This page is read only. You can view the source, but not change it. Ask your administrator if you think this is wrong. ====== Dealing with Graphs ====== Graphs are ubiquitous in geometric combinatorics. Hence they occur a lot throughout the ''%%polymake%%'' system, explicitly and implicitly. It is important to understand that the user encounters graphs on two distinct layers in the object hierarchy. It is the purpose of this tutorial to explore the various features. For the sake of simplicity here we restrict our attention to undirected graphs. ===== Graphs of Polytopes ===== Coming from polytopes the first situation in which a graph occurs is the vertex-edge graph of such a polytope. <code perl> > $p=rand_sphere(3,20); > print $p->GRAPH->N_NODES; 20 </code> Here ''%%GRAPH%%'' is a property of the polytope object ''%%$p%%'' which happens to be of the object type ''%%Graph%%''. The following is a fragment of the file ''%%apps/polytopes/rules/polytope_properties.rules%%''. This is where all the standard properties of polytopes are declared. <code> property GRAPH : Graph { # Difference of the vertices for each edge (only defined up to signs). property EDGE_DIRECTIONS : EdgeMap<Undirected, Vector<Scalar>>; } </code> In fact, this ''%%Graph%%'' is the main object class of another application, called ''%%graph%%''. This application ''%%graph%%'' is defined in ''%%apps/graph%%'' and its subdirectories. Although the application ''%%graph%%'' has much fewer features than the application ''%%polytope%%'' the overall mechanism of interaction is the same. In particular, there are properties, rules, clients, and such. Non-trivial features include the algorithm for computing the diameter and the visualization of graphs based on a pseudo-physical model (described in [[https://arxiv.org/abs/0711.2397|this paper]]). <code perl> > print $p->DIAMETER; 3 > $p->VISUAL; </code> <HTML> <div class="threejs_output"><!-- polymake for knusper Thu May 7 12:08:41 2020 p --> <html> <head> <title>p</title> <style> /* // COMMON_CODE_BLOCK_BEGIN */ html{overflow: scroll;} .threejs_output { font-family: Arial, Helvetica, sans-serif} strong{font-size: 18px;} canvas { z-index: 8; } input[type='range'] {} input[type='radio'] {margin-left:0;} input[type='checkbox'] {margin-right:7px; margin-left: 0px; padding-left:0px;} .group{padding-bottom: 40px;} .settings * {z-index: 11; } .settings{z-index: 10; margin-left: 30px; display: none; width: 14em; height: 90%; border: solid 1px silver; padding: 2px; overflow-y: scroll; background-color: white } .indented{margin-left: 20px; margin-top: 15px; padding-bottom: 0px;} .shownObjectsList{overflow: auto; max-width: 150px; max-height: 150px;} .showSettingsButton{display: block; z-index: 12; position: absolute } .hideSettingsButton{display: none; z-index: 12; position: absolute; opacity: 0.5} .resetButton{margin-top: 20px;} button{margin-left: 0;} img{cursor: pointer;} .suboption{padding-top: 30px;} .transparency{display: none;} .labelsCheckbox{margin-top: 10px;} input[type=range] { -webkit-appearance: none; padding:0; width:90%; margin-left: auto; margin-right: auto; margin-top: 20px; display: block; } input[type=range]:focus { outline: none; } input[type=range]::-webkit-slider-runnable-track { height: 4px; cursor: pointer; animate: 0.2s; box-shadow: 0px 0px 0px #000000; background: #E3E3E3; border-radius: 0px; border: 0px solid #000000; } input[type=range]::-webkit-slider-thumb { box-shadow: 1px 1px 2px #B8B8B8; border: 1px solid #ABABAB; height: 13px; width: 25px; border-radius: 20px; background: #E0E0E0; cursor: pointer; -webkit-appearance: none; margin-top: -5px; } input[type=range]:focus::-webkit-slider-runnable-track { background: #E3E3E3; } input[type=range]::-moz-range-track { height: 4px; cursor: pointer; animate: 0.2s; box-shadow: 0px 0px 0px #000000; background: #E3E3E3; border-radius: 0px; border: 0px solid #000000; } input[type=range]::-moz-range-thumb { box-shadow: 1px 1px 2px #B8B8B8; border: 1px solid #ABABAB; height: 13px; width: 25px; border-radius: 20px; background: #E0E0E0; cursor: pointer; } input[type=range]::-ms-track { height: 4px; cursor: pointer; animate: 0.2s; background: transparent; border-color: transparent; color: transparent; } input[type=range]::-ms-fill-lower { background: #E3E3E3; border: 0px solid #000000; border-radius: 0px; box-shadow: 0px 0px 0px #000000; } input[type=range]::-ms-fill-upper { background: #E3E3E3; border: 0px solid #000000; border-radius: 0px; box-shadow: 0px 0px 0px #000000; } input[type=range]::-ms-thumb { box-shadow: 1px 1px 2px #B8B8B8; border: 1px solid #ABABAB; height: 13px; width: 25px; border-radius: 20px; background: #E0E0E0; cursor: pointer; } input[type=range]:focus::-ms-fill-lower { background: #E3E3E3; } input[type=range]:focus::-ms-fill-upper { background: #E3E3E3; } /* // COMMON_CODE_BLOCK_END */ </style> </head> <body> <div id='settings_0' class='settings'> <div class=group id='transparency_0' class='transparency'> <strong>Transparency</strong> <input id='transparencyRange_0' type='range' min=0 max=1 step=0.01 value=0> </div> <div class=group id='rotation_0'> <strong>Rotation</strong> <div class=indented> <div><input type='checkbox' id='changeRotationX_0'> x-axis</div> <div><input type='checkbox' id='changeRotationY_0'> y-axis</div> <div><input type='checkbox' id='changeRotationZ_0'> z-axis</div> <button id='resetButton_0' class='resetButton' >Reset</button> </div> <div class=suboption>Rotation speed</div> <input id='rotationSpeedRange_0' type='range' min=0 max=5 step=0.01 value=2> </div> <div class=group id='display_0'> <strong>Display</strong> <div class=indented> <div id='shownObjectsList_0' class='shownObjectsList'></div> <div class='labelsCheckbox'><input type='checkbox' id='labelsCheckboxInput_0' checked>Labels</div> </div> </div> <div class=group id='svg_0'> <strong>SVG</strong> <div class=indented> <form> <input type="radio" name='screenshotMode' value='download' id='download_0' checked> Download<br> <input type="radio" name='screenshotMode' value='tab' id='tab_0' > New tab<br> </form> <button id='takeScreenshot_0'>Screenshot</button> </div> </div> </div> <!-- end of settings --> <img id='hideSettingsButton_0' style="display: none" class='hideSettingsButton' src='/kernelspecs/polymake/close.svg' width=20px"> <img id='showSettingsButton_0' class='showSettingsButton' src='/kernelspecs/polymake/menu.svg' width=20px"> <div id="model18222438667"></div> <script> requirejs.config({ paths: { three: '/kernelspecs/polymake/three', Detector: '/kernelspecs/polymake/Detector', SVGRenderer: '/kernelspecs/polymake/SVGRenderer', CanvasRenderer: '/kernelspecs/polymake/CanvasRenderer', Projector: '/kernelspecs/polymake/Projector', TrackballControls: '/kernelspecs/polymake/TrackballControls' }, shim: { 'three': { exports: 'THREE' }, 'Detector': { deps: [ 'three' ], exports: 'Detector' }, 'SVGRenderer': { deps: [ 'three' ], exports: 'THREE.SVGRenderer' }, 'CanvasRenderer': { deps: [ 'three' ], exports: 'THREE.CanvasRenderer' }, 'Projector': { deps: [ 'three' ], exports: 'THREE.Projector' }, 'TrackballControls': { deps: [ 'three' ], exports: 'THREE.TrackballControls' } } }); require(['three'],function(THREE){ window.THREE = THREE; require(['Detector','SVGRenderer','CanvasRenderer','Projector','TrackballControls'],function(Detector,SVGRenderer,CanvasRenderer,Projector,TrackballControls){ THREE.SVGRenderer = SVGRenderer; THREE.CanvasRenderer = CanvasRenderer; THREE.Projector = Projector; THREE.TrackballControls = TrackballControls; // COMMON_CODE_BLOCK_BEGIN var foldable = false; var container = document.getElementById( 'model18222438667' ); var renderer = Detector.webgl? new THREE.WebGLRenderer({antialias: true}): new THREE.CanvasRenderer({antialias: true}); var svgRenderer = new THREE.SVGRenderer({antialias: true}); var box = document.getElementsByClassName( 'threejs_output' )[0]; var width = box.clientWidth; var height = box.clientHeight; renderer.setSize(width, height); svgRenderer.setSize(width, height); renderer.setClearColor(0xFFFFFF, 1); svgRenderer.setClearColor(0xFFFFFF, 1); container.appendChild(renderer.domElement); var scene = new THREE.Scene(); var camera = new THREE.PerspectiveCamera(75, width/height, 0.1, 1000); var renderid; camera.position.set(0, 0, 5); camera.lookAt(0, 0, 0); camera.up.set(0, 1, 0); // class to allow move points together with labels and spheres var PMPoint = function (x,y,z) { this.vector = new THREE.Vector3(x,y,z); this.sprite = null; this.sphere = null; } PMPoint.prototype.makelabel = function(label) { this.sprite = textSprite( label ); this.sprite.position.copy(this.vector); } PMPoint.prototype.makesphere = function(radius,material) { this.sphere = new THREE.Mesh(new THREE.SphereGeometry(radius), material); this.sphere.position.copy(this.vector); } PMPoint.prototype.setX = function(x) { this.vector.setX(x); if (this.sprite) { this.sprite.position.setX(x); } if (this.sphere) { this.sphere.position.setX(x); } }; PMPoint.prototype.setY = function(y) { this.vector.setY(y); if (this.sprite) { this.sprite.position.setY(y); } if (this.sphere) { this.sphere.position.setY(y); } }; PMPoint.prototype.setZ = function(z) { this.vector.setZ(z); if (this.sprite) { this.sprite.position.setZ(z); } if (this.sphere) { this.sphere.position.setZ(z); } }; PMPoint.prototype.set = function(x,y,z) { this.vector.set(x,y,z); if (this.sprite) { this.sprite.position.set(x,y,z); } if (this.sphere) { this.sphere.position.set(x,y,z); } }; PMPoint.prototype.add = function(o) { if (this.sprite) { o.add(this.sprite); } if (this.sphere) { o.add(this.sphere); } }; var controls = new THREE.TrackballControls(camera, container); controls.zoomSpeed = 0.2; controls.rotateSpeed = 4; var all_objects = []; var centroids = []; // select the target node var target = document.querySelector('#model18222438667'); // create an observer instance var observer = new MutationObserver(function(mutations) { mutations.forEach(function(mutation) { if (mutation.removedNodes && mutation.removedNodes.length > 0) { cancelAnimationFrame(renderId); observer.disconnect(); console.log("cancelled frame "+renderId); } }); }); // configuration of the observer: var config = { childList: true, characterData: true } // pass in the target node, as well as the observer options while (target) { if (target.className=="output") { observer.observe(target, config); break; } target = target.parentNode; } // COMMON_CODE_BLOCK_END var objectnames = ["p"]; var obj = new THREE.Object3D(); var allpoints = []; allpoints.push(new PMPoint(0.0880654, 0.550636, -0.830087)); allpoints.push(new PMPoint(-0.464577, 0.586602, 0.663375)); allpoints.push(new PMPoint(-0.750212, 0.373797, 0.545396)); allpoints.push(new PMPoint(-0.939747, 0.270387, -0.209203)); allpoints.push(new PMPoint(-0.806957, 0.242463, -0.538547)); allpoints.push(new PMPoint(-0.038505, 0.963403, 0.265278)); allpoints.push(new PMPoint(-0.985171, -0.171317, 0.00944524)); allpoints.push(new PMPoint(-0.317729, -0.566055, 0.760677)); allpoints.push(new PMPoint(0.59098, 0.277889, -0.757312)); allpoints.push(new PMPoint(0.501039, 0.161787, -0.850168)); allpoints.push(new PMPoint(-0.0390277, 0.0113995, -0.999173)); allpoints.push(new PMPoint(-0.62088, -0.421985, 0.660633)); allpoints.push(new PMPoint(-0.757662, 0.65048, -0.0531356)); allpoints.push(new PMPoint(-0.349907, 0.921397, 0.169092)); allpoints.push(new PMPoint(0.270145, 0.0815514, 0.95936)); allpoints.push(new PMPoint(-0.111087, 0.0587341, 0.992074)); allpoints.push(new PMPoint(-0.382034, -0.71526, -0.585196)); allpoints.push(new PMPoint(0.415059, 0.664095, 0.621855)); allpoints.push(new PMPoint(0.415932, -0.377311, -0.827428)); allpoints.push(new PMPoint(0.611113, -0.603533, -0.512142)); <!-- Vertex style --> var points_material = new THREE.MeshBasicMaterial ( {color: 0xFF0000, } ); points_material.side = THREE.DoubleSide; points_material.transparent = true; <!-- POINTS --> allpoints[0].makesphere(0.02,points_material); allpoints[1].makesphere(0.02,points_material); allpoints[2].makesphere(0.02,points_material); allpoints[3].makesphere(0.02,points_material); allpoints[4].makesphere(0.02,points_material); allpoints[5].makesphere(0.02,points_material); allpoints[6].makesphere(0.02,points_material); allpoints[7].makesphere(0.02,points_material); allpoints[8].makesphere(0.02,points_material); allpoints[9].makesphere(0.02,points_material); allpoints[10].makesphere(0.02,points_material); allpoints[11].makesphere(0.02,points_material); allpoints[12].makesphere(0.02,points_material); allpoints[13].makesphere(0.02,points_material); allpoints[14].makesphere(0.02,points_material); allpoints[15].makesphere(0.02,points_material); allpoints[16].makesphere(0.02,points_material); allpoints[17].makesphere(0.02,points_material); allpoints[18].makesphere(0.02,points_material); allpoints[19].makesphere(0.02,points_material); allpoints[0].makelabel("0"); allpoints[1].makelabel("1"); allpoints[2].makelabel("2"); allpoints[3].makelabel("3"); allpoints[4].makelabel("4"); allpoints[5].makelabel("5"); allpoints[6].makelabel("6"); allpoints[7].makelabel("7"); allpoints[8].makelabel("8"); allpoints[9].makelabel("9"); allpoints[10].makelabel("10"); allpoints[11].makelabel("11"); allpoints[12].makelabel("12"); allpoints[13].makelabel("13"); allpoints[14].makelabel("14"); allpoints[15].makelabel("15"); allpoints[16].makelabel("16"); allpoints[17].makelabel("17"); allpoints[18].makelabel("18"); allpoints[19].makelabel("19"); for (index = 0; index < allpoints.length; ++index) { allpoints[index].add(obj); } var faces = new THREE.Geometry(); <!-- VERTICES --> faces.vertices.push(allpoints[0].vector); faces.vertices.push(allpoints[1].vector); faces.vertices.push(allpoints[2].vector); faces.vertices.push(allpoints[3].vector); faces.vertices.push(allpoints[4].vector); faces.vertices.push(allpoints[5].vector); faces.vertices.push(allpoints[6].vector); faces.vertices.push(allpoints[7].vector); faces.vertices.push(allpoints[8].vector); faces.vertices.push(allpoints[9].vector); faces.vertices.push(allpoints[10].vector); faces.vertices.push(allpoints[11].vector); faces.vertices.push(allpoints[12].vector); faces.vertices.push(allpoints[13].vector); faces.vertices.push(allpoints[14].vector); faces.vertices.push(allpoints[15].vector); faces.vertices.push(allpoints[16].vector); faces.vertices.push(allpoints[17].vector); faces.vertices.push(allpoints[18].vector); faces.vertices.push(allpoints[19].vector); centroids.push(computeCentroid(faces)); <!-- Facet style --> var faces_material = new THREE.MeshBasicMaterial ( {color: 0x77EC9E, transparent: true, opacity: 1, side: THREE.DoubleSide , depthWrite: true, depthTest: true, } ); faces_material.side = THREE.DoubleSide; faces_material.transparent = true; <!-- FACETS --> faces.faces.push(new THREE.Face3(18, 19, 16, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(18, 9, 19, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(19, 9, 8, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(17, 19, 8, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(14, 17, 15, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(14, 19, 17, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(7, 19, 14, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(7, 14, 15, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(11, 7, 15, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(16, 7, 11, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(16, 19, 7, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(17, 5, 1, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(15, 17, 1, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(2, 15, 1, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(1, 5, 13, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(16, 11, 6, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(6, 11, 2, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(11, 15, 2, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(2, 1, 12, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(13, 12, 1, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(18, 10, 9, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(17, 8, 5, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(8, 0, 5, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(0, 13, 5, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(9, 10, 0, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(9, 0, 8, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(18, 16, 10, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(4, 16, 6, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(10, 16, 4, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(4, 3, 12, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(4, 6, 3, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(3, 2, 12, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(6, 2, 3, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(0, 4, 12, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(0, 12, 13, undefined, undefined, 0)); faces.faces.push(new THREE.Face3(10, 4, 0, undefined, undefined, 0)); faces.computeFaceNormals(); faces.computeVertexNormals(); var object = new THREE.Mesh(faces, faces_material); obj.add(object); <!-- Edge style --> var line_material = new THREE.LineBasicMaterial ( {color: 0x000000, linewidth: 1.5, } ); line_material.side = THREE.DoubleSide; line_material.transparent = true; <!-- EDGES --> var line = new THREE.Geometry(); line.vertices.push(allpoints[18].vector); line.vertices.push(allpoints[19].vector); line.vertices.push(allpoints[16].vector); line.vertices.push(allpoints[18].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[18].vector); line.vertices.push(allpoints[9].vector); line.vertices.push(allpoints[19].vector); line.vertices.push(allpoints[18].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[19].vector); line.vertices.push(allpoints[9].vector); line.vertices.push(allpoints[8].vector); line.vertices.push(allpoints[19].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[17].vector); line.vertices.push(allpoints[19].vector); line.vertices.push(allpoints[8].vector); line.vertices.push(allpoints[17].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[14].vector); line.vertices.push(allpoints[17].vector); line.vertices.push(allpoints[15].vector); line.vertices.push(allpoints[14].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[14].vector); line.vertices.push(allpoints[19].vector); line.vertices.push(allpoints[17].vector); line.vertices.push(allpoints[14].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[7].vector); line.vertices.push(allpoints[19].vector); line.vertices.push(allpoints[14].vector); line.vertices.push(allpoints[7].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[7].vector); line.vertices.push(allpoints[14].vector); line.vertices.push(allpoints[15].vector); line.vertices.push(allpoints[7].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[11].vector); line.vertices.push(allpoints[7].vector); line.vertices.push(allpoints[15].vector); line.vertices.push(allpoints[11].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[16].vector); line.vertices.push(allpoints[7].vector); line.vertices.push(allpoints[11].vector); line.vertices.push(allpoints[16].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[16].vector); line.vertices.push(allpoints[19].vector); line.vertices.push(allpoints[7].vector); line.vertices.push(allpoints[16].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[17].vector); line.vertices.push(allpoints[5].vector); line.vertices.push(allpoints[1].vector); line.vertices.push(allpoints[17].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[15].vector); line.vertices.push(allpoints[17].vector); line.vertices.push(allpoints[1].vector); line.vertices.push(allpoints[15].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[2].vector); line.vertices.push(allpoints[15].vector); line.vertices.push(allpoints[1].vector); line.vertices.push(allpoints[2].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[1].vector); line.vertices.push(allpoints[5].vector); line.vertices.push(allpoints[13].vector); line.vertices.push(allpoints[1].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[16].vector); line.vertices.push(allpoints[11].vector); line.vertices.push(allpoints[6].vector); line.vertices.push(allpoints[16].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[6].vector); line.vertices.push(allpoints[11].vector); line.vertices.push(allpoints[2].vector); line.vertices.push(allpoints[6].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[11].vector); line.vertices.push(allpoints[15].vector); line.vertices.push(allpoints[2].vector); line.vertices.push(allpoints[11].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[2].vector); line.vertices.push(allpoints[1].vector); line.vertices.push(allpoints[12].vector); line.vertices.push(allpoints[2].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[13].vector); line.vertices.push(allpoints[12].vector); line.vertices.push(allpoints[1].vector); line.vertices.push(allpoints[13].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[18].vector); line.vertices.push(allpoints[10].vector); line.vertices.push(allpoints[9].vector); line.vertices.push(allpoints[18].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[17].vector); line.vertices.push(allpoints[8].vector); line.vertices.push(allpoints[5].vector); line.vertices.push(allpoints[17].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[8].vector); line.vertices.push(allpoints[0].vector); line.vertices.push(allpoints[5].vector); line.vertices.push(allpoints[8].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[0].vector); line.vertices.push(allpoints[13].vector); line.vertices.push(allpoints[5].vector); line.vertices.push(allpoints[0].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[9].vector); line.vertices.push(allpoints[10].vector); line.vertices.push(allpoints[0].vector); line.vertices.push(allpoints[9].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[9].vector); line.vertices.push(allpoints[0].vector); line.vertices.push(allpoints[8].vector); line.vertices.push(allpoints[9].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[18].vector); line.vertices.push(allpoints[16].vector); line.vertices.push(allpoints[10].vector); line.vertices.push(allpoints[18].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[4].vector); line.vertices.push(allpoints[16].vector); line.vertices.push(allpoints[6].vector); line.vertices.push(allpoints[4].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[10].vector); line.vertices.push(allpoints[16].vector); line.vertices.push(allpoints[4].vector); line.vertices.push(allpoints[10].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[4].vector); line.vertices.push(allpoints[3].vector); line.vertices.push(allpoints[12].vector); line.vertices.push(allpoints[4].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[4].vector); line.vertices.push(allpoints[6].vector); line.vertices.push(allpoints[3].vector); line.vertices.push(allpoints[4].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[3].vector); line.vertices.push(allpoints[2].vector); line.vertices.push(allpoints[12].vector); line.vertices.push(allpoints[3].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[6].vector); line.vertices.push(allpoints[2].vector); line.vertices.push(allpoints[3].vector); line.vertices.push(allpoints[6].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[0].vector); line.vertices.push(allpoints[4].vector); line.vertices.push(allpoints[12].vector); line.vertices.push(allpoints[0].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[0].vector); line.vertices.push(allpoints[12].vector); line.vertices.push(allpoints[13].vector); line.vertices.push(allpoints[0].vector); obj.add(new THREE.Line(line, line_material)); var line = new THREE.Geometry(); line.vertices.push(allpoints[10].vector); line.vertices.push(allpoints[4].vector); line.vertices.push(allpoints[0].vector); line.vertices.push(allpoints[10].vector); obj.add(new THREE.Line(line, line_material)); scene.add(obj); all_objects.push(obj); // COMMON_CODE_BLOCK_BEGIN var xRotationEnabled = false; var yRotationEnabled = false; var zRotationEnabled = false; var rotationSpeedFactor = 1; var settingsShown = false; var labelsShown = true; var intervals = []; var timeouts = []; var explodingSpeed = 0.05; var explodeScale = 0; var XMLS = new XMLSerializer(); var svgElement; var renderId; var render = function () { renderId = requestAnimationFrame(render); // comment in for automatic explosion // explode(updateFactor()); var phi = 0.02 * rotationSpeedFactor; if (xRotationEnabled){ scene.rotation.x += phi; } if(yRotationEnabled){ scene.rotation.y += phi; } if(zRotationEnabled){ scene.rotation.z += phi; } controls.update(); renderer.render(scene, camera); }; render(); function computeCentroid(geom) { centroid = new THREE.Vector3(); geom.vertices.forEach(function(v) { centroid.add(v); }); centroid.divideScalar(geom.vertices.length); return centroid; } function changeTransparency(event){ var opacity = 1-Number(event.currentTarget.value); for (var i=0; i<all_objects.length; i++){ for (var j=0; j<all_objects[i].children.length; j++){ if (all_objects[i].children[j].material.type == "MultiMaterial") { for (var k=0; k<all_objects[i].children[j].material.materials.length; k++){ all_objects[i].children[j].material.materials[k].opacity = opacity; all_objects[i].children[j].material.materials[k].depthWrite = opacity < 0.5 ? false : true; all_objects[i].children[j].material.materials[k].depthTest = opacity < 0.5 ? false : true; } } else if (all_objects[i].children[j].material.transparent && all_objects[i].children[j].material.type == "MeshBasicMaterial" && all_objects[i].children[j].geometry.type == "Geometry"){ all_objects[i].children[j].material.opacity = opacity; all_objects[i].children[j].material.depthWrite = opacity < 0.5 ? false : true; all_objects[i].children[j].material.depthTest = opacity < 0.5 ? false : true; } } } } function changeRotationX(event){ xRotationEnabled = event.currentTarget.checked; } function changeRotationY(event){ yRotationEnabled = event.currentTarget.checked; } function changeRotationZ(event){ zRotationEnabled = event.currentTarget.checked; } function changeRotationSpeedFactor(event){ rotationSpeedFactor = Number(event.currentTarget.value); } function resetScene(){ scene.rotation.set(0,0,0); camera.position.set(0,0,5); camera.up.set(0,1,0); } function showSettings(event){ event.currentTarget.style.display = 'none'; document.getElementById('settings_0').style.position = 'absolute'; document.getElementById('settings_0').style.display = 'block'; document.getElementById('showSettingsButton_0').style.display = 'none'; document.getElementById('hideSettingsButton_0').style.display = 'block'; settingsShown = true; } function hideSettings(event){ event.currentTarget.style.display = 'none'; document.getElementById('settings_0').style.display = 'none'; document.getElementById('hideSettingsButton_0').style.display = 'none'; document.getElementById('showSettingsButton_0').style.display = 'block'; settingsShown = false; } var pos = 150* Math.PI; function updateFactor() { pos++; return Math.sin(.01*pos)+1; } function makelabel(message, x, y, z, params) { var spritey = textSprite( message, params ); spritey.position.set(x, y, z); obj.add(spritey); } function textSprite(message, parameters) { if ( parameters === undefined ) parameters = {}; var fontface = "Helvetica"; var fontsize = parameters.hasOwnProperty("fontsize") ? parameters["fontsize"] : 18; fontsize = fontsize*10; var canvas = document.createElement('canvas'); var size = 1024; canvas.width = size; canvas.height = size; var context = canvas.getContext('2d'); context.font = fontsize + "px " + fontface; // text color context.fillStyle = "rgba(0, 0, 0, 1.0)"; context.fillText(message, size/2, size/2); // canvas contents will be used for a texture var texture = new THREE.Texture(canvas); texture.needsUpdate = true; var spriteMaterial = new THREE.SpriteMaterial( {map: texture, useScreenCoordinates: false}); var sprite = new THREE.Sprite(spriteMaterial); return sprite; } function takeSvgScreenshot(){ if (labelsShown){ hideLabels(); } svgRenderer.render(scene,camera); svgElement = XMLS.serializeToString(svgRenderer.domElement); if (labelsShown){ displayLabels(); } if (document.getElementById('tab_0').checked){ //show in new tab var myWindow = window.open("",""); myWindow.document.body.innerHTML = svgElement; } else{ // download svg file download("screenshot.svg", svgElement); } } function showOrHideObject(event){ var nr = Number(event.currentTarget.name); all_objects[nr].visible = event.currentTarget.checked; } function displayOrHideOptionsRecursive( obj ) { for (var j=0; j<obj.children.length; j++) { var child = obj.children[j]; if (child.material===undefined && child) { displayOrHideOptionsRecursive( child ); } else { if (child.material.type == "MultiMaterial") { for (var k=0; k<child.material.materials.length; k++) { if (child.material.materials[k].transparent) { document.getElementById('transparency_0').style.display = 'block'; document.getElementById('transparencyRange_0').value = 1 - child.material.materials[k].opacity; return; } } } else if ( child.material.transparent && child.material.type == "MeshBasicMaterial" && child.geometry.type == "Geometry"){ document.getElementById('transparency_0').style.display = 'block'; return; } } } } function displayOrHideOptions() { for (var i=0; i<all_objects.length; i++) { var obj = all_objects[i]; displayOrHideOptionsRecursive( obj ); } } displayOrHideOptions() // ---------------------- EXPLOSION ------------------------------------------------ // --------------------------------------------------------------------------------- function explode(factor) { var obj, c; var c0 = centroids[0]; for (var i = 0; i<centroids.length; ++i) { c = centroids[i]; obj = all_objects[all_objects.length - centroids.length + i]; obj.position.set(c.x*factor, c.y*factor, c.z*factor); } } function triggerExplode(event){ explodeScale = Number(event.currentTarget.value); explode(explodeScale); } function setExplodingSpeed(event){ explodingSpeed = Number(event.currentTarget.value); } function triggerAutomaticExplode(event){ if (event.currentTarget.checked){ startExploding(); } else { clearIntervals(); } } function startExploding(){ intervals.push(setInterval(explodingInterval, 25)); } function explodingInterval(){ explodeScale += explodingSpeed; if (explodeScale <= 6){ explode(explodeScale); } else{ explode(6); explodeScale = 6; clearIntervals(); timeouts.push(setTimeout(startUnexploding, 3000)); } document.getElementById('explodeRange_0').value = explodeScale; } function startUnexploding(){ intervals.push(setInterval(unexplodingInterval, 25)); } function unexplodingInterval(){ explodeScale -= explodingSpeed; if (explodeScale >= 0){ explode(explodeScale); } else { explode(0); explodeScale = 0; clearIntervals(); timeouts.push(setTimeout(startExploding, 3000)); } document.getElementById('explodeRange_0').value = explodeScale; } function clearIntervals(){ intervals.forEach(function(interval){ clearInterval(interval); }); intervals = []; timeouts.forEach(function(timeout){ clearTimeout(timeout); }); timeouts = []; } // append checkboxes for displaying or hiding objects var shownObjectsList = document.getElementById('shownObjectsList_0'); for (var i=0; i<all_objects.length; i++){ var objNode = document.createElement('span'); objNode.innerHTML = objectnames[i] + '<br>'; var checkbox = document.createElement('input'); checkbox.type = 'checkbox'; checkbox.checked = true; checkbox.name = String(i); checkbox.onchange = showOrHideObject; shownObjectsList.appendChild(checkbox); shownObjectsList.appendChild(objNode); } function displayLabels(){ for (var i=0; i<all_objects.length; i++){ for (var j=0; j<all_objects[i].children.length; j++){ var child = all_objects[i].children[j]; if (child.type == 'Sprite'){ child.visible = true; } } } } function hideLabels(){ for (var i=0; i<all_objects.length; i++){ for (var j=0; j<all_objects[i].children.length; j++){ var child = all_objects[i].children[j]; if (child.type == 'Sprite'){ child.visible = false; } } } } function displayOrHideLabels(event){ if (event.currentTarget.checked){ displayLabels(); labelsShown = true; } else { hideLabels(); labelsShown = false; } } function download(filename, text) { var element = document.createElement('a'); element.setAttribute('href', 'data:text/plain;charset=utf-8,' + encodeURIComponent(text)); element.setAttribute('download', filename); element.style.display = 'none'; document.body.appendChild(element); element.click(); document.body.removeChild(element); } var tempobj; tempobj = document.getElementById('explodeRange_0'); if (tempobj) { tempobj.oninput = triggerExplode; document.getElementById('explodeCheckbox_0').onchange = triggerAutomaticExplode; document.getElementById('explodingSpeedRange_0').oninput = setExplodingSpeed; } tempobj = document.getElementById('foldRange_0'); if (tempobj) { tempobj.oninput = fold; } document.getElementById('transparencyRange_0').oninput = changeTransparency; document.getElementById('changeRotationX_0').onchange = changeRotationX; document.getElementById('changeRotationY_0').onchange = changeRotationY; document.getElementById('changeRotationZ_0').onchange = changeRotationZ; document.getElementById('resetButton_0').onclick = resetScene; document.getElementById('rotationSpeedRange_0').oninput = changeRotationSpeedFactor; document.getElementById('labelsCheckboxInput_0').onchange = displayOrHideLabels; document.getElementById('takeScreenshot_0').onclick = takeSvgScreenshot; document.getElementById('showSettingsButton_0').onclick = showSettings; document.getElementById('hideSettingsButton_0').onclick = hideSettings; // ------------------ SHORTCUTS -------------------------------------------- // ------------------------------------------------------------------------- /** * http://www.openjs.com/scripts/events/keyboard_shortcuts/ * Version : 2.01.B * By Binny V A * License : BSD */ shortcut = { 'all_shortcuts':{},//All the shortcuts are stored in this array 'add': function(shortcut_combination,callback,opt) { //Provide a set of default options var default_options = { 'type':'keydown', 'propagate':false, 'disable_in_input':false, 'target':document, 'keycode':false } if(!opt) opt = default_options; else { for(var dfo in default_options) { if(typeof opt[dfo] == 'undefined') opt[dfo] = default_options[dfo]; } } var ele = opt.target; if(typeof opt.target == 'string') ele = document.getElementById(opt.target); var ths = this; shortcut_combination = shortcut_combination.toLowerCase(); //The function to be called at keypress var func = function(e) { e = e || window.event; if(opt['disable_in_input']) { //Don't enable shortcut keys in Input, Textarea fields var element; if(e.target) element=e.target; else if(e.srcElement) element=e.srcElement; if(element.nodeType==3) element=element.parentNode; if(element.tagName == 'INPUT' || element.tagName == 'TEXTAREA') return; } //Find Which key is pressed if (e.keyCode) code = e.keyCode; else if (e.which) code = e.which; var character = String.fromCharCode(code).toLowerCase(); if(code == 188) character=","; //If the user presses , when the type is onkeydown if(code == 190) character="."; //If the user presses , when the type is onkeydown var keys = shortcut_combination.split("+"); //Key Pressed - counts the number of valid keypresses - if it is same as the number of keys, the shortcut function is invoked var kp = 0; //Work around for stupid Shift key bug created by using lowercase - as a result the shift+num combination was broken var shift_nums = { "`":"~", "1":"!", "2":"@", "3":"#", "4":"$", "5":"%", "6":"^", "7":"&", "8":"*", "9":"(", "0":")", "-":"_", "=":"+", ";":":", "'":"\"", ",":"<", ".":">", "/":"?", "\\":"|" } //Special Keys - and their codes var special_keys = { 'esc':27, 'escape':27, 'tab':9, 'space':32, 'return':13, 'enter':13, 'backspace':8, 'scrolllock':145, 'scroll_lock':145, 'scroll':145, 'capslock':20, 'caps_lock':20, 'caps':20, 'numlock':144, 'num_lock':144, 'num':144, 'pause':19, 'break':19, 'insert':45, 'home':36, 'delete':46, 'end':35, 'pageup':33, 'page_up':33, 'pu':33, 'pagedown':34, 'page_down':34, 'pd':34, 'left':37, 'up':38, 'right':39, 'down':40, 'f1':112, 'f2':113, 'f3':114, 'f4':115, 'f5':116, 'f6':117, 'f7':118, 'f8':119, 'f9':120, 'f10':121, 'f11':122, 'f12':123 } var modifiers = { shift: { wanted:false, pressed:false}, ctrl : { wanted:false, pressed:false}, alt : { wanted:false, pressed:false}, meta : { wanted:false, pressed:false} //Meta is Mac specific }; if(e.ctrlKey) modifiers.ctrl.pressed = true; if(e.shiftKey) modifiers.shift.pressed = true; if(e.altKey) modifiers.alt.pressed = true; if(e.metaKey) modifiers.meta.pressed = true; for(var i=0; k=keys[i],i<keys.length; i++) { //Modifiers if(k == 'ctrl' || k == 'control') { kp++; modifiers.ctrl.wanted = true; } else if(k == 'shift') { kp++; modifiers.shift.wanted = true; } else if(k == 'alt') { kp++; modifiers.alt.wanted = true; } else if(k == 'meta') { kp++; modifiers.meta.wanted = true; } else if(k.length > 1) { //If it is a special key if(special_keys[k] == code) kp++; } else if(opt['keycode']) { if(opt['keycode'] == code) kp++; } else { //The special keys did not match if(character == k) kp++; else { if(shift_nums[character] && e.shiftKey) { //Stupid Shift key bug created by using lowercase character = shift_nums[character]; if(character == k) kp++; } } } } if(kp == keys.length && modifiers.ctrl.pressed == modifiers.ctrl.wanted && modifiers.shift.pressed == modifiers.shift.wanted && modifiers.alt.pressed == modifiers.alt.wanted && modifiers.meta.pressed == modifiers.meta.wanted) { callback(e); if(!opt['propagate']) { //Stop the event //e.cancelBubble is supported by IE - this will kill the bubbling process. e.cancelBubble = true; e.returnValue = false; //e.stopPropagation works in Firefox. if (e.stopPropagation) { e.stopPropagation(); e.preventDefault(); } return false; } } } this.all_shortcuts[shortcut_combination] = { 'callback':func, 'target':ele, 'event': opt['type'] }; //Attach the function with the event if(ele.addEventListener) ele.addEventListener(opt['type'], func, false); else if(ele.attachEvent) ele.attachEvent('on'+opt['type'], func); else ele['on'+opt['type']] = func; }, //Remove the shortcut - just specify the shortcut and I will remove the binding 'remove':function(shortcut_combination) { shortcut_combination = shortcut_combination.toLowerCase(); var binding = this.all_shortcuts[shortcut_combination]; delete(this.all_shortcuts[shortcut_combination]) if(!binding) return; var type = binding['event']; var ele = binding['target']; var callback = binding['callback']; if(ele.detachEvent) ele.detachEvent('on'+type, callback); else if(ele.removeEventListener) ele.removeEventListener(type, callback, false); else ele['on'+type] = false; } } shortcut.add("Alt+Left",function() { var event = new Event('click'); if (settingsShown){ document.getElementById('hideSettingsButton_0').dispatchEvent(event); } else{ document.getElementById('showSettingsButton_0').dispatchEvent(event); } }); if (foldable) moveToBaryCenter(); });}); // COMMON_CODE_BLOCK_END </script> </body> </html> </div> </HTML> As polytopes and all other objects in ''%%polymake%%'''s object hierarchy the graphs from the application ''%%graph%%'' are immutable objects. It is not possible to add a node or to delete an edge. It is instructive to look at the beginning of the file ''%%apps/graph/graph_properties.rules%%'' which reads like this. <code> declare object Graph<Dir=Undirected> { # combinatorial description of the Graph in the form of adjacency matrix property ADJACENCY : GraphAdjacency<Dir>; ... </code> The key property of a graph object is its ''%%ADJACENCY%%''. For each node all neighbors are listed. Here we want to focus on the type ''%%GraphAdjacency%%'' of this property. This refers to C++ class from the Polymake Template Library named ''%%Graph%%'', and this is where the data structure and most algorithms reside. It is possible to directly manipulate objects of this type, and these are //not// immutable, they can be changed. The following shows how one can create a 5-cycle. Calling the method ''%%edge%%'' creates an edge if it did not exist before. The output is the ordered list of neighbors per node. <code perl> > $g=new GraphAdjacency(5); > for (my $i=0; $i<5; ++$i) { $g->edge($i,($i+1)%5) }; > print $g; {1 4} {0 2} {1 3} {2 4} {0 3} </code> If a graph has many nodes it is convenient to know which line of the output refers to which node. If an array of labels is given this could also be used instead of the numbers which are the default. <code perl> > print rows_labeled($g); 0:1 4 1:0 2 2:1 3 3:2 4 4:0 3 </code> There are other ways to change such a graph. Contracting the edge //(x,y)// where //x// is smaller than //y// implies that the node //y// is destroyed. <code perl> > $g->delete_edge(0,1); > $g->contract_edge(2,3); > $g->squeeze(); </code> However, most of our graph algorithms expect a graph with consecutively numbered nodes. The function ''%%squeeze%%'' takes care of a proper renumbering, but this takes linear time in the number of nodes. <code perl> > print rows_labeled($g); 0:3 1:2 2:1 3 3:0 2 </code> How do I iterate over the adjacent nodes to a given node? <code> foreach (@{$g->adjacent_nodes(0)}) { print "node number $_ is adjacent to node number 0\n"; } </code> It is also legal to copy all adjacent nodes to an array as in: <code> @x = @{$g->adjacent_nodes(0)}; </code> Subsequently, the individial neighbors can be accessed, for instance, as ''%%$x[1]%%''. However, for technical reasons too difficult to explain here, it is //not// legal to write ''%%$g->adjacent_nodes(0)->[1]%%''!\\ Usually it is preferred to avoid copying; so use constructions like ''%%foreach%%'' and ''%%map%%'' if possible. ===== Defining a Graph from Scratch ===== You can also work with graphs independent of their connection to polytopes. We will switch to ''%%application "graph"%%'' for the following commands, but this is not strictly necessary. We want to define a new object of type ''%%Graph%%'' in ''%%polymake%%''. The key property of a graph is its adjacency matrix, which is stored in the property ''%%ADJACENCY%%''. It lists the neighbors of each node. We use again the above example of a 5-cycle C<sub>5</sub> with consecutively numbered nodes. Then one can define C<sub>5</sub> by <code perl> > application "graph"; > $g=new Graph(ADJACENCY=>[[1,4],[0,2],[1,3],[2,4],[0,3]]); </code> The list of edges of the graph is induced by the adjacency matrix (please note that in a undirected graph each edge appears twice). You can get an explicit list of the edges with the user function ''%%EDGES%%''. <code perl> > print $g->EDGES; {0 1} {1 2} {2 3} {0 4} {3 4} </code> Note however, that this list is not stored in the object, as it is just a different view on the adjacency matrix. Most often when you define a graph you would not write it down as a list of adjacencies, but as a list of edges. For convenience, ''%%polymake%%'' provides a way to create a graph from a list of edges. The same 5-cycle as above could also be defined via <code perl> > $g=graph_from_edges([[0,1],[1,2],[2,3],[0,4],[3,4]]); </code> The order of the edges, and the order of the nodes for each edge in a undirected case, is not important. We can check the adjacency matrix, <code perl> > print $g->ADJACENCY; {1 4} {0 2} {1 3} {2 4} {0 3} </code> and continue to work with the graph by e.g. checking its ''%%DIAMETER%%'', ''%%BIPARTITE%%''-ness or other properties: <code perl> > print $g->DIAMETER; 2 > print $g->BIPARTITE; false > print $g->MAX_CLIQUES; {{0 1} {0 4} {1 2} {2 3} {3 4}} </code> ===== Directed Graphs ===== By specifying the template parameter ''%%Directed%%'' a graph is born as a directed graph. Properties which make sense for directed graphs work as expected. A directed graph may have two arcs between any two nodes with opposite orientations. <code perl> > $g=new Graph<Directed>(ADJACENCY=>[[1],[2],[3],[2,4],[0]]); > print $g->DIAMETER; 4 </code> Some properties of graphs do not make sense for directed graph. Here is an example of an undirected graph property which does not make sense for directed graphs. <code perl> > #print $g->MAX_CLIQUES; polymake: ERROR: Object Graph<Directed> does not have a property or method MAX_CLIQUES </code> Graphs with multiple edges/arcs are currently not supported. ===== Visualizing Graphs ===== Like other "big" ''%%polymake%%'' objects the ''%%Graph%%'' class has a member (function) ''%%VISUAL%%'' which returns an abstract visualization object. Depending on the configuration it typically uses ''%%JReality%%'' or ''%%JavaView%%''. Particularly interesting for graph drawing is the visualization via ''%%Graphviz%%''. <code perl> > svg($g->VISUAL); > graphviz($g->VISUAL); </code> {{ tutorials:release:4.1:apps_graph:output_0.svg }} <code> Undefined subroutine &Polymake::User::graphviz called </code> Note that the latter starts a postscript viewer with the ''%%Graphviz%%'' output. Make sure that the custom variable ''%%$Postscript::viewer%%'' is set to something reasonable (like, e.g., ''%%/usr/bin/evince%%''). user_guide/tutorials/release/4.1/apps_graph.txt Last modified: 2020/06/15 13:05by 127.0.0.1