n",
       "0: x2 >= 0\n",
       "1: x4 >= 0\n",
       "2: x3 >= 0\n",
       "3: x1 >= 0\n",
       "4: -x1 - x2 - x3 - x4 >= -1\n",
       "\n"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "$M = new Matrix<Rational>([[1,0,0,0,0],[1,1,0,0,0],[1,0,1,0,0],[1,0,0,1,0],[1,0,0,0,1],[1,1/2,1/2,1/2,1/2]]);\n",
    "$p = new Polytope<Rational>(POINTS => $M);\n",
    "$q = lpclosure($p);\n",
    "$q->FACETS;\n",
    "print_constraints($q);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus, we have obtained the integral hull in a single step of the lift-and-project closure as opposed to two steps in the CG-closure.\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Alternative Input: mps2poly\n",
    "\n",
    "Another input format that `polymake` can handle is the MPS-format used by soplex or gurobi. This format is more focused on the columns then on the rows of a linear program.\n",
    "\n",
    "\n",
    "    Name          0/1-cube\n",
    "    Rows\n",
    "     N  C0000000\n",
    "     G  R0000000\n",
    "    COLUMNS\n",
    "        x1        C0000000  1                        R0000000  -1\n",
    "        x2        C0000000  1                        R0000000  -1\n",
    "        x3        C0000000  1                        R0000000  -1\n",
    "    RHS\n",
    "        B         R0000000  -2\n",
    "    BOUNDS\n",
    "     LO BND       x1        0\n",
    "     UP BND       x1        1\n",
    "     LO BND       x2        0\n",
    "     UP BND       x2        1\n",
    "     LO BND       x3        0\n",
    "     UP BND       x3        1\n",
    "    ENDDATA\n",
    "    \n",
    "\n",
    "This is again a description of the same 0/1-Cube like above.  \n",
    "If the example is contained in the file `c3t.mps`, we can create again a polytope from it via:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [],
   "source": [
    "$p = mps2poly('files/optimization/c3t.mps');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It gives use the same polytope with the same linear program as above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
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       "\t\t\t\t<div class=indented>\n",
       "\t\t\t\t\t<form>\n",
       "\t\t\t\t\t\t<input type=\"radio\" name='screenshotMode' value='download' id='download_2' checked> Download<br>\n",
       "\t\t\t\t\t\t<input type=\"radio\" name='screenshotMode' value='tab' id='tab_2' > New tab<br>\n",
       "\t\t\t\t\t</form>\n",
       "\t\t\t\t\t<button id='takeScreenshot_2'>Screenshot</button>\n",
       "\t\t\t\t</div>\n",
       "\t\t\t</div>\n",
       "\n",
       "\t\t</div>\t<!-- end of settings -->\n",
       "\t\t<img id='hideSettingsButton_2' class='hideSettingsButton' src='/kernelspecs/polymake/close.svg' width=20px\">\n",
       "\t\t<img id='showSettingsButton_2' class='showSettingsButton' src='/kernelspecs/polymake/menu.svg' width=20px\">\n",
       "<div id=\"model39615074437\"></div>\n",
       "</div>\n",
       "   <script>\n",
       "    requirejs.config({\n",
       "      paths: {\n",
       "        three: '/kernelspecs/polymake/three',\n",
       "        TrackballControls: '/kernelspecs/polymake/TrackballControls',\n",
       "        OrbitControls: '/kernelspecs/polymake/OrbitControls',\n",
       "        Projector: '/kernelspecs/polymake/Projector',\n",
       "        SVGRenderer: '/kernelspecs/polymake/SVGRenderer',\n",
       "        WEBGL: '/kernelspecs/polymake/WebGL',\n",
       "      },\n",
       "      shim: {\n",
       "        'three': { exports: 'THREE'},\n",
       "        'SVGRenderer': { deps: [ 'three' ], exports: 'THREE.SVGRenderer' },\n",
       "        'WEBGL': { deps: [ 'three' ], exports: 'THREE.WEBGL' },\n",
       "        'Projector': { deps: [ 'three' ], exports: 'THREE.Projector' },\n",
       "        'TrackballControls': { deps: [ 'three' ], exports: 'THREE.TrackballControls' },\n",
       "        'OrbitControls': { deps: [ 'three' ], exports: 'THREE.OrbitControls' },\n",
       "      }\n",
       "    });\n",
       "    \n",
       "    require(['three'],function(THREE){\n",
       "        window.THREE = THREE;\n",
       "      require(['TrackballControls', 'OrbitControls', 'Projector', 'SVGRenderer', 'WEBGL'],\n",
       "               function(TrackballControls, OrbitControls, Projector, SVGRenderer, WEBGL) {\n",
       "    THREE.TrackballControls = TrackballControls;\n",
       "    THREE.OrbitControls = OrbitControls;\n",
       "    THREE.Projector = Projector;\n",
       "    THREE.SVGRenderer = SVGRenderer;\n",
       "    THREE.WEBGL = WEBGL;\n",
       "\n",
       "// COMMON_CODE_BLOCK_BEGIN\n",
       "\n",
       "const intervalLength = 25; // for automatic animations\n",
       "const explodableModel = true; \n",
       "const modelContains = { points: false, pointlabels: false, lines: false, edgelabels: false, faces: false, arrowheads: false };\n",
       "const foldables = [];\n",
       "\n",
       "var three = document.getElementById(\"model39615074437\");\n",
       "var scene = new THREE.Scene();\n",
       "var renderer = new THREE.WebGLRenderer( { antialias: true } );\n",
       "var svgRenderer = new THREE.SVGRenderer( { antialias: true } );\n",
       "renderer.setPixelRatio( window.devicePixelRatio );\n",
       "renderer.setClearColor(0xFFFFFF, 1);\n",
       "svgRenderer.setClearColor(0xFFFFFF, 1);\n",
       "three.appendChild(renderer.domElement);\n",
       "\n",
       "var frustumSize = 4;\n",
       "var cameras = [new THREE.PerspectiveCamera(75, 1, 0.1, 1000), new THREE.OrthographicCamera()];\n",
       "cameras.forEach(function(cam) {\n",
       "    cam.position.set(0, 0, 5);\n",
       "    cam.lookAt(0, 0, 0);  \n",
       "    cam.up.set(0, 1, 0);         \n",
       "});\n",
       "var controls = [new THREE.TrackballControls(cameras[0], three), new THREE.OrbitControls(cameras[1], three)];\n",
       "var camera, control;\n",
       "\n",
       "controls[0].zoomSpeed = 0.2;\n",
       "controls[0].rotateSpeed = 4;\n",
       "\n",
       "\n",
       "// class to allow move points together with labels and spheres\n",
       "var PMPoint = function (x,y,z) {\n",
       "   this.vector = new THREE.Vector3(x,y,z);\n",
       "   this.sprite = null;\n",
       "   this.sphere = null;\n",
       "}\n",
       "PMPoint.prototype.addLabel = function(labelsprite) {\n",
       "   this.sprite = labelsprite;\n",
       "   this.sprite.position.copy(this.vector);\n",
       "}\n",
       "PMPoint.prototype.addSphere = function(spheremesh) {\n",
       "   this.sphere = spheremesh;\n",
       "   this.sphere.position.copy(this.vector);\n",
       "}\n",
       "PMPoint.prototype.set = function(x,y,z) {\n",
       "   this.vector.set(x,y,z);\n",
       "   if (this.sprite) {\n",
       "      this.sprite.position.copy(this.vector);\n",
       "   }\n",
       "   if (this.sphere) {\n",
       "      this.sphere.position.copy(this.vector);\n",
       "   }\n",
       "}\n",
       "PMPoint.prototype.radius = function() {\n",
       "   if (this.sphere) {\n",
       "      return this.sphere.geometry.parameters.radius;\n",
       "   } else {\n",
       "      return 0;\n",
       "   }\n",
       "};\n",
       "// select the target node\n",
       "var target = document.querySelector('#model39615074437');\n",
       "\n",
       "// create an observer instance\n",
       "var observer = new MutationObserver(function(mutations) {\n",
       "   mutations.forEach(function(mutation) {\n",
       "      if (mutation.removedNodes && mutation.removedNodes.length > 0) {\n",
       "         cancelAnimationFrame(renderId);\n",
       "         observer.disconnect();\n",
       "         console.log(\"cancelled frame \"+renderId);\n",
       "      }\n",
       "   });\n",
       "});\n",
       "\n",
       "// configuration of the observer:\n",
       "var config = { childList: true, characterData: true }\n",
       "\n",
       "// pass in the target node, as well as the observer options\n",
       "while (target) {\n",
       "   if (target.className==\"output\") {\n",
       "      observer.observe(target, config);\n",
       "      break;\n",
       "   }\n",
       "   target = target.parentNode;\n",
       "}\n",
       "\n",
       "// COMMON_CODE_BLOCK_END\n",
       "\n",
       "var obj0 = new THREE.Object3D();\n",
       "obj0.name = \"p\";\n",
       "obj0.userData.explodable = 1;\n",
       "obj0.userData.points = [];\n",
       "obj0.userData.points.push(new PMPoint(0, 1, 1));\n",
       "obj0.userData.points.push(new PMPoint(1, 0, 1));\n",
       "obj0.userData.points.push(new PMPoint(1, 0, 0));\n",
       "obj0.userData.points.push(new PMPoint(1, 1, 0));\n",
       "obj0.userData.points.push(new PMPoint(0, 1, 0));\n",
       "obj0.userData.points.push(new PMPoint(0, 0, 1));\n",
       "obj0.userData.points.push(new PMPoint(0, 0, 0));\n",
       "\n",
       "obj0.userData.pointradii = 0.02;\n",
       "   <!-- Vertex style -->\n",
       "obj0.userData.pointmaterial = new THREE.MeshBasicMaterial( { color: 0xFF0000, side: THREE.DoubleSide, transparent: false } );\n",
       "obj0.userData.pointlabels = [\"0\", \"1\", \"2\", \"3\", \"4\", \"5\", \"6\"];\n",
       "obj0.userData.edgeindices = [0, 1, 1, 2, 0, 3, 1, 3, 2, 3, 0, 4, 3, 4, 0, 5, 1, 5, 2, 6, 4, 6, 5, 6];\n",
       "obj0.userData.facets = [[1, 3, 0], [0, 4, 6, 5], [2, 3, 1], [1, 5, 6, 2], [0, 3, 4], [4, 3, 2, 6], [1, 0, 5]];\n",
       "   <!-- Facet style -->\n",
       "obj0.userData.facetmaterial = new THREE.MeshBasicMaterial( { color: 0x77EC9E, depthFunc: THREE.LessDepth, opacity: 1, polygonOffset: true, polygonOffsetFactor: 1, polygonOffsetUnits: 0.5, side: THREE.DoubleSide, transparent: true } );\n",
       "init_object(obj0);\n",
       "scene.add(obj0);\n",
       "\n",
       "var obj1 = new THREE.Object3D();\n",
       "obj1.name = \"GRAPH directed with LP \";\n",
       "obj1.userData.explodable = 1;\n",
       "obj1.userData.points = [];\n",
       "obj1.userData.points.push(new PMPoint(0, 1, 1));\n",
       "obj1.userData.points.push(new PMPoint(1, 0, 1));\n",
       "obj1.userData.points.push(new PMPoint(1, 0, 0));\n",
       "obj1.userData.points.push(new PMPoint(1, 1, 0));\n",
       "obj1.userData.points.push(new PMPoint(0, 1, 0));\n",
       "obj1.userData.points.push(new PMPoint(0, 0, 1));\n",
       "obj1.userData.points.push(new PMPoint(0, 0, 0));\n",
       "\n",
       "obj1.userData.edgeindices = [2, 1, 2, 3, 4, 0, 4, 3, 5, 0, 5, 1, 6, 2, 6, 4, 6, 5];\n",
       "   <!-- Edge style -->\n",
       "obj1.userData.edgematerial = new THREE.LineBasicMaterial( { color: 0x000000, depthTest: true, linewidth: 1.5, transparent: false } );\n",
       "obj1.userData.arrowstyle = 1;\n",
       "init_object(obj1);\n",
       "scene.add(obj1);\n",
       "\n",
       "// COMMON_CODE_BLOCK_BEGIN\n",
       "function textSpriteMaterial(message, parameters) {\n",
       "    if ( parameters === undefined ) parameters = {};\n",
       "    var fontface = \"Helvetica\";\n",
       "    var fontsize = parameters.hasOwnProperty(\"fontsize\") ? parameters[\"fontsize\"] : 15;\n",
       "    fontsize = fontsize*10;\n",
       "    var lines = message.split('\\\\n');\n",
       "    var size = 512;\n",
       "    for(var i = 0; i<lines.length; i++){\n",
       "        var tmp = lines[i].length;\n",
       "        while(tmp*fontsize > size){\n",
       "           fontsize--;\n",
       "        }\n",
       "    }\n",
       "    \n",
       "    var canvas = document.createElement('canvas');\n",
       "    canvas.width = size;\n",
       "    canvas.height = size;\n",
       "    var context = canvas.getContext('2d');\n",
       "    context.fillStyle = \"rgba(255, 255, 255, 0)\";\n",
       "    context.fill();\n",
       "    context.font = fontsize + \"px \" + fontface;\n",
       "    \n",
       "    // text color\n",
       "    context.fillStyle = \"rgba(0, 0, 0, 1.0)\";\n",
       "     for(var i = 0; i<lines.length; i++){\n",
       "        context.fillText(lines[i], size/2, size/2+i*fontsize);\n",
       "     }\n",
       "    \n",
       "    // canvas contents will be used for a texture\n",
       "    var texture = new THREE.Texture(canvas);\n",
       "    texture.needsUpdate = true;\n",
       "    \n",
       "    var spriteMaterial = new THREE.SpriteMaterial({map: texture, depthTest: true, depthWrite: false, polygonOffset: true, polygonOffsetFactor: -1, polygonOffsetUnits: 1 });\n",
       "    return spriteMaterial;\n",
       "}\n",
       "\n",
       "\n",
       "// ---------------------- INITIALIZING OBJECTS--------------------------------------\n",
       "// ---------------------------------------------------------------------------------\n",
       "\n",
       "function init_object(obj) {\n",
       "    if (obj.userData.hasOwnProperty(\"pointmaterial\")) {\n",
       "        init_points(obj);\n",
       "        modelContains.points = true;\n",
       "    }\n",
       "    if (obj.userData.hasOwnProperty(\"pointlabels\")) {\n",
       "        init_pointlabels(obj);\n",
       "        modelContains.pointlabels = true;\n",
       "    }\n",
       "    if (obj.userData.hasOwnProperty(\"edgematerial\")) {\n",
       "        init_lines(obj);\n",
       "        modelContains.lines = true;\n",
       "    }\n",
       "    if (obj.userData.hasOwnProperty(\"edgelabels\")) {\n",
       "        init_edgelabels(obj);\n",
       "        modelContains.edgelabels = true;\n",
       "    }\n",
       "    if (obj.userData.hasOwnProperty(\"arrowstyle\")) {\n",
       "        init_arrowheads(obj);\n",
       "        modelContains.arrowheads = true;\n",
       "    }\n",
       "    if (obj.userData.hasOwnProperty(\"facetmaterial\")) {\n",
       "        init_faces(obj);\n",
       "        modelContains.faces = true;\n",
       "    }\n",
       "}\n",
       "\n",
       "function init_points(obj) {\n",
       "    var pointgroup = new THREE.Group();\n",
       "    pointgroup.name = \"points\";\n",
       "    var points = obj.userData.points;\n",
       "    var radii = obj.userData.pointradii;\n",
       "    var materials = obj.userData.pointmaterial;\n",
       "    var geometry,material;\n",
       "    if (!Array.isArray(radii)) {\n",
       "        geometry = new THREE.SphereBufferGeometry(radii);  \n",
       "    }\n",
       "    if (!Array.isArray(materials)) {\n",
       "        material = materials;\n",
       "    }\n",
       "    for (var i=0; i<points.length; i++) {\n",
       "        var point = points[i];\n",
       "        if (Array.isArray(radii)) {\n",
       "            if (radii[i] == 0) {\n",
       "                continue;\n",
       "            }\n",
       "            geometry = new THREE.SphereBufferGeometry(radii[i]);  \n",
       "        } \n",
       "        if (Array.isArray(materials)) {\n",
       "            material = materials[i];     \n",
       "        } \n",
       "        var sphere = new THREE.Mesh(geometry, material);\n",
       "        point.addSphere(sphere);\n",
       "        pointgroup.add(sphere);\n",
       "    }\n",
       "    obj.add(pointgroup);\n",
       "}\n",
       "\n",
       "function init_pointlabels(obj) {\n",
       "    var points = obj.userData.points;\n",
       "    var labels = obj.userData.pointlabels;\n",
       "    var pointlabels = new THREE.Group();\n",
       "    pointlabels.name = \"pointlabels\";\n",
       "    if (Array.isArray(labels)) {\n",
       "        for (var i=0; i<points.length; i++) {\n",
       "            var point = points[i];\n",
       "            var spriteMaterial = textSpriteMaterial( labels[i] );\n",
       "\t        var sprite = new THREE.Sprite(spriteMaterial);\n",
       "            point.addLabel(sprite);\n",
       "            pointlabels.add(sprite);\n",
       "        }\n",
       "    } else {\n",
       "        var spriteMaterial = textSpriteMaterial( labels );\n",
       "        for (var i=0; i<points.length; i++) {\n",
       "            var point = points[i];\n",
       "\t        var sprite = new THREE.Sprite(spriteMaterial);\n",
       "            point.addLabel(sprite);\n",
       "            pointlabels.add(sprite);\n",
       "        }\n",
       "    }\n",
       "    obj.add(pointlabels);\n",
       "}\n",
       "\n",
       "function init_lines(obj) {\n",
       "    var edgeindices = obj.userData.edgeindices;\n",
       "    var points = obj.userData.points;\n",
       "    var materials = obj.userData.edgematerial;\n",
       "    var geometry = new THREE.BufferGeometry();\n",
       "    var bufarr = new Float32Array( obj.userData.edgeindices.length * 3 );\n",
       "    var bufattr = new THREE.Float32BufferAttribute( bufarr, 3 );\n",
       "    var geometry = new THREE.BufferGeometry();\n",
       "    geometry.setAttribute('position', bufattr);\n",
       "    if (Array.isArray(materials)) {     \n",
       "        for (var i=0; i<materials.length; i++) {\n",
       "            geometry.addGroup(2*i,2,i);\n",
       "        }\n",
       "    }\n",
       "    var lines = new THREE.LineSegments(geometry, materials);\n",
       "    lines.name = \"lines\";\n",
       "    obj.add(lines);\n",
       "    updateEdgesPosition(obj);\n",
       "}\n",
       "\n",
       "function init_edgelabels(obj) {\n",
       "    var points = obj.userData.points;\n",
       "    var edgeindices = obj.userData.edgeindices;\n",
       "    var labels = obj.userData.edgelabels;\n",
       "    var edgelabels = new THREE.Group();\n",
       "    edgelabels.name = \"edgelabels\";\n",
       "    if (Array.isArray(labels)) {\n",
       "        for (var i=0; i<edgeindices.length/2; i++) {\n",
       "            var spriteMaterial = textSpriteMaterial( labels[i] );\n",
       "            var sprite = new THREE.Sprite(spriteMaterial);\n",
       "            sprite.position.copy(new THREE.Vector3().addVectors(points[edgeindices[2*i]].vector,points[edgeindices[2*i+1]].vector).multiplyScalar(0.5));\n",
       "            edgelabels.add(sprite);\n",
       "        }\n",
       "    } else {\n",
       "        var spriteMaterial = textSpriteMaterial( labels );\n",
       "        for (var i=0; i<edgeindices.length/2; i++) {\n",
       "            var sprite = new THREE.Sprite(spriteMaterial);\n",
       "            sprite.position.copy(new THREE.Vector3().addVectors(points[edgeindices[2*i]].vector,points[edgeindices[2*i+1]].vector).multiplyScalar(0.5));\n",
       "            edgelabels.add(sprite);\n",
       "        }\n",
       "    }\n",
       "    obj.add(edgelabels);\n",
       "}\n",
       "\n",
       "function init_arrowheads(obj) {\n",
       "    var arrowheads = new THREE.Group();\n",
       "    arrowheads.name = \"arrowheads\";\n",
       "    var arrowstyle = obj.userData.arrowstyle;\n",
       "    var edgeindices = obj.userData.edgeindices;\n",
       "    var edgematerials = obj.userData.edgematerial;\n",
       "    var points = obj.userData.points;\n",
       "    var material;\n",
       "    if (!Array.isArray(edgematerials)) {\n",
       "        material = new THREE.MeshBasicMaterial( {color: edgematerials.color} );\n",
       "    }\n",
       "\n",
       "    for (var i=0; i<edgeindices.length; i=i+2) {\n",
       "        var start = points[edgeindices[i]];\n",
       "        var end = points[edgeindices[i+1]];\n",
       "        var dist = start.vector.distanceTo( end.vector ) - start.radius() - end.radius();\n",
       "        if (dist <= 0) {\n",
       "            continue;\n",
       "        }\n",
       "        var dir = new THREE.Vector3().subVectors(end.vector,start.vector);\n",
       "        dir.normalize();\n",
       "        var axis = new THREE.Vector3().set(dir.z,0,-dir.x);\n",
       "        axis.normalize();\n",
       "        var radians = Math.acos( dir.y );\n",
       "        var radius = dist/25;\n",
       "        var height = dist/5;\n",
       "        var geometry = new THREE.ConeBufferGeometry(radius,height);\n",
       "        var position = new THREE.Vector3().addVectors(start.vector,dir.clone().multiplyScalar(start.radius()+dist-height/2));\n",
       "        if (Array.isArray(edgematerials)) {\n",
       "            material = new THREE.MeshBasicMaterial( {color: edgematerials[i].color} );\n",
       "        }\n",
       "        var cone = new THREE.Mesh( geometry, material );\n",
       "        cone.quaternion.setFromAxisAngle(axis,radians);;\n",
       "        cone.position.copy(position);;\n",
       "        arrowheads.add(cone);\n",
       "    }\n",
       "    obj.add(arrowheads);\n",
       "}\n",
       "\n",
       "function init_faces(obj) {\n",
       "    var points = obj.userData.points;\n",
       "    var facets = obj.userData.facets;\n",
       "    obj.userData.triangleindices = [];\n",
       "    for (var i=0; i<facets.length; i++) {\n",
       "        facet = facets[i];\n",
       "        for (var t=0; t<facet.length-2; t++) {\n",
       "            obj.userData.triangleindices.push(facet[0],facet[t+1],facet[t+2]);  \n",
       "        }\n",
       "    }\n",
       "    var bufarr = new Float32Array( obj.userData.triangleindices.length * 3 );\n",
       "    var bufattr = new THREE.Float32BufferAttribute(bufarr,3);\n",
       "    \n",
       "    var materials = obj.userData.facetmaterial;\n",
       "    var geometry = new THREE.BufferGeometry();\n",
       "    geometry.setAttribute('position',bufattr);\n",
       "    if (Array.isArray(materials)) {\n",
       "        var tricount = 0;\n",
       "        var facet;\n",
       "        for (var i=0; i<facets.length; i++) {\n",
       "            facet = facets[i];\n",
       "            geometry.addGroup(tricount,(facet.length-2)*3,i);\n",
       "            tricount += (facet.length-2)*3;\n",
       "        }\n",
       "    }\n",
       "    var mesh = new THREE.Mesh(geometry, materials);\n",
       "    mesh.name = \"faces\";\n",
       "    obj.add(mesh); \n",
       "    updateFacesPosition(obj);\n",
       "}\n",
       "// //INITIALIZING\n",
       "\n",
       "\n",
       "function updateFacesPosition(obj) {\n",
       "    var points = obj.userData.points;\n",
       "    var indices = obj.userData.triangleindices;\n",
       "    var faces = obj.getObjectByName(\"faces\");\n",
       "    var ba = faces.geometry.getAttribute(\"position\");\n",
       "    for (var i=0; i<indices.length; i++) {\n",
       "        ba.setXYZ(i, points[indices[i]].vector.x, points[indices[i]].vector.y ,points[indices[i]].vector.z); \n",
       "    }\n",
       "    faces.geometry.attributes.position.needsUpdate = true;\n",
       "    \n",
       "}\n",
       "\n",
       "function updateEdgesPosition(obj) {\n",
       "    var points = obj.userData.points;\n",
       "    var indices = obj.userData.edgeindices;\n",
       "    var lines = obj.getObjectByName(\"lines\");\n",
       "    var ba = lines.geometry.getAttribute(\"position\"); \n",
       "    for (var i=0; i<indices.length; i++) {\n",
       "        ba.setXYZ(i, points[indices[i]].vector.x, points[indices[i]].vector.y ,points[indices[i]].vector.z); \n",
       "    }\n",
       "    lines.geometry.attributes.position.needsUpdate = true;\n",
       "}\n",
       "\n",
       "function onWindowResize() {\n",
       "    renderer.setSize( three.clientWidth, three.clientHeight );\n",
       "    svgRenderer.setSize( three.clientWidth, three.clientHeight );\n",
       "    updateCamera();\n",
       "}\n",
       "\n",
       "function updateCamera() {\n",
       "    var width = three.clientWidth;\n",
       "    var height = three.clientHeight;\n",
       "    var aspect = width / height;\n",
       "    if (camera.type == \"OrthographicCamera\") {\n",
       "        camera.left = frustumSize * aspect / - 2;\n",
       "        camera.right = frustumSize * aspect / 2;\n",
       "        camera.top = frustumSize / 2;\n",
       "        camera.bottom = - frustumSize / 2;\n",
       "    } else if (camera.type == \"PerspectiveCamera\") {\n",
       "        camera.aspect = aspect;\n",
       "    }\n",
       "    camera.updateProjectionMatrix();\n",
       "}\n",
       "\n",
       "function changeCamera(event) {\n",
       "    var selindex = event.currentTarget.selectedIndex;\n",
       "    camera = cameras[selindex];\n",
       "    control = controls[selindex];\n",
       "    control.enabled = true; \n",
       "    for (var i=0; i<controls.length; i++) {\n",
       "        if (i!=selindex) {\n",
       "            controls[i].enabled = false;\n",
       "        }\n",
       "    }\n",
       "    updateCamera();\n",
       "}\n",
       "\n",
       "var camtypenode = document.getElementById('cameraType_2');\n",
       "camtypenode.onchange = changeCamera;\n",
       "camtypenode.dispatchEvent(new Event('change'));\n",
       "\n",
       "onWindowResize();\n",
       "window.addEventListener('resize', onWindowResize);\t\n",
       "\n",
       "\n",
       "var xRotationEnabled = false;\n",
       "var yRotationEnabled = false;\n",
       "var zRotationEnabled = false;\n",
       "var rotationSpeedFactor = 1;\n",
       "var settingsShown = false;\n",
       "var labelsShown = true;\n",
       "var intervals = [];\n",
       "var timeouts = [];\n",
       "var explodingSpeed = 0.05;\n",
       "var explodeScale = 0;\n",
       "var XMLS = new XMLSerializer();\n",
       "var svgElement;\n",
       "var renderId;\n",
       "\n",
       "var render = function () {\n",
       "\n",
       "\trenderId = requestAnimationFrame(render);\n",
       "\n",
       "//\tcomment in for automatic explosion\n",
       "//\texplode(updateFactor());\n",
       "\n",
       "    var phi = 0.02 * rotationSpeedFactor;\n",
       "\n",
       "    if (xRotationEnabled) {\n",
       "        scene.rotation.x += phi;\n",
       "    }\n",
       "    if (yRotationEnabled) {\n",
       "        scene.rotation.y += phi;\n",
       "    }\n",
       "    if (zRotationEnabled) {\n",
       "        scene.rotation.z += phi;\n",
       "    }\n",
       "\n",
       "    control.update();\n",
       "    renderer.render(scene, camera);\n",
       "};\n",
       "\n",
       "if ( THREE.WEBGL.isWebGLAvailable() ) {\n",
       "\trender();\n",
       "} else {\n",
       "\tvar warning = WEBGL.getWebGLErrorMessage();\n",
       "\tthree.appendChild( warning );\n",
       "}\n",
       "    \n",
       "function changeTransparency() {\n",
       "    var opacity = 1-Number(event.currentTarget.value);\n",
       "    for (var i=0; i<scene.children.length; i++) {\n",
       "        child = scene.children[i];\n",
       "        if ( child.userData.hasOwnProperty(\"facetmaterial\") ) {\n",
       "            if (Array.isArray(child.userData.facetmaterial)) {\n",
       "                for (var j=0; j<child.userData.facetmaterial.length; j++) {\n",
       "                    child.userData.facetmaterial[j].opacity = opacity;\n",
       "                }\n",
       "            } else {\n",
       "                child.userData.facetmaterial.opacity = opacity;\n",
       "            }    \n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "function changeRotationX(event){\n",
       "    xRotationEnabled = event.currentTarget.checked;\n",
       "}\t\n",
       "\n",
       "function changeRotationY(event){\n",
       "    yRotationEnabled = event.currentTarget.checked;\n",
       "}\t\n",
       "\n",
       "function changeRotationZ(event){\n",
       "    zRotationEnabled = event.currentTarget.checked;\n",
       "}\t\n",
       "\n",
       "\n",
       "function changeRotationSpeedFactor(event){\n",
       "    rotationSpeedFactor = Number(event.currentTarget.value);\n",
       "}\n",
       "\n",
       "function resetScene(){\n",
       "    scene.rotation.set(0,0,0);\n",
       "    camera.position.set(0,0,5);\n",
       "    camera.up.set(0,1,0);\n",
       "}\n",
       "\n",
       "function showSettings(event){\n",
       "    document.getElementById('settings_2').style.visibility = 'visible';\n",
       "    document.getElementById('showSettingsButton_2').style.visibility = 'hidden';\n",
       "    document.getElementById('hideSettingsButton_2').style.visibility = 'visible';\n",
       "    settingsShown = true;\n",
       "}\n",
       "\n",
       "function hideSettings(event){\n",
       "    document.getElementById('settings_2').style.visibility = 'hidden';\n",
       "    document.getElementById('showSettingsButton_2').style.visibility = 'visible';\n",
       "    document.getElementById('hideSettingsButton_2').style.visibility = 'hidden';\n",
       "    settingsShown = false;\n",
       "}\n",
       "\n",
       "\n",
       "\n",
       "var pos = 150* Math.PI;\n",
       "\n",
       "function updateFactor() {\n",
       "    pos++;\n",
       "    return Math.sin(.01*pos)+1;\n",
       "}\n",
       "\n",
       "// ------------------------ FOLDING ------------------------------------------------\n",
       "// ---------------------------------------------------------------------------------\n",
       "// rotate point p around axis defined by points p1 and p2 by given angle\n",
       "function rotate(p, p1, p2, angle ){   \n",
       "    angle = -angle;\n",
       "    var x = p.x, y = p.y, z = p.z, \n",
       "    a = p1.x, b = p1.y, c = p1.z, \n",
       "    u = p2.x-p1.x, v = p2.y-p1.y, w = p2.z-p1.z;\n",
       "    var result = [];\n",
       "    var L = u*u + v*v + w*w;\n",
       "    var sqrt = Math.sqrt;\n",
       "    var cos = Math.cos;\n",
       "    var sin = Math.sin;\n",
       "\n",
       "    result[0] = ((a*(v*v+w*w)-u*(b*v+c*w-u*x-v*y-w*z))*(1-cos(angle))+L*x*cos(angle)+sqrt(L)*(-c*v+b*w-w*y+v*z)*sin(angle))/L;\n",
       "    result[1] = ((b*(u*u+w*w)-v*(a*u+c*w-u*x-v*y-w*z))*(1-cos(angle))+L*y*cos(angle)+sqrt(L)*(c*u-a*w+w*x-u*z)*sin(angle))/L;\n",
       "    result[2] = ((c*(u*u+v*v)-w*(a*u+b*v-u*x-v*y-w*z))*(1-cos(angle))+L*z*cos(angle)+sqrt(L)*(-b*u+a*v-v*x+u*y)*sin(angle))/L;\n",
       "\n",
       "    return result;\n",
       "}\n",
       "\n",
       "var fold = function(event){\n",
       "    var obj = foldables[Number(event.currentTarget.name)];\n",
       "    var foldvalue = Number(event.currentTarget.value);\n",
       "    var scale = foldvalue - obj.userData.oldscale;\n",
       "\n",
       "    for (var j=0; j<obj.userData.axes.length; j++) {\n",
       "        rotateVertices(obj, j, scale);\n",
       "    }\n",
       "    update(obj);\n",
       "    obj.userData.oldscale += scale;\n",
       "    lookAtBarycenter(obj);\n",
       "}\n",
       "\n",
       "function lookAtBarycenter(obj){\n",
       "    control.target = barycenter(obj);\n",
       "}\n",
       "\n",
       "function barycenter(obj) {\n",
       "    var center = new THREE.Vector3(0,0,0);\n",
       "    var points = obj.userData.points;\n",
       "    for (var i=0; i<points.length; i++){\n",
       "        center.add(points[i].vector);\n",
       "    }\n",
       "    center.divideScalar(points.length);\n",
       "    return center;\n",
       "}\n",
       "\n",
       "function rotateVertices(obj, edge, scale) {\n",
       "    var axes = obj.userData.axes;\n",
       "    var subtrees = obj.userData.subtrees;\n",
       "    var points = obj.userData.points;\n",
       "    var angles = obj.userData.angles;\n",
       "    if (edge < axes.length){\n",
       "        for (var j=0; j<subtrees[edge].length; j++){\n",
       "            var rotP = rotate(points[subtrees[edge][j]].vector, points[axes[edge][0]].vector,points[axes[edge][1]].vector, scale * (Math.PI - angles[edge]));\n",
       "            points[subtrees[edge][j]].set(rotP[0],rotP[1],rotP[2]);\n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "function update(obj) {\n",
       "   updateFacesPosition(obj);\n",
       "   updateEdgesPosition(obj);\n",
       "}\n",
       "\n",
       "if (foldables.length) {\n",
       "    var settings = document.getElementById('settings_2');\n",
       "    var foldDiv = document.createElement('div');\n",
       "    foldDiv.id = 'fold_2';\n",
       "    var title = document.createElement('strong');\n",
       "    title.innerHTML = 'Fold';\n",
       "    foldDiv.appendChild(title);\n",
       "    foldDiv.className = 'group';\n",
       "    for (var i=0; i<foldables.length; i++) {\n",
       "        var range = document.createElement('input');\n",
       "        range.type = 'range';\n",
       "        range.min = 0;\n",
       "        range.max = 1;\n",
       "        range.value = 0;\n",
       "        range.step = 0.001;\n",
       "        range.name = String(i);\n",
       "        range.oninput = fold;\n",
       "        foldDiv.appendChild(range);\n",
       "    }\n",
       "    lookAtBarycenter(foldables[0]);\n",
       "    settings.insertBefore(foldDiv,settings.childNodes[0]);\n",
       "}\n",
       "\n",
       "    \n",
       "// ---------------------- EXPLOSION ------------------------------------------------\n",
       "// ---------------------------------------------------------------------------------\n",
       "\n",
       "if (explodableModel) {\n",
       "    for (var i=0; i<scene.children.length; i++) {\n",
       "        obj = scene.children[i];\n",
       "        if ( obj.userData.explodable ) {\n",
       "            computeCentroid(obj);\n",
       "        }\n",
       "    }\n",
       "    document.getElementById('explodeRange_2').oninput = triggerExplode;\n",
       "    document.getElementById('explodeCheckbox_2').onchange = triggerAutomaticExplode;\n",
       "    document.getElementById('explodingSpeedRange_2').oninput = setExplodingSpeed;\n",
       "}\n",
       "\n",
       "function computeCentroid(obj) {\n",
       "    centroid = new THREE.Vector3();\n",
       "    obj.userData.points.forEach(function(pmpoint) {\n",
       "        centroid.add(pmpoint.vector);\t\t\t\n",
       "    });\n",
       "    centroid.divideScalar(obj.userData.points.length);\n",
       "    obj.userData.centroid = centroid;\n",
       "}\n",
       "\n",
       "function explode(factor) {\n",
       "    for (var i=0; i<scene.children.length; i++) {\n",
       "        var obj = scene.children[i];\n",
       "        if (obj.userData.hasOwnProperty(\"centroid\")) { \n",
       "            var c = obj.userData.centroid;\n",
       "            obj.position.set(c.x*factor, c.y*factor, c.z*factor);\n",
       "        }\n",
       "    }\t\n",
       "}\n",
       "\n",
       "function triggerExplode(event){\n",
       "    explodeScale = Number(event.currentTarget.value);\n",
       "    explode(explodeScale);\n",
       "}\n",
       "\n",
       "function setExplodingSpeed(event){\n",
       "    explodingSpeed = Number(event.currentTarget.value);\n",
       "}\n",
       "\n",
       "function triggerAutomaticExplode(event){\n",
       "    if (event.currentTarget.checked){\n",
       "        startExploding();\n",
       "    } else {\n",
       "        clearIntervals();\n",
       "    }\t\n",
       "}\n",
       "\n",
       "function startExploding(){\n",
       "    intervals.push(setInterval(explodingInterval, 25));\n",
       "}\n",
       "\n",
       "\n",
       "function explodingInterval(){\n",
       "    explodeScale += explodingSpeed;\n",
       "    if (explodeScale <= 6){ \n",
       "        explode(explodeScale);\n",
       "    }\n",
       "    else{\n",
       "        explode(6);\n",
       "        explodeScale = 6;\n",
       "        clearIntervals();\n",
       "        timeouts.push(setTimeout(startUnexploding, 3000));\n",
       "    }\n",
       "    document.getElementById('explodeRange_2').value = explodeScale;\n",
       "}\n",
       "\n",
       "\n",
       "function startUnexploding(){\n",
       "    intervals.push(setInterval(unexplodingInterval, 25));\n",
       "}\n",
       "\n",
       "function unexplodingInterval(){\n",
       "    explodeScale -= explodingSpeed;\n",
       "    if (explodeScale >= 0){\t\n",
       "        explode(explodeScale);\n",
       "    }\n",
       "    else {\n",
       "        explode(0);\n",
       "        explodeScale = 0;\n",
       "        clearIntervals();\n",
       "        timeouts.push(setTimeout(startExploding, 3000));\n",
       "    }\n",
       "    document.getElementById('explodeRange_2').value = explodeScale;\n",
       "}\n",
       "\n",
       "function clearIntervals(){\n",
       "    intervals.forEach(function(interval){\n",
       "        clearInterval(interval);\n",
       "    });\n",
       "    intervals = [];\n",
       "    timeouts.forEach(function(timeout){\n",
       "        clearTimeout(timeout);\n",
       "    });\n",
       "    timeouts = [];\n",
       "}\n",
       "\n",
       "// ---------------------- DISPLAY --------------------------------------------------\n",
       "// ---------------------------------------------------------------------------------\n",
       "\n",
       "const objectTypeInnerHTMLs = { points: \"Points\", pointlabels: \"Point labels\", lines: \"Edges\", edgelabels: \"Edge labels\", faces: \"Faces\", arrowheads: \"Arrow heads\" };\n",
       "const objectTypeVisible = {};\n",
       "Object.assign(objectTypeVisible,modelContains);\n",
       "const sortedObjectTypeKeys = Object.keys(objectTypeInnerHTMLs).sort();\n",
       "const shownObjectTypesList = document.getElementById('shownObjectTypesList_2');\n",
       "\n",
       "function setVisibility(bool,objname) {\n",
       "    for (var i=0; i<scene.children.length; i++){\n",
       "        var obj = scene.children[i].getObjectByName(objname);\n",
       "        if (obj) {\n",
       "            obj.visible = bool;\n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "function toggleObjectTypeVisibility(event){\n",
       "    var name = event.currentTarget.name;\n",
       "    var checked = event.currentTarget.checked;\n",
       "    objectTypeVisible[name] = checked;\n",
       "    setVisibility(checked,name);\n",
       "}\n",
       "\n",
       "for (var i=0; i<sortedObjectTypeKeys.length; i++){\n",
       "    var key = sortedObjectTypeKeys[i];\n",
       "    if (modelContains[key]) {\n",
       "        var objTypeNode = document.createElement('span');\n",
       "        objTypeNode.innerHTML = objectTypeInnerHTMLs[key] + '<br>';\n",
       "        var checkbox = document.createElement('input');\n",
       "        checkbox.type = 'checkbox';\n",
       "        checkbox.checked = true;\n",
       "        checkbox.name = key;\n",
       "        checkbox.onchange = toggleObjectTypeVisibility;\n",
       "        shownObjectTypesList.appendChild(checkbox);\n",
       "        shownObjectTypesList.appendChild(objTypeNode);\n",
       "    }\n",
       "}\n",
       "\n",
       "// ------------------------------------------------------\n",
       "\n",
       "function toggleObjectVisibility(event){\n",
       "    var nr = Number(event.currentTarget.name);\n",
       "    scene.children[nr].visible = event.currentTarget.checked;\n",
       "}\n",
       "\n",
       "// append checkboxes for displaying or hiding objects\n",
       "var shownObjectsList = document.getElementById('shownObjectsList_2');\n",
       "for (var i=0; i<scene.children.length; i++){\n",
       "    obj = scene.children[i];\n",
       "    var objNode = document.createElement('span');\n",
       "    objNode.innerHTML = obj.name + '<br>';\n",
       "    var checkbox = document.createElement('input');\n",
       "    checkbox.type = 'checkbox';\n",
       "    checkbox.checked = true;\n",
       "    checkbox.name = String(i);\n",
       "    checkbox.onchange = toggleObjectVisibility;\n",
       "    shownObjectsList.appendChild(checkbox);\n",
       "    shownObjectsList.appendChild(objNode);\n",
       "}\n",
       "\n",
       "// ---------------------- SVG ------------------------------------------------------\n",
       "// ---------------------------------------------------------------------------------\n",
       "\n",
       "function takeSvgScreenshot() {\n",
       "    if (objectTypeVisible[\"pointlabels\"]) {\n",
       "        setVisibility(false,\"pointlabels\");\n",
       "    }\n",
       "    if (objectTypeVisible[\"edgelabels\"]) {\n",
       "        setVisibility(false,\"edgelabels\");\n",
       "    }\n",
       "    svgRenderer.render(scene,camera);\n",
       "    svgElement = XMLS.serializeToString(svgRenderer.domElement);\n",
       "    \n",
       "    if (objectTypeVisible[\"pointlabels\"]) {\n",
       "        setVisibility(true,\"pointlabels\");\n",
       "    }\n",
       "    if (objectTypeVisible[\"edgelabels\"]) {\n",
       "        setVisibility(true,\"edgelabels\");\n",
       "    }\n",
       "\n",
       "    if (document.getElementById('tab_2').checked){\n",
       "        //show in new tab\n",
       "        var myWindow = window.open(\"\",\"\");\n",
       "        myWindow.document.body.innerHTML = svgElement;\n",
       "    } else{\n",
       "        // download svg file \n",
       "        download(\"screenshot.svg\", svgElement);\n",
       "    }\n",
       "}\n",
       "\n",
       "function download(filename, text) {\n",
       "    var element = document.createElement('a');\n",
       "    element.setAttribute('href', 'data:text/plain;charset=utf-8,' + encodeURIComponent(text));\n",
       "    element.setAttribute('download', filename);\n",
       "\n",
       "    element.style.display = 'none';\n",
       "    document.body.appendChild(element);\n",
       "\n",
       "    element.click();\n",
       "\n",
       "    document.body.removeChild(element);\n",
       "}\n",
       "\n",
       "\n",
       "document.getElementById('transparencyRange_2').oninput = changeTransparency;\n",
       "document.getElementById('changeRotationX_2').onchange = changeRotationX;\n",
       "document.getElementById('changeRotationY_2').onchange = changeRotationY;\n",
       "document.getElementById('changeRotationZ_2').onchange = changeRotationZ;\n",
       "document.getElementById('resetButton_2').onclick = resetScene;\n",
       "document.getElementById('rotationSpeedRange_2').oninput = changeRotationSpeedFactor;\n",
       "document.getElementById('takeScreenshot_2').onclick = takeSvgScreenshot;\n",
       "document.getElementById('showSettingsButton_2').onclick = showSettings;\n",
       "document.getElementById('hideSettingsButton_2').onclick = hideSettings;\n",
       "\n",
       "\n",
       "// ------------------ SHORTCUTS --------------------------------------------\n",
       "// -------------------------------------------------------------------------\n",
       "\n",
       "/**\n",
       " * http://www.openjs.com/scripts/events/keyboard_shortcuts/\n",
       " * Version : 2.01.B\n",
       " * By Binny V A\n",
       " * License : BSD\n",
       " */\n",
       "shortcut = {\n",
       "\t'all_shortcuts':{},//All the shortcuts are stored in this array\n",
       "\t'add': function(shortcut_combination,callback,opt) {\n",
       "\t\t//Provide a set of default options\n",
       "\t\tvar default_options = {\n",
       "\t\t\t'type':'keydown',\n",
       "\t\t\t'propagate':false,\n",
       "\t\t\t'disable_in_input':false,\n",
       "\t\t\t'target':document,\n",
       "\t\t\t'keycode':false\n",
       "\t\t}\n",
       "\t\tif(!opt) opt = default_options;\n",
       "\t\telse {\n",
       "\t\t\tfor(var dfo in default_options) {\n",
       "\t\t\t\tif(typeof opt[dfo] == 'undefined') opt[dfo] = default_options[dfo];\n",
       "\t\t\t}\n",
       "\t\t}\n",
       "\n",
       "\t\tvar ele = opt.target;\n",
       "\t\tif(typeof opt.target == 'string') ele = document.getElementById(opt.target);\n",
       "\t\tvar ths = this;\n",
       "\t\tshortcut_combination = shortcut_combination.toLowerCase();\n",
       "\n",
       "\t\t//The function to be called at keypress\n",
       "\t\tvar func = function(e) {\n",
       "\t\t\te = e || window.event;\n",
       "\t\t\t\n",
       "\t\t\tif(opt['disable_in_input']) { //Don't enable shortcut keys in Input, Textarea fields\n",
       "\t\t\t\tvar element;\n",
       "\t\t\t\tif(e.target) element=e.target;\n",
       "\t\t\t\telse if(e.srcElement) element=e.srcElement;\n",
       "\t\t\t\tif(element.nodeType==3) element=element.parentNode;\n",
       "\n",
       "\t\t\t\tif(element.tagName == 'INPUT' || element.tagName == 'TEXTAREA') return;\n",
       "\t\t\t}\n",
       "\t\n",
       "\t\t\t//Find Which key is pressed\n",
       "\t\t\tif (e.keyCode) code = e.keyCode;\n",
       "\t\t\telse if (e.which) code = e.which;\n",
       "\t\t\tvar character = String.fromCharCode(code).toLowerCase();\n",
       "\t\t\t\n",
       "\t\t\tif(code == 188) character=\",\"; //If the user presses , when the type is onkeydown\n",
       "\t\t\tif(code == 190) character=\".\"; //If the user presses , when the type is onkeydown\n",
       "\n",
       "\t\t\tvar keys = shortcut_combination.split(\"+\");\n",
       "\t\t\t//Key Pressed - counts the number of valid keypresses - if it is same as the number of keys, the shortcut function is invoked\n",
       "\t\t\tvar kp = 0;\n",
       "\t\t\t\n",
       "\t\t\t//Work around for stupid Shift key bug created by using lowercase - as a result the shift+num combination was broken\n",
       "\t\t\tvar shift_nums = {\n",
       "\t\t\t\t\"`\":\"~\",\n",
       "\t\t\t\t\"1\":\"!\",\n",
       "\t\t\t\t\"2\":\"@\",\n",
       "\t\t\t\t\"3\":\"#\",\n",
       "\t\t\t\t\"4\":\"$\",\n",
       "\t\t\t\t\"5\":\"%\",\n",
       "\t\t\t\t\"6\":\"^\",\n",
       "\t\t\t\t\"7\":\"&\",\n",
       "\t\t\t\t\"8\":\"*\",\n",
       "\t\t\t\t\"9\":\"(\",\n",
       "\t\t\t\t\"0\":\")\",\n",
       "\t\t\t\t\"-\":\"_\",\n",
       "\t\t\t\t\"=\":\"+\",\n",
       "\t\t\t\t\";\":\":\",\n",
       "\t\t\t\t\"'\":\"\\\"\",\n",
       "\t\t\t\t\",\":\"<\",\n",
       "\t\t\t\t\".\":\">\",\n",
       "\t\t\t\t\"/\":\"?\",\n",
       "\t\t\t\t\"\\\\\":\"|\"\n",
       "\t\t\t}\n",
       "\t\t\t//Special Keys - and their codes\n",
       "\t\t\tvar special_keys = {\n",
       "\t\t\t\t'esc':27,\n",
       "\t\t\t\t'escape':27,\n",
       "\t\t\t\t'tab':9,\n",
       "\t\t\t\t'space':32,\n",
       "\t\t\t\t'return':13,\n",
       "\t\t\t\t'enter':13,\n",
       "\t\t\t\t'backspace':8,\n",
       "\t\n",
       "\t\t\t\t'scrolllock':145,\n",
       "\t\t\t\t'scroll_lock':145,\n",
       "\t\t\t\t'scroll':145,\n",
       "\t\t\t\t'capslock':20,\n",
       "\t\t\t\t'caps_lock':20,\n",
       "\t\t\t\t'caps':20,\n",
       "\t\t\t\t'numlock':144,\n",
       "\t\t\t\t'num_lock':144,\n",
       "\t\t\t\t'num':144,\n",
       "\t\t\t\t\n",
       "\t\t\t\t'pause':19,\n",
       "\t\t\t\t'break':19,\n",
       "\t\t\t\t\n",
       "\t\t\t\t'insert':45,\n",
       "\t\t\t\t'home':36,\n",
       "\t\t\t\t'delete':46,\n",
       "\t\t\t\t'end':35,\n",
       "\t\t\t\t\n",
       "\t\t\t\t'pageup':33,\n",
       "\t\t\t\t'page_up':33,\n",
       "\t\t\t\t'pu':33,\n",
       "\t\n",
       "\t\t\t\t'pagedown':34,\n",
       "\t\t\t\t'page_down':34,\n",
       "\t\t\t\t'pd':34,\n",
       "\t\n",
       "\t\t\t\t'left':37,\n",
       "\t\t\t\t'up':38,\n",
       "\t\t\t\t'right':39,\n",
       "\t\t\t\t'down':40,\n",
       "\t\n",
       "\t\t\t\t'f1':112,\n",
       "\t\t\t\t'f2':113,\n",
       "\t\t\t\t'f3':114,\n",
       "\t\t\t\t'f4':115,\n",
       "\t\t\t\t'f5':116,\n",
       "\t\t\t\t'f6':117,\n",
       "\t\t\t\t'f7':118,\n",
       "\t\t\t\t'f8':119,\n",
       "\t\t\t\t'f9':120,\n",
       "\t\t\t\t'f10':121,\n",
       "\t\t\t\t'f11':122,\n",
       "\t\t\t\t'f12':123\n",
       "\t\t\t}\n",
       "\t\n",
       "\t\t\tvar modifiers = { \n",
       "\t\t\t\tshift: { wanted:false, pressed:false},\n",
       "\t\t\t\tctrl : { wanted:false, pressed:false},\n",
       "\t\t\t\talt  : { wanted:false, pressed:false},\n",
       "\t\t\t\tmeta : { wanted:false, pressed:false}\t//Meta is Mac specific\n",
       "\t\t\t};\n",
       "                        \n",
       "\t\t\tif(e.ctrlKey)\tmodifiers.ctrl.pressed = true;\n",
       "\t\t\tif(e.shiftKey)\tmodifiers.shift.pressed = true;\n",
       "\t\t\tif(e.altKey)\tmodifiers.alt.pressed = true;\n",
       "\t\t\tif(e.metaKey)   modifiers.meta.pressed = true;\n",
       "                        \n",
       "\t\t\tfor(var i=0; k=keys[i],i<keys.length; i++) {\n",
       "\t\t\t\t//Modifiers\n",
       "\t\t\t\tif(k == 'ctrl' || k == 'control') {\n",
       "\t\t\t\t\tkp++;\n",
       "\t\t\t\t\tmodifiers.ctrl.wanted = true;\n",
       "\n",
       "\t\t\t\t} else if(k == 'shift') {\n",
       "\t\t\t\t\tkp++;\n",
       "\t\t\t\t\tmodifiers.shift.wanted = true;\n",
       "\n",
       "\t\t\t\t} else if(k == 'alt') {\n",
       "\t\t\t\t\tkp++;\n",
       "\t\t\t\t\tmodifiers.alt.wanted = true;\n",
       "\t\t\t\t} else if(k == 'meta') {\n",
       "\t\t\t\t\tkp++;\n",
       "\t\t\t\t\tmodifiers.meta.wanted = true;\n",
       "\t\t\t\t} else if(k.length > 1) { //If it is a special key\n",
       "\t\t\t\t\tif(special_keys[k] == code) kp++;\n",
       "\t\t\t\t\t\n",
       "\t\t\t\t} else if(opt['keycode']) {\n",
       "\t\t\t\t\tif(opt['keycode'] == code) kp++;\n",
       "\n",
       "\t\t\t\t} else { //The special keys did not match\n",
       "\t\t\t\t\tif(character == k) kp++;\n",
       "\t\t\t\t\telse {\n",
       "\t\t\t\t\t\tif(shift_nums[character] && e.shiftKey) { //Stupid Shift key bug created by using lowercase\n",
       "\t\t\t\t\t\t\tcharacter = shift_nums[character]; \n",
       "\t\t\t\t\t\t\tif(character == k) kp++;\n",
       "\t\t\t\t\t\t}\n",
       "\t\t\t\t\t}\n",
       "\t\t\t\t}\n",
       "\t\t\t}\n",
       "\t\t\t\n",
       "\t\t\tif(kp == keys.length && \n",
       "\t\t\t\t\t\tmodifiers.ctrl.pressed == modifiers.ctrl.wanted &&\n",
       "\t\t\t\t\t\tmodifiers.shift.pressed == modifiers.shift.wanted &&\n",
       "\t\t\t\t\t\tmodifiers.alt.pressed == modifiers.alt.wanted &&\n",
       "\t\t\t\t\t\tmodifiers.meta.pressed == modifiers.meta.wanted) {\n",
       "\t\t\t\tcallback(e);\n",
       "\t\n",
       "\t\t\t\tif(!opt['propagate']) { //Stop the event\n",
       "\t\t\t\t\t//e.cancelBubble is supported by IE - this will kill the bubbling process.\n",
       "\t\t\t\t\te.cancelBubble = true;\n",
       "\t\t\t\t\te.returnValue = false;\n",
       "\t\n",
       "\t\t\t\t\t//e.stopPropagation works in Firefox.\n",
       "\t\t\t\t\tif (e.stopPropagation) {\n",
       "\t\t\t\t\t\te.stopPropagation();\n",
       "\t\t\t\t\t\te.preventDefault();\n",
       "\t\t\t\t\t}\n",
       "\t\t\t\t\treturn false;\n",
       "\t\t\t\t}\n",
       "\t\t\t}\n",
       "\t\t}\n",
       "\t\tthis.all_shortcuts[shortcut_combination] = {\n",
       "\t\t\t'callback':func, \n",
       "\t\t\t'target':ele, \n",
       "\t\t\t'event': opt['type']\n",
       "\t\t};\n",
       "\t\t//Attach the function with the event\n",
       "\t\tif(ele.addEventListener) ele.addEventListener(opt['type'], func, false);\n",
       "\t\telse if(ele.attachEvent) ele.attachEvent('on'+opt['type'], func);\n",
       "\t\telse ele['on'+opt['type']] = func;\n",
       "\t},\n",
       "\n",
       "\t//Remove the shortcut - just specify the shortcut and I will remove the binding\n",
       "\t'remove':function(shortcut_combination) {\n",
       "\t\tshortcut_combination = shortcut_combination.toLowerCase();\n",
       "\t\tvar binding = this.all_shortcuts[shortcut_combination];\n",
       "\t\tdelete(this.all_shortcuts[shortcut_combination])\n",
       "\t\tif(!binding) return;\n",
       "\t\tvar type = binding['event'];\n",
       "\t\tvar ele = binding['target'];\n",
       "\t\tvar callback = binding['callback'];\n",
       "\n",
       "\t\tif(ele.detachEvent) ele.detachEvent('on'+type, callback);\n",
       "\t\telse if(ele.removeEventListener) ele.removeEventListener(type, callback, false);\n",
       "\t\telse ele['on'+type] = false;\n",
       "\t}\n",
       "}\n",
       "\n",
       "shortcut.add(\"Alt+Left\",function() {\n",
       "\tvar event = new Event('click');\n",
       "\tif (settingsShown){\n",
       "\t\tdocument.getElementById('hideSettingsButton_2').dispatchEvent(event);\n",
       "\t} else {\n",
       "\t\tdocument.getElementById('showSettingsButton_2').dispatchEvent(event);\n",
       "\t}\n",
       "});\n",
       "\n",
       "\n",
       "// COMMON_CODE_BLOCK_END\n",
       "\n",
       "});});\n",
       "      </script>\n",
       "   </body>\n",
       "</html>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print $p->VISUAL->DIRECTED_GRAPH;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Multiple LP's for one Polytope\n",
    "\n",
    "You can add more than one linear program to one polytope. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "$p->add(\"LP\", LINEAR_OBJECTIVE => [0,1,1,-1]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The different linear programs are saved in an array. The first one can still be accessed like befor:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0 1 1 1"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print $p->LP->LINEAR_OBJECTIVE;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The other ones can be accessed like the entries of an array."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0 1 1 -1"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print $p->LP->[1]->LINEAR_OBJECTIVE;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can find more about this [here](https://polymake.org/doku.php/user_guide/howto/scripting#multiple_subobjects)."
   ]
  }
 ],
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   "file_extension": ".pl",
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