2D PGA project reject in clifford + pyganja https://enkimute.github.io/ganja.js/examples/coffeeshop.html#pga2d_project_and_reject

project_reject.ipynb

{"cells":[{"metadata":{"trusted":true},"cell_type":"code","source":"from clifford import Cl\nfrom pyganja import *\nimport numpy as np\n\nlayout, blades = Cl(2, 0, 1, firstIdx=0)\nlocals().update(blades)\n\n# for shorter reprs\nlayout.__name__ = 'layout'\nlayout.__module__ = __name__\n\ndef point(x, y):\n    return (e0 + x*e1 + y*e2).dual()\n\ndef line(a, b, c):\n    return a*e1 + b*e2 + c*e0\n\ndef dist(x, y):\n     return abs(x & y)\n\ndef angle(x, y):\n    return np.arccos((x|y).value[0])\n\ndef project(a, b):\n    return (a|b)/b\n\ndef reject(a, b):\n    return (a|b)\n\n\n# Define some points and lines.\nA = point(1, 1)\nB = point(-1, 1)\nC = point(-1, -1)\nAC = A & C","execution_count":1,"outputs":[{"output_type":"stream","text":"/srv/conda/envs/notebook/lib/python3.7/site-packages/pyganja/__init__.py:2: UserWarning: Failed to import cef_gui, cef functions will be unavailable\n  from .script_api import *\n","name":"stderr"}]},{"metadata":{"trusted":true},"cell_type":"code","source":"from pyganja.script_api import _to_scene_string\n\ndef generate_notebook_js(scene, sig=None, *,\n                         default_color=Color.DEFAULT, default_static=False, **kwargs):\n    script_json = _to_scene_string(scene, default_color=default_color, default_static=default_static)\n    if sig is not None:\n        p = (sig > 0).sum().item()  # convert to json-compatible scalar\n        q = (sig < 0).sum().item()  # convert to json-compatible scalar\n        r = len(sig) - p - q\n    else:\n        p = 4\n        q = 1\n        r = 0\n    mv_length = str(2 ** (p + q + r))\n\n    # not the best way to test conformal, as it prevents non-euclidean  geometry\n    if q != 0:\n        conformal = True\n    else:\n        conformal = False\n\n    opts = dict(\n        p=p, q=q, r=r, mv_length=mv_length,\n        graph=dict(\n            conformal=conformal,\n            gl=True,\n            useUnnaturalLineDisplayForPointPairs=True,\n        )\n    )\n    if p - q == 2:\n        kwargs['gl'] = False  # 2d\n    opts[\"graph\"].update(kwargs)\n    js = \"\"\"\n    // We load ganja.js with requireJS, since `module.exports` / require\n    // only work if things are in separate files. In most situations\n    // `module` is already undefined, but in VSCode is is not (gh-46).\n    // Explicitly clearing it makes ganja.js fall back to RequireJS.\n    let module = undefined;\n    \"\"\"\n    js += read_ganja()\n    js += \"\"\"\n    // take a closure on element before the next cell replaces it\n    (function(element) {\n        (requirejs||require)(['Algebra'], function(Algebra) {\n            var opts = \"\"\" + json.dumps(opts) + \"\"\";  // injected from python\n            var output = Algebra({p: opts.p, q: opts.q, r: opts.r, baseType: Float64Array}).inline((opts)=>{\n                var data = \"\"\" + script_json + \"\"\";  // injected from python\n                // When we get a file, we load and display.\n                var canvas;\n                var h=0, p=0;\n                // convert arrays of floats back to CGA elements.\n                data = data.map(x=>x.length==opts.mv_length?new Element(x):x);\n                // add the graph to the page.\n                canvas = this.graph(data, opts.graph);\n                canvas.options.h = h;\n                canvas.options.p = p;\n                // make it big.\n                canvas.style.width = '100%';\n                canvas.style.height = '50vh';\n                return canvas;\n            })(opts);\n            element.append(output);\n            var a = document.createElement(\"button\");\n            var t = document.createTextNode(\"\\N{FLOPPY DISK} Save\");\n            a.appendChild(t);\n            function screenshot(){\n                //output.width = 1920;  output.height = 1080;\n                output.update(output.value);\n                output.toBlob(function(blob) {\n                    var url = URL.createObjectURL(blob);\n                    window.open(url, '_blank');\n                });\n            }\n            window.addEventListener('resize', function() {\n                output.update(output.value);\n            });\n            a.onclick = screenshot\n            var butnelem = element.append(a);\n        });\n    })(element);\n    \"\"\"\n    return Javascript(js)","execution_count":2,"outputs":[]},{"metadata":{"trusted":true},"cell_type":"code","source":"gs = GanjaScene()\ngs.add_facet([A, B, C], color=0xD0FFE1)\ngs.add_object(project(B, AC), label=\"project B onto AC\", color=0x882288)\ngs.add_object(project(AC, B), label=\"project AC onto B\", color=0x882288)\ngs.add_object(reject(AC, B), label=\"reject AC from B\", color=0x008844)\ngs.add_object(AC, label=\"AC\", color=0x00AA88)\ngs.add_object(A, label=\"A\", color=0x224488)\ngs.add_object(B, label=\"B\", color=0x224488)\ngs.add_object(C, label=\"C\", color=0x224488)\n\ngenerate_notebook_js(gs, sig=layout.sig, gl=False, grid=True, labels=True)","execution_count":3,"outputs":[{"output_type":"execute_result","execution_count":3,"data":{"text/plain":"<IPython.core.display.Javascript object>","application/javascript":"\n    // We load ganja.js with requireJS, since `module.exports` / require\n    // only work if things are in separate files. In most situations\n    // `module` is already undefined, but in VSCode is is not (gh-46).\n    // Explicitly clearing it makes ganja.js fall back to RequireJS.\n    let module = undefined;\n    /** Ganja.js - Geometric Algebra - Not Just Algebra.\n  * @author Enki\n  * @link   https://github.com/enkimute/ganja.js\n  */\n\n/*********************************************************************************************************************/\n//\n// Ganja.js is an Algebra generator for javascript. It generates a wide variety of Algebra's and supports operator\n// overloading, algebraic literals and a variety of graphing options.\n//\n// Ganja.js is designed with prototyping and educational purposes in mind. Clean mathematical syntax is the primary\n// target.\n//\n// Ganja.js exports only one function called *Algebra*. This function is used to generate Algebra classes. (say complex\n// numbers, minkowski or 3D CGA). The returned class can be used to create, add, multiply etc, but also to upgrade\n// javascript functions with algebraic literals, operator overloading, vectors, matrices and much more.\n//\n// As a simple example, multiplying two complex numbers 3+2i and 1+4i could be done like this :\n//\n//  var complex = Algebra(0,1);\n//  var a = new complex([3,2]);\n//  var b = new complex([1,3]);\n//  var result = a.Mul(b);\n//\n// But the same can be written using operator overloading and algebraic literals. (where scientific notation with\n// lowercase e is overloaded to directly specify generators (e1, e2, e12, ...))\n//\n//   var result = Algebra(0,1,()=>(3+2e1)*(1+4e1));\n//\n// Please see github for user documentation and examples.\n//\n/*********************************************************************************************************************/\n\n// Documentation below is for implementors. I'll assume you know about Clifford Algebra's, grades, its products, etc ..\n// I'll also assume you are familiar with ES6. My style may feel a bith mathematical, advise is to read slow.\n\n(function (name, context, definition) {\n  if (typeof module != 'undefined' && module.exports) module.exports = definition();\n  else if (typeof define == 'function' && define.amd) define(name, definition);\n  else context[name] = definition();\n}('Algebra', this, function () {\n\n/** The Algebra class generator. Possible calling signatures :\n  *   Algebra([func])                      => algebra with no dimensions, i.e. R. Optional function for the translator.\n  *   Algebra(p,[func])                    => 'p' positive dimensions and an optional function to pass to the translator.\n  *   Algebra(p,q,[func])                  => 'p' positive and 'q' negative dimensions and optional function.\n  *   Algebra(p,q,r,[func])                => 'p' positive, 'q' negative and 'r' zero dimensions and optional function.\n  *   Algebra({                            => for custom basis, cayley, mixing, etc pass in an object as first parameter.\n  *     [p:p],                             => optional 'p' for # of positive dimensions\n  *     [q:q],                             => optional 'q' for # of negative dimensions\n  *     [r:r],                             => optional 'r' for # of zero dimensions\n  *     [metric:array],                    => alternative for p,q,r. e.g. ([1,1,1,-1] for spacetime)\n  *     [basis:array],                     => array of strings with basis names. (e.g. ['1','e1','e2','e12'])\n  *     [Cayley:Cayley],                   => optional custom Cayley table (strings). (e.g. [['1','e1'],['e1','-1']])\n  *     [mix:boolean],                     => Allows mixing of various algebras. (for space efficiency).\n  *     [graded:boolean],                  => Use a graded algebra implementation. (automatic for +6D)\n  *     [baseType:Float32Array]            => optional basetype to use. (only for flat generator)\n  *   },[func])                            => optional function for the translator.\n **/\n  return function Algebra(p,q,r) {\n  // Resolve possible calling signatures so we know the numbers for p,q,r. Last argument can always be a function.\n    var fu=arguments[arguments.length-1],options=p; if (options instanceof Object) {\n      q = (p.q || (p.metric && p.metric.filter(x=>x==-1).length))| 0;\n      r = (p.r || (p.metric && p.metric.filter(x=>x==0).length)) | 0;\n      p = p.p === undefined ? (p.metric && p.metric.filter(x=>x==1).length) || 0 : p.p || 0;\n    } else { options={}; p=p|0; r=r|0; q=q|0; };\n\n  // Support for multi-dual-algebras\n    if (options.dual || (p==0 && q==0 && r<0)) { r=options.dual=options.dual||-r; // Create a dual number algebra if r<0 (old) or options.dual set(new)\n      options.basis  = [...Array(r+1)].map((a,i)=>i?'e0'+i:'1');  options.metric = [1,...Array(r)]; options.tot=r+1;\n      options.Cayley = [...Array(r+1)].map((a,i)=>[...Array(r+1)].map((y,j)=>i*j==0?((i+j)?'e0'+(i+j):'1'):'0'));\n    }\n    if (options.over) options.baseType = Array;\n\n  // Calculate the total number of dimensions.\n    var tot = options.tot = (options.tot||(p||0)+(q||0)+(r||0)||(options.basis&&options.basis.length))|0;\n\n  // Unless specified, generate a full set of Clifford basis names. We generate them as an array of strings by starting\n  // from numbers in binary representation and changing the set bits into their relative position.\n  // Basis names are ordered first per grade, then lexically (not cyclic!).\n  // For 10 or more dimensions all names will be double digits ! 1e01 instead of 1e1 ..\n    var basis=(options.basis&&(options.basis.length==2**tot||r<0||options.Cayley)&&options.basis)||[...Array(2**tot)]           // => [undefined, undefined, undefined, undefined, undefined, undefined, undefined, undefined]\n              .map((x,xi)=>(((1<<30)+xi).toString(2)).slice(-tot||-1)                                                           // => [\"000\", \"001\", \"010\", \"011\", \"100\", \"101\", \"110\", \"111\"]  (index of array in base 2)\n              .replace(/./g,(a,ai)=>a=='0'?'':String.fromCharCode(66+ai-(r!=0))))                                               // => [\"\", \"3\", \"2\", \"23\", \"1\", \"13\", \"12\", \"123\"] (1 bits replaced with their positions, 0's removed)\n              .sort((a,b)=>(a.toString().length==b.toString().length)?(a>b?1:b>a?-1:0):a.toString().length-b.toString().length) // => [\"\", \"1\", \"2\", \"3\", \"12\", \"13\", \"23\", \"123\"] (sorted numerically)\n              .map(x=>x&&'e'+(x.replace(/./g,x=>('0'+(x.charCodeAt(0)-65)).slice(tot>9?-2:-1) ))||'1')                          // => [\"1\", \"e1\", \"e2\", \"e3\", \"e12\", \"e13\", \"e23\", \"e123\"] (converted to commonly used basis names)\n\n  // See if the basis names start from 0 or 1, store grade per component and lowest component per grade.\n    var low=basis.length==1?1:basis[1].match(/\\d+/g)[0]*1,\n        grades=options.grades||(options.dual&&basis.map((x,i)=>i?1:0))||basis.map(x=>tot>9?(x.length-1)/2:x.length-1),\n        grade_start=grades.map((a,b,c)=>c[b-1]!=a?b:-1).filter(x=>x+1).concat([basis.length]);\n\n  // String-simplify a concatenation of two basis blades. (and supports custom basis names e.g. e21 instead of e12)\n  // This is the function that implements e1e1 = +1/-1/0 and e1e2=-e2e1. The brm function creates the remap dictionary.\n    var simplify = (s,p,q,r)=>{\n          var sign=1,c,l,t=[],f=true,ss=s.match(tot>9?/(\\d\\d)/g:/(\\d)/g);if (!ss) return s; s=ss; l=s.length;\n          while (f) { f=false;\n          // implement Ex*Ex = metric.\n            for (var i=0; i<l;) if (s[i]===s[i+1]) { if (options.metric) sign*=options.metric[s[i]]; else if ((s[i]-low)>=(p+r)) sign*=-1; else if ((s[i]-low)<r) sign=0;i+=2; f=true; } else t.push(s[i++]);\n          // implement Ex*Ey = -Ey*Ex while sorting basis vectors.\n            for (var i=0; i<t.length-1; i++) if (t[i]>t[i+1]) { c=t[i];t[i]=t[i+1];t[i+1]=c;sign*=-1;f=true; break;} if (f) { s=t;t=[];l=s.length; }\n          }\n          var ret=(sign==0)?'0':((sign==1)?'':'-')+(t.length?'e'+t.join(''):'1'); return (brm&&brm[ret])||(brm&&brm['-'+ret]&&'-'+brm['-'+ret])||ret;\n        },\n        brm=(x=>{ var ret={}; for (var i in basis) ret[basis[i]=='1'?'1':simplify(basis[i],p,q,r)] = basis[i]; return ret; })(basis);\n\n  // As an alternative to the string fiddling, one can also bit-fiddle. In this case the basisvectors are represented by integers with 1 bit per generator set.\n    var simplify_bits = (A,B,p2)=>{ var n=p2||(p+q+r),t=0,ab=A&B,res=A^B; if (ab&((1<<r)-1)) return [0,0]; while (n--) t^=(A=A>>1); t&=B; t^=ab>>(p+r); t^=t>>16; t^=t>>8; t^=t>>4; return [1-2*(27030>>(t&15)&1),res]; },\n        bc = (v)=>{ v=v-((v>>1)& 0x55555555); v=(v&0x33333333)+((v>>2)&0x33333333); c=((v+(v>>4)&0xF0F0F0F)*0x1010101)>>24; return c };\n\n  if (!options.graded && tot <= 6 || options.graded===false || options.Cayley) {\n  // Faster and degenerate-metric-resistant dualization. (a remapping table that maps items into their duals).\n    var drm=basis.map((a,i)=>{ return {a:a,i:i} })\n                 .sort((a,b)=>a.a.length>b.a.length?1:a.a.length<b.a.length?-1:(+a.a.slice(1).split('').sort().join(''))-(+b.a.slice(1).split('').sort().join('')) )\n                 .map(x=>x.i).reverse(),\n        drms=drm.map((x,i)=>(x==0||i==0)?1:simplify(basis[x]+basis[i])[0]=='-'?-1:1);\n\n  /// Store the full metric (also for bivectors etc ..)\n    var metric = options.Cayley&&options.Cayley.map((x,i)=>x[i]) || basis.map((x,xi)=>simplify(x+x,p,q,r)|0);\n\n  /// Generate multiplication tables for the outer and geometric products.\n    var mulTable   = options.Cayley||basis.map(x=>basis.map(y=>(x==1)?y:(y==1)?x:simplify(x+y,p,q,r)));\n\n  // subalgebra support. (must be bit-order basis blades, does no error checking.)\n    if (options.even) options.basis = basis.filter(x=>x.length%2==1);\n    if (options.basis && !options.Cayley && r>=0 && options.basis.length != 2**tot) {\n      metric = metric.filter((x,i)=>options.basis.indexOf(basis[i])!=-1);\n      mulTable = mulTable.filter((x,i)=>options.basis.indexOf(basis[i])!=-1).map(x=>x.filter((x,i)=>options.basis.indexOf(basis[i])!=-1));\n      basis  = options.basis;\n    }\n\n  /// Convert Cayley table to product matrices. The outer product selects the strict sum of the GP (but without metric), the inner product\n  /// is the left contraction.\n    var gp=basis.map(x=>basis.map(x=>'0')), cp=gp.map(x=>gp.map(x=>'0')), cps=gp.map(x=>gp.map(x=>'0')), op=gp.map(x=>gp.map(x=>'0')), gpo={};          // Storage for our product tables.\n    basis.forEach((x,xi)=>basis.forEach((y,yi)=>{ var n = mulTable[xi][yi].replace(/^-/,''); if (!gpo[n]) gpo[n]=[]; gpo[n].push([xi,yi]); }));\n    basis.forEach((o,oi)=>{\n      gpo[o].forEach(([xi,yi])=>op[oi][xi]=(grades[oi]==grades[xi]+grades[yi])?((mulTable[xi][yi]=='0')?'0':((mulTable[xi][yi][0]!='-')?'':'-')+'b['+yi+']*this['+xi+']'):'0');\n      gpo[o].forEach(([xi,yi])=>{\n        gp[oi][xi] =((gp[oi][xi]=='0')?'':gp[oi][xi]+'+')   + ((mulTable[xi][yi]=='0')?'0':((mulTable[xi][yi][0]!='-')?'':'-')+'b['+yi+']*this['+xi+']');\n        cp[oi][xi] =((cp[oi][xi]=='0')?'':cp[oi][xi]+'+')   + ((grades[oi]==grades[yi]-grades[xi])?gp[oi][xi]:'0');\n        cps[oi][xi]=((cps[oi][xi]=='0')?'':cps[oi][xi]+'+') + ((grades[oi]==Math.abs(grades[yi]-grades[xi]))?gp[oi][xi]:'0');\n      });\n    });\n\n  /// Flat Algebra Multivector Base Class.\n    var generator = class MultiVector extends (options.baseType||Float32Array) {\n    /// constructor - create a floating point array with the correct number of coefficients.\n      constructor(a) { super(a||basis.length); return this; }\n\n    /// grade selection - return a only the part of the input with the specified grade.\n      Grade(grade,res) { res=res||new this.constructor(); for (var i=0,l=res.length; i<l; i++) if (grades[i]==grade) res[i]=this[i]; else res[i]=0; return res; }\n      Even(res) { res=res||new this.constructor(); for (var i=0,l=res.length; i<l; i++) if (grades[i]%2==0) res[i]=this[i]; else res[i]=0; return res; }\n\n    /// grade creation - convert array with just one grade to full multivector.\n      nVector(grade,...args) { this.set(args,grade_start[grade]); return this; }\n\n    /// Fill in coordinates (accepts sequence of index,value as arguments)\n      Coeff() { for (var i=0,l=arguments.length; i<l; i+=2) this[arguments[i]]=arguments[i+1]; return this; }\n\n    /// Negates specific grades (passed in as args)\n      Map(res, ...a) { for (var i=0, l=res.length; i<l; i++) res[i] = (~a.indexOf(grades[i]))?-this[i]:this[i]; return res; }\n\n    /// Returns the vector grade only.\n      get Vector ()    { return this.slice(grade_start[1],grade_start[2]); };\n\n      toString() { var res=[]; for (var i=0; i<basis.length; i++) if (Math.abs(this[i])>1e-10) res.push(((this[i]==1)&&i?'':((this[i]==-1)&&i)?'-':(this[i].toFixed(10)*1))+(i==0?'':tot==1&&q==1?'i':basis[i].replace('e','e_'))); return res.join('+').replace(/\\+-/g,'-')||'0'; }\n\n    /// Reversion, Involutions, Conjugation for any number of grades, component acces shortcuts.\n      get Negative (){ var res = new this.constructor(); for (var i=0; i<this.length; i++) res[i]= -this[i]; return res; };\n      get Reverse (){ var res = new this.constructor(); for (var i=0; i<this.length; i++) res[i]= this[i]*[1,1,-1,-1][grades[i]%4]; return res; };\n      get Involute (){ var res = new this.constructor(); for (var i=0; i<this.length; i++) res[i]= this[i]*[1,-1,1,-1][grades[i]%4]; return res; };\n      get Conjugate (){ var res = new this.constructor(); for (var i=0; i<this.length; i++) res[i]= this[i]*[1,-1,-1,1][grades[i]%4]; return res; };\n\n    /// The Dual, Length, non-metric length and normalized getters.\n      get Dual (){ if (r) return this.map((x,i,a)=>a[drm[i]]*drms[i]); var res = new this.constructor(); res[res.length-1]=1; return res.Mul(this); };\n      get Length (){ return options.over?Math.sqrt(Math.abs(this.Mul(this.Conjugate).s.s)):Math.sqrt(Math.abs(this.Mul(this.Conjugate).s)); };\n      get VLength (){ var res = 0; for (var i=0; i<this.length; i++) res += this[i]*this[i]; return Math.sqrt(res); };\n      get Normalized (){ var res = new this.constructor(),l=this.Length; if (!l) return this; l=1/l; for (var i=0; i<this.length; i++) if (options.over) {res[i]=this[i].Scale(l);} else {res[i]=this[i]*l}; return res; };\n    }\n\n  /// Convert symbolic matrices to code. (skipping zero's on dot and wedge matrices).\n  /// These all do straightforward string fiddling. If the 'mix' option is set they reference basis components using e.g. '.e1' instead of eg '[3]' .. so that\n  /// it will work for elements of subalgebras etc.\n    generator.prototype.Add   = new Function('b,res','res=res||new this.constructor();\\n'+basis.map((x,xi)=>'res['+xi+']=b['+xi+']+this['+xi+']').join(';\\n').replace(/(b|this)\\[(.*?)\\]/g,(a,b,c)=>options.mix?'('+b+'.'+(c|0?basis[c]:'s')+'||0)':a)+';\\nreturn res')\n    generator.prototype.Scale = new Function('b,res','res=res||new this.constructor();\\n'+basis.map((x,xi)=>'res['+xi+']=b*this['+xi+']').join(';\\n').replace(/(b|this)\\[(.*?)\\]/g,(a,b,c)=>options.mix?'('+b+'.'+(c|0?basis[c]:'s')+'||0)':a)+';\\nreturn res')\n    generator.prototype.Sub   = new Function('b,res','res=res||new this.constructor();\\n'+basis.map((x,xi)=>'res['+xi+']=this['+xi+']-b['+xi+']').join(';\\n').replace(/(b|this)\\[(.*?)\\]/g,(a,b,c)=>options.mix?'('+b+'.'+(c|0?basis[c]:'s')+'||0)':a)+';\\nreturn res')\n    generator.prototype.Mul   = new Function('b,res','res=res||new this.constructor();\\n'+gp.map((r,ri)=>'res['+ri+']='+r.join('+').replace(/\\+\\-/g,'-').replace(/(\\w*?)\\[(.*?)\\]/g,(a,b,c)=>options.mix?'('+b+'.'+(c|0?basis[c]:'s')+'||0)':a).replace(/\\+0/g,'')+';').join('\\n')+'\\nreturn res;');\n    generator.prototype.LDot  = new Function('b,res','res=res||new this.constructor();\\n'+cp.map((r,ri)=>'res['+ri+']='+r.join('+').replace(/\\+\\-/g,'-').replace(/\\+0/g,'').replace(/(\\w*?)\\[(.*?)\\]/g,(a,b,c)=>options.mix?'('+b+'.'+(c|0?basis[c]:'s')+'||0)':a)+';').join('\\n')+'\\nreturn res;');\n    generator.prototype.Dot   = new Function('b,res','res=res||new this.constructor();\\n'+cps.map((r,ri)=>'res['+ri+']='+r.join('+').replace(/\\+\\-/g,'-').replace(/\\+0/g,'').replace(/(\\w*?)\\[(.*?)\\]/g,(a,b,c)=>options.mix?'('+b+'.'+(c|0?basis[c]:'s')+'||0)':a)+';').join('\\n')+'\\nreturn res;');\n    generator.prototype.Wedge = new Function('b,res','res=res||new this.constructor();\\n'+op.map((r,ri)=>'res['+ri+']='+r.join('+').replace(/\\+\\-/g,'-').replace(/\\+0/g,'').replace(/(\\w*?)\\[(.*?)\\]/g,(a,b,c)=>options.mix?'('+b+'.'+(c|0?basis[c]:'s')+'||0)':a)+';').join('\\n')+'\\nreturn res;');\n    generator.prototype.Vee   = new Function('b,res','res=res||new this.constructor();\\n'+op.map((r,ri)=>'res['+drm[ri]+']='+r.map(x=>x.replace(/\\[(.*?)\\]/g,function(a,b){return '['+(drm[b|0])+']'})).join('+').replace(/\\+\\-/g,'-').replace(/\\+0/g,'').replace(/(\\w*?)\\[(.*?)\\]/g,(a,b,c)=>options.mix?'('+b+'.'+(c|0?basis[c]:'s')+'||0)':a)+';').join('\\n')+'\\nreturn res;');\n\n  /// Add getter and setters for the basis vectors/bivectors etc ..\n    basis.forEach((b,i)=>Object.defineProperty(generator.prototype, i?b:'s', {\n      configurable: true, get(){ return this[i] }, set(x){ this[i]=x; }\n    }));\n\n  /// Graded generator for high-dimensional algebras.\n  } else {\n\n  /// extra graded lookups.\n    var basisg = grade_start.slice(0,grade_start.length-1).map((x,i)=>basis.slice(x,grade_start[i+1]));\n    var counts = grade_start.map((x,i,a)=>i==a.length-1?0:a[i+1]-x).slice(0,tot+1);\n    var basis_bits = basis.map(x=>x=='1'?0:x.slice(1).match(tot>9?/\\d\\d/g:/\\d/g).reduce((a,b)=>a+(1<<(b-low)),0)),\n        bits_basis = []; basis_bits.forEach((b,i)=>bits_basis[b]=i);\n    var metric = basisg.map((x,xi)=>x.map((y,yi)=>simplify_bits(basis_bits[grade_start[xi]+yi],basis_bits[grade_start[xi]+yi])[0]));\n    var drms   = basisg.map((x,xi)=>x.map((y,yi)=>simplify_bits(basis_bits[grade_start[xi]+yi],(~basis_bits[grade_start[xi]+yi])&((1<<tot)-1))[0]));\n\n  /// Flat Algebra Multivector Base Class.\n    var generator = class MultiVector extends Array {\n    /// constructor - create a floating point array with the correct number of coefficients.\n      constructor(a) { super(a||tot); return this; }\n\n    /// grade selection - return a only the part of the input with the specified grade.\n      Grade(grade,res) { res=new this.constructor(); res[grade] = this[grade]; return res; }\n\n    /// grade creation - convert array with just one grade to full multivector.\n      nVector(grade,...args) { this[grade]=args; return this; }\n\n    /// Fill in coordinates (accepts sequence of index,value as arguments)\n      Coeff() {\n                for (var i=0,l=arguments.length; i<l; i+=2) if (arguments[i+1]) {\n                 var gi = grades[arguments[i]];\n                 if (this[gi]==undefined) this[gi]=[];\n                 this[gi][arguments[i]-grade_start[gi]]=arguments[i+1];\n                }\n                return this;\n              }\n\n    /// Negates specific grades (passed in as args)\n      Map(res, ...a) {  /* tbc */ }\n\n    /// Returns the vector grade only.\n      get Vector ()    { return this[1] };\n\n    /// multivector addition, subtraction and scalar multiplication.\n      Add(b,r) {\n        r=r||new this.constructor();\n        for (var i=0,l=Math.max(this.length,b.length);i<l;i++)\n          if (!this[i] ^ !b[i]) r[i] = (!this[i]) ? b[i].slice():this[i].slice();\n          else if (!(this[i]||b[i])) {}\n          else { if (r[i]==undefined) r[i]=[]; for(var j=0,m=Math.max(this[i].length,b[i].length);j<m;j++)\n          {\n            if (typeof this[i][j]==\"string\" || typeof r[i][j]==\"string\" || typeof b[i][j]==\"string\") {\n             if (!this[i][j]) r[i][j] = \"\"+b[i][j];\n             else if (!b[i][j]) r[i][j] = \"\"+this[i][j];\n             else r[i][j]=\"(\"+(this[i][j]||\"0\")+(b[i][j][0]==\"-\"?\"\":\"+\")+(b[i][j]||\"0\")+\")\";\n            } else r[i][j]=(this[i][j]||0)+(b[i][j]||0);\n          }}\n        return r;\n      }\n      Sub(b,r) {\n        r=r||new this.constructor();\n        for (var i=0,l=Math.max(this.length,b.length);i<l;i++)\n          if (!this[i] || !b[i]) r[i] = (!this[i]) ? (b[i]?b[i].map(x=>(typeof x==\"string\")?\"-\"+x:-x):undefined):this[i];\n          else { if (r[i]==undefined) r[i]=[]; for(var j=0,m=Math.max(this[i].length,b[i].length);j<m;j++)\n            if (typeof this[i][j]==\"string\" || typeof r[i][j]==\"string\" || typeof b[i][j]==\"string\") r[i][j]=\"(\"+(this[i][j]||\"0\")+\"-\"+(b[i][j]||\"0\")+\")\";\n            else r[i][j]=(this[i][j]||0)-(b[i][j]||0);\n          }\n        return r;\n      }\n      Scale(s) { return this.map(x=>x&&x.map(y=>typeof y==\"string\"?y+\"*\"+s:y*s)); }\n\n    // geometric product.\n      Mul(b,r) {\n        r=r||new this.constructor(); var gotstring=false;\n        for (var i=0,x,gsx; gsx=grade_start[i],x=this[i],i<this.length; i++) if (x) for (var j=0,y,gsy;gsy=grade_start[j],y=b[j],j<b.length; j++) if (y) for (var a=0; a<x.length; a++) if (x[a]) for (var bb=0; bb<y.length; bb++) if (y[bb]) {\n          if (i==j && a==bb) { r[0] = r[0]||(typeof x[0]==\"string\" || typeof y[bb]==\"string\"?[\"\"]:[0]);\n            if (typeof x[a]==\"string\" || typeof r[0][0]==\"string\" || typeof y[bb]==\"string\") {\n            r[0][0] = (r[0][0]?(r[0][0]+(x[a][0]==\"-\"?\"\":\"+\")):\"\")+ x[a]+\"*\"+y[bb]+(metric[i][a]!=1?\"*\"+metric[i][a]:\"\");  gotstring=true;\n            } else r[0][0] += x[a]*y[bb]*metric[i][a];\n          } else {\n             var rn=simplify_bits(basis_bits[gsx+a],basis_bits[gsy+bb]), g=bc(rn[1]), e=bits_basis[rn[1]]-grade_start[g];\n             if (!r[g])r[g]=[];\n               if (typeof r[g][e]==\"string\"||typeof x[a]==\"string\"||typeof y[bb]==\"string\") {\n                 r[g][e] = (r[g][e]?r[g][e]+\"+\":\"\") + (rn[0]!=1?rn[0]+\"*\":\"\")+ x[a]+(y[bb]!=1?\"*\"+y[bb]:\"\"); gotstring=true;\n               } else r[g][e] = (r[g][e]||0) + rn[0]*x[a]*y[bb];\n          }\n        }\n        if (gotstring) return r.map(g=>g.map(e=>e&&'('+e+')'))\n        return r;\n      }\n    // outer product.\n      Wedge(b,r) {\n        r=r||new this.constructor();\n        for (var i=0,x,gsx; gsx=grade_start[i],x=this[i],i<this.length; i++) if (x) for (var j=0,y,gsy;gsy=grade_start[j],y=b[j],j<b.length; j++) if (y) for (var a=0; a<x.length; a++) if (x[a]) for (var bb=0; bb<y.length; bb++) if (y[bb]) {\n          if (i!=j || a!=bb) {\n             var n1=basis_bits[gsx+a], n2=basis_bits[gsy+bb], rn=simplify_bits(n1,n2,tot), g=bc(rn[1]), e=bits_basis[rn[1]]-grade_start[g];\n             if (g == i+j) { if (!r[g]) r[g]=[]; r[g][e] = (r[g][e]||0) + rn[0]*x[a]*y[bb]; }\n          }\n        }\n        return r;\n      }\n    // outer product glsl output.\n      OPNS_GLSL(b,point_source) {\n        var r='',count=0,curg;\n        for (var i=0,x,gsx; gsx=grade_start[i],x=this[i],i<this.length; i++) if (x) for (var j=0,y,gsy;gsy=grade_start[j],y=b[j],j<b.length; j++) if (y) for (var a=0; a<counts[i]; a++) for (var bb=0; bb<counts[j]; bb++) {\n          if (i!=j || a!=bb) {\n             var n1=basis_bits[gsx+a], n2=basis_bits[gsy+bb], rn=simplify_bits(n1,n2,tot), g=bc(rn[1]), e=bits_basis[rn[1]]-grade_start[g];\n             if (g == i+j) { curg=g; r += `res[${e}]${rn[0]=='1'?\"+=\":\"-=\"}(${point_source[a]})*b[${bb}]; //${count++}\\n`;  }\n          }\n        }\n        r=r.split('\\n').filter(x=>x).sort((a,b)=>((a.match(/\\d+/)[0]|0)-(b.match(/\\d+/)[0]|0))||((a.match(/\\d+$/)[0]|0)-(b.match(/\\d+$/)[0]|0))).map(x=>x.replace(/\\/\\/\\d+$/,''));\n        var r2 = 'float sum=0.0; float res=0.0;\\n', g=0;\n        r.forEach(x=>{\n          var cg = x.match(/\\d+/)[0]|0;\n          if (cg != g) r2 += \"sum \"+(((metric[curg][g]==-1))?\"-=\":\"+=\")+\" res*res;\\nres = 0.0;\\n\";\n          r2 += x.replace(/\\[\\d+\\]/,'') + '\\n';\n          g=cg;\n        });\n        r2+= \"sum \"+((metric[curg][g]==-1)?\"-=\":\"+=\")+\" res*res;\\n\";\n        return r2;\n      }\n    // Left contraction.\n      LDot(b,r) {\n        r=r||new this.constructor();\n        for (var i=0,x,gsx; gsx=grade_start[i],x=this[i],i<this.length; i++) if (x) for (var j=0,y,gsy;gsy=grade_start[j],y=b[j],j<b.length; j++) if (y) for (var a=0; a<x.length; a++) if (x[a]) for (var bb=0; bb<y.length; bb++) if (y[bb]) {\n          if (i==j && a==bb) { r[0] = r[0]||[0]; r[0][0] += x[a]*y[bb]*metric[i][a]; }\n          else {\n             var rn=simplify_bits(basis_bits[gsx+a],basis_bits[gsy+bb]), g=bc(rn[1]), e=bits_basis[rn[1]]-grade_start[g];\n             if (g == j-i) { if (!r[g])r[g]=[]; r[g][e] = (r[g][e]||0) + rn[0]*x[a]*y[bb]; }\n          }\n        }\n        return r;\n      }\n    // Symmetric contraction.\n      Dot(b,r) {\n        r=r||new this.constructor();\n        for (var i=0,x,gsx; gsx=grade_start[i],x=this[i],i<this.length; i++) if (x) for (var j=0,y,gsy;gsy=grade_start[j],y=b[j],j<b.length; j++) if (y) for (var a=0; a<x.length; a++) if (x[a]) for (var bb=0; bb<y.length; bb++) if (y[bb]) {\n          if (i==j && a==bb) { r[0] = r[0]||[0]; r[0][0] += x[a]*y[bb]*metric[i][a]; }\n          else {\n             var rn=simplify_bits(basis_bits[gsx+a],basis_bits[gsy+bb]), g=bc(rn[1]), e=bits_basis[rn[1]]-grade_start[g];\n             if (g == Math.abs(j-i)) { if (!r[g])r[g]=[]; r[g][e] = (r[g][e]||0) + rn[0]*x[a]*y[bb]; }\n          }\n        }\n        return r;\n      }\n    // Should be optimized..\n      Vee(b,r)          { return (this.Dual.Wedge(b.Dual)).Dual; }\n    // Output, lengths, involutions, normalized, dual.\n      toString()        { return [...this].map((g,gi)=>g&&g.map((c,ci)=>!c?undefined:c+basisg[gi][ci]).filter(x=>x).join('+')).filter(x=>x).join('+').replace(/\\+\\-/g,'-'); }\n      get s ()          { if (this[0]) return this[0][0]||0; return 0; }\n      get Length ()     { var res=0; this.forEach((g,gi)=>g&&g.forEach((e,ei)=>res+=(e||0)**2*metric[gi][ei])); return Math.abs(res)**.5; }\n      get VLength ()    { var res=0; this.forEach((g,gi)=>g&&g.forEach((e,ei)=>res+=(e||0)**2)); return Math.abs(res)**.5; }\n      get Reverse ()    { var r=new this.constructor(); this.forEach((x,gi)=>x&&x.forEach((e,ei)=>{if(!r[gi])r[gi]=[]; r[gi][ei] = this[gi][ei]*[1,1,-1,-1][gi%4]; })); return r; }\n      get Involute ()   { var r=new this.constructor(); this.forEach((x,gi)=>x&&x.forEach((e,ei)=>{if(!r[gi])r[gi]=[]; r[gi][ei] = this[gi][ei]*[1,-1,1,-1][gi%4]; })); return r; }\n      get Conjugate ()  { var r=new this.constructor(); this.forEach((x,gi)=>x&&x.forEach((e,ei)=>{if(!r[gi])r[gi]=[]; r[gi][ei] = this[gi][ei]*[1,-1,-1,1][gi%4]; })); return r; }\n      get Dual()        { var r=new this.constructor(); this.forEach((g,gi)=>{ if (!g) return; r[tot-gi]=[]; g.forEach((e,ei)=>r[tot-gi][counts[gi]-1-ei]=drms[gi][ei]*e); }); return r; }\n      get Normalized () { return this.Scale(1/this.Length); }\n    }\n\n\n   // This generator is UNDER DEVELOPMENT - I'm publishing it so I can test on observable.\n  }\n\n  // Generate a new class for our algebra. It extends the javascript typed arrays (default float32 but can be specified in options).\n    var res = class Element extends generator {\n\n    // constructor - create a floating point array with the correct number of coefficients.\n      constructor(a) { super(a); if (this.upgrade) this.upgrade(); return this; }\n\n    // Grade selection. (implemented by parent class).\n      Grade(grade,res) { res=res||new Element(); return super.Grade(grade,res); }\n\n    // Right and Left divide - Defined on the elements, shortcuts to multiplying with the inverse.\n      Div  (b,res) { return this.Mul(b.Inverse,res); }\n      LDiv (b,res) { return b.Inverse.Mul(this,res); }\n\n    // Taylor exp - for PGA bivectors in 2D and 3D closed form solution is used.\n      Exp  ()      {\n        if (options.dual) { var f=Math.exp(this.s); return this.map((x,i)=>i?x*f:f); }\n        if (r==1 && tot<=4 && Math.abs(this[0])<1E-9 && !options.over) {\n           var u = Math.sqrt(Math.abs(this.Dot(this).s)); if (Math.abs(u)<1E-5) return this.Add(Element.Scalar(1));\n           var v = this.Wedge(this).Scale(-1/(2*u)); \n           var res2 = Element.Add(Element.Sub(Math.cos(u),v.Scale(Math.sin(u))),Element.Div(Element.Mul((Element.Add(Math.sin(u),v.Scale(Math.cos(u)))),this),(Element.Add(u,v))));\n           return res2;       \n        }\n        var res = Element.Scalar(1), y=1, M= this.Scale(1), N=this.Scale(1); for (var x=1; x<15; x++) { res=res.Add(M.Scale(1/y)); M=M.Mul(N); y=y*(x+1); }; return res;\n      }\n      \n    // Log - only for up to 3D PGA for now\n      Log () {\n        if (r!=1 || tot>4 || options.over) return;\n        var b = this.Grade(2), bdb = Element.Dot(b,b);\n        if (Math.abs(bdb.s)<=1E-5) return this.s<0?b.Scale(-1):b;\n        var s = Math.sqrt(-bdb), bwb = Element.Wedge(b,b); \n        if (Math.abs(bwb.e0123)<=1E-5) return b.Scale(Math.atan2(s,this.s)/s);\n        var p = bwb.Scale(-1/(2*s));\n        return Element.Div(Element.Mul(Element.Mul((Element.Add(Math.atan2(s,this.s),Element.Div(p,this.s))),b),(Element.Sub(s,p))),(Element.Mul(s,s)));        \n      }  \n\n    // Helper for efficient inverses. (custom involutions - negates grades in arguments).\n      Map () { var res=new Element(); return super.Map(res,...arguments); }\n\n    // Factories - Make it easy to generate vectors, bivectors, etc when using the functional API. None of the examples use this but\n    // users that have used other GA libraries will expect these calls. The Coeff() is used internally when translating algebraic literals.\n      static Element()   { return new Element([...arguments]); };\n      static Coeff()     { return (new Element()).Coeff(...arguments); }\n      static Scalar(x)   { return (new Element()).Coeff(0,x); }\n      static Vector()    { return (new Element()).nVector(1,...arguments); }\n      static Bivector()  { return (new Element()).nVector(2,...arguments); }\n      static Trivector() { return (new Element()).nVector(3,...arguments); }\n      static nVector(n)  { return (new Element()).nVector(...arguments); }\n\n    // Static operators. The parser will always translate operators to these static calls so that scalars, vectors, matrices and other non-multivectors can also be handled.\n    // The static operators typically handle functions and matrices, calling through to element methods for multivectors. They are intended to be flexible and allow as many\n    // types of arguments as possible. If performance is a consideration, one should use the generated element methods instead. (which only accept multivector arguments)\n      static toEl(x)        { if (x instanceof Function) x=x(); if (!(x instanceof Element)) x=Element.Scalar(x); return x; }\n\n    // Addition and subtraction. Subtraction with only one parameter is negation.\n      static Add(a,b,res)   {\n      // Resolve expressions passed in.\n        while(a.call)a=a(); while(b.call)b=b(); if (a.Add && b.Add) return a.Add(b,res);\n      // If either is a string, the result is a string.\n        if ((typeof a=='string')||(typeof b=='string')) return a.toString()+b.toString();\n      // If only one is an array, add the other element to each of the elements.\n        if ((a instanceof Array && !a.Add)^(b instanceof Array && !b.Add)) return (a instanceof Array)?a.map(x=>Element.Add(x,b)):b.map(x=>Element.Add(a,x));\n      // If both are equal length arrays, add elements one-by-one\n        if ((a instanceof Array)&&(b instanceof Array)&&a.length==b.length) return a.map((x,xi)=>Element.Add(x,b[xi]));\n      // If they're both not elements let javascript resolve it.\n        if (!(a instanceof Element || b instanceof Element)) return a+b;\n      // Here we're left with scalars and multivectors, call through to generated code.\n        a=Element.toEl(a); b=Element.toEl(b); return a.Add(b,res);\n      }\n\n      static Sub(a,b,res)   {\n      // Resolve expressions passed in.\n        while(a.call)a=a(); while(b&&b.call) b=b(); if (a.Sub && b && b.Sub) return a.Sub(b,res);\n      // If only one is an array, add the other element to each of the elements.\n        if (b&&((a instanceof Array)^(b instanceof Array))) return (a instanceof Array)?a.map(x=>Element.Sub(x,b)):b.map(x=>Element.Sub(a,x));\n      // If both are equal length arrays, add elements one-by-one\n        if (b&&(a instanceof Array)&&(b instanceof Array)&&a.length==b.length) return a.map((x,xi)=>Element.Sub(x,b[xi]));\n      // Negation\n        if (arguments.length==1) return Element.Mul(a,-1);\n      // If none are elements here, let js do it.\n        if (!(a instanceof Element || b instanceof Element)) return a-b;\n      // Here we're left with scalars and multivectors, call through to generated code.\n        a=Element.toEl(a); b=Element.toEl(b); return a.Sub(b,res);\n      }\n\n    // The geometric product. (or matrix*matrix, matrix*vector, vector*vector product if called with 1D and 2D arrays)\n      static Mul(a,b,res)   {\n      // Resolve expressions\n        while(a.call&&!a.length)a=a(); while(b.call&&!b.length)b=b(); if (a.Mul && b.Mul) return a.Mul(b,res);\n      // still functions -> experimental curry style (dont use this.)\n        if (a.call && b.call) return (ai,bi)=>Element.Mul(a(ai),b(bi));\n      // scalar mul.\n        if (Number.isFinite(a) && b.Scale) return b.Scale(a); else if (Number.isFinite(b) && a.Scale) return a.Scale(b);\n      // Handle matrices and vectors.\n        if ((a instanceof Array)&&(b instanceof Array)) {\n        // vector times vector performs a dot product. (which internally uses the GP on each component)\n          if((!(a[0] instanceof Array) || (a[0] instanceof Element)) &&(!(b[0] instanceof Array) || (b[0] instanceof Element))) { var r=tot?Element.Scalar(0):0; a.forEach((x,i)=>r=Element.Add(r,Element.Mul(x,b[i]),r)); return r; }\n        // Array times vector\n          if(!(b[0] instanceof Array)) return a.map((x,i)=>Element.Mul(a[i],b));\n        // Array times Array\n          var r=a.map((x,i)=>b[0].map((y,j)=>{ var r=tot?Element.Scalar(0):0; x.forEach((xa,k)=>r=Element.Add(r,Element.Mul(xa,b[k][j]))); return r; }));\n        // Return resulting array or scalar if 1 by 1.\n          if (r.length==1 && r[0].length==1) return r[0][0]; else return r;\n        }\n      // Only one is an array multiply each of its elements with the other.\n        if ((a instanceof Array)^(b instanceof Array)) return (a instanceof Array)?a.map(x=>Element.Mul(x,b)):b.map(x=>Element.Mul(a,x));\n      // Try js multiplication, else call through to geometric product.\n        var r=a*b; if (!isNaN(r)) return r;\n        a=Element.toEl(a); b=Element.toEl(b); return a.Mul(b,res);\n      }\n\n    // The inner product. (default is left contraction).\n      static LDot(a,b,res)   {\n      // Expressions\n        while(a.call)a=a(); while(b.call)b=b(); //if (a.LDot) return a.LDot(b,res);\n      // Map elements in array\n        if (b instanceof Array && !(a instanceof Array)) return b.map(x=>Element.LDot(a,x));\n        if (a instanceof Array && !(b instanceof Array)) return a.map(x=>Element.LDot(x,b));\n      // js if numbers, else contraction product.\n        if (!(a instanceof Element || b instanceof Element)) return a*b;\n        a=Element.toEl(a);b=Element.toEl(b); return a.LDot(b,res);\n      }\n\n    // The symmetric inner product. (default is left contraction).\n      static Dot(a,b,res)   {\n      // Expressions\n        while(a.call)a=a(); while(b.call)b=b(); //if (a.LDot) return a.LDot(b,res);\n      // js if numbers, else contraction product.\n        if (!(a instanceof Element || b instanceof Element)) return a|b;\n        a=Element.toEl(a);b=Element.toEl(b); return a.Dot(b,res);\n      }\n\n    // The outer product. (Grassman product - no use of metric)\n      static Wedge(a,b,res) {\n      // Expressions\n        while(a.call)a=a(); while(b.call)b=b(); if (a.Wedge) return a.Wedge(Element.toEl(b),res);\n      // The outer product of two vectors is a matrix .. internally Mul not Wedge !\n        if (a instanceof Array && b instanceof Array) return a.map(xa=>b.map(xb=>Element.Mul(xa,xb)));\n      // js, else generated wedge product.\n        if (!(a instanceof Element || b instanceof Element)) return a*b;\n        a=Element.toEl(a);b=Element.toEl(b); return a.Wedge(b,res);\n      }\n\n    // The regressive product. (Dual of the outer product of the duals).\n      static Vee(a,b,res) {\n      // Expressions\n        while(a.call)a=a(); while(b.call)b=b(); if (a.Vee) return a.Vee(Element.toEl(b),res);\n      // js, else generated vee product. (shortcut for dual of wedge of duals)\n        if (!(a instanceof Element || b instanceof Element)) return 0;\n        a=Element.toEl(a);b=Element.toEl(b); return a.Vee(b,res);\n      }\n\n    // The sandwich product. Provided for convenience (>>> operator)\n      static sw(a,b) {\n      // Expressions\n        while(a.call)a=a(); while(b.call)b=b(); if (a.sw) return a.sw(b);\n      // Map elements in array\n        if (b instanceof Array && !b.Add) return b.map(x=>Element.sw(a,x));\n      // Call through. no specific generated code for it so just perform the muls.\n        a=Element.toEl(a); b=Element.toEl(b); return a.Mul(b).Mul(a.Reverse);\n      }\n\n    // Division - scalars or cal through to element method.\n      static Div(a,b,res) {\n      // Expressions\n        while(a.call)a=a(); while(b.call)b=b();\n      // js or call through to element divide.\n        if (!(a instanceof Element || b instanceof Element)) return a/b;\n        a=Element.toEl(a);\n        if (Number.isFinite(b)) { return a.Scale(1/b,res); }\n        b=Element.toEl(b); return a.Div(b,res);\n      }\n\n    // Pow - needs obvious extensions for natural powers. (exponentiation by squaring)\n      static Pow(a,b,res) {\n      // Expressions\n        while(a.call)a=a(); while(b.call)b=b(); if (a.Pow) return a.Pow(b,res);\n      // Exponentiation.\n        if (a===Math.E && b.Exp) return b.Exp();\n      // Squaring\n        if (b===2) return this.Mul(a,a,res);\n      // No elements, call through to js\n        if (!(a instanceof Element || b instanceof Element)) return a**b;\n      // Inverse\n        if (b===-1) return a.Inverse;\n      // Call through to element pow.\n        a=Element.toEl(a); return a.Pow(b);\n      }\n\n    // Handles scalars and calls through to element method.\n      static exp(a) {\n      // Expressions.\n        while(a.call)a=a();\n      // If it has an exp callthrough, use it, else call through to math.\n        if (a.Exp) return a.Exp();\n        return Math.exp(a);\n      }\n\n    // Dual, Involute, Reverse, Conjugate, Normalize and length, all direct call through. Conjugate handles matrices.\n      static Dual(a)      { return Element.toEl(a).Dual; };\n      static Involute(a)  { return Element.toEl(a).Involute; };\n      static Reverse(a)   { return Element.toEl(a).Reverse; };\n      static Conjugate(a) { if (a.Conjugate) return a.Conjugate; if (a instanceof Array) return a[0].map((c,ci)=>a.map((r,ri)=>Element.Conjugate(a[ri][ci]))); return Element.toEl(a).Conjugate; }\n      static Normalize(a) { return Element.toEl(a).Normalized; };\n      static Length(a)    { return Element.toEl(a).Length };\n\n    // Comparison operators always use length. Handle expressions, then js or length comparison\n      static eq(a,b)  { if (!(a instanceof Element)||!(b instanceof Element)) return a==b; while(a.call)a=a(); while(b.call)b=b(); for (var i=0; i<a.length; i++) if (a[i]!=b[i]) return false; return true; }\n      static neq(a,b) { if (!(a instanceof Element)||!(b instanceof Element)) return a!=b; while(a.call)a=a(); while(b.call)b=b(); for (var i=0; i<a.length; i++) if (a[i]!=b[i]) return true; return false; }\n      static lt(a,b)  { while(a.call)a=a(); while(b.call)b=b(); return (a instanceof Element?a.Length:a)<(b instanceof Element?b.Length:b); }\n      static gt(a,b)  { while(a.call)a=a(); while(b.call)b=b(); return (a instanceof Element?a.Length:a)>(b instanceof Element?b.Length:b); }\n      static lte(a,b) { while(a.call)a=a(); while(b.call)b=b(); return (a instanceof Element?a.Length:a)<=(b instanceof Element?b.Length:b); }\n      static gte(a,b) { while(a.call)a=a(); while(b.call)b=b(); return (a instanceof Element?a.Length:a)>=(b instanceof Element?b.Length:b); }\n\n    // Debug output and printing multivectors.\n      static describe(x) { if (x===true) console.log(`Basis\\n${basis}\\nMetric\\n${metric.slice(1,1+tot)}\\nCayley\\n${mulTable.map(x=>(x.map(x=>('           '+x).slice(-2-tot)))).join('\\n')}\\nMatrix Form:\\n`+gp.map(x=>x.map(x=>x.match(/(-*b\\[\\d+\\])/)).map(x=>x&&((x[1].match(/-/)||' ')+String.fromCharCode(65+1*x[1].match(/\\d+/)))||' 0')).join('\\n')); return {basis:basisg||basis,metric,mulTable} }\n\n    // Direct sum of algebras - experimental\n      static sum(B){\n        var A = Element;\n        // Get the multiplication tabe and basis.\n        var T1 = A.describe().mulTable, T2 = B.describe().mulTable;\n        var B1 = A.describe().basis, B2 = B.describe().basis;\n        // Get the maximum index of T1, minimum of T2 and rename T2 if needed.\n        var max_T1 = B1.filter(x=>x.match(/e/)).map(x=>x.match(/\\d/g)).flat().map(x=>x|0).sort((a,b)=>b-a)[0];\n        var max_T2 = B2.filter(x=>x.match(/e/)).map(x=>x.match(/\\d/g)).flat().map(x=>x|0).sort((a,b)=>b-a)[0];\n        var min_T2 = B2.filter(x=>x.match(/e/)).map(x=>x.match(/\\d/g)).flat().map(x=>x|0).sort((a,b)=>a-b)[0];\n        // remapping ..\n        T2 = T2.map(x=>x.map(y=>y.match(/e/)?y.replace(/(\\d)/g,(x)=>(x|0)+max_T1):y.replace(\"1\",\"e\"+(1+max_T2+max_T1))));\n        B2 = B2.map((y,i)=>i==0?y.replace(\"1\",\"e\"+(1+max_T2+max_T1)):y.replace(/(\\d)/g,(x)=>(x|0)+max_T1));\n        // Build the new basis and multable..\n        var basis = [...B1,...B2];\n        var Cayley = T1.map((x,i)=>[...x,...T2[0].map(x=>\"0\")]).concat(T2.map((x,i)=>[...T1[0].map(x=>\"0\"),...x]))\n        // Build the new algebra.\n        var grades = [...B1.map(x=>x==\"1\"?0:x.length-1),...B2.map((x,i)=>i?x.length-1:0)];\n        var a = Algebra({basis,Cayley,grades,tot:Math.log2(B1.length)+Math.log2(B2.length)})\n        // And patch up ..\n        a.Scalar = function(x) {\n          var res = new a();\n          for (var i=0; i<res.length; i++) res[i] = basis[i] == Cayley[i][i] ? x:0;\n          return res;\n        }\n        return a;\n      }\n\n    // The graphing function supports several modes. It can render 1D functions and 2D functions on canvas, and PGA2D, PGA3D and CGA2D functions using SVG.\n    // It handles animation and interactivity.\n    //   graph(function(x))     => function of 1 parameter will be called with that parameter from -1 to 1 and graphed on a canvas. Returned values should also be in the [-1 1] range\n    //   graph(function(x,y))   => functions of 2 parameters will be called from -1 to 1 on both arguments. Returned values can be 0-1 for greyscale or an array of three RGB values.\n    //   graph(array)           => array of algebraic elements (points, lines, circles, segments, texts, colors, ..) is graphed.\n    //   graph(function=>array) => same as above, for animation scenario's this function is called each frame.\n    // An optional second parameter is an options object { width, height, animate, camera, scale, grid, canvas }\n      static graph(f,options) {\n      // Store the original input\n        if (!f) return; var origf=f;\n      // generate default options.\n        options=options||{}; options.scale=options.scale||1; options.camera=options.camera||(tot<4?Element.Scalar(1):new Element([0.7071067690849304, 0, 0, 0, 0, 0, 0, 0, 0, 0.7071067690849304, 0, 0, 0, 0, 0, 0]));\n        if (options.conformal && tot==4) var ni = options.ni||this.Coeff(4,1,3,1), no = options.no||this.Coeff(4,0.5,3,-0.5), minus_no = no.Scale(-1);\n        var ww=options.width, hh=options.height, cvs=options.canvas, tpcam=new Element([0,0,0,0,0,0,0,0,0,0,0,-5,0,0,1,0]),tpy=this.Coeff(4,1),tp=new Element(),\n      // project 3D to 2D. This allows to render 3D and 2D PGA with the same code.\n        project=(o)=>{ if (!o) return o; while (o.call) o=o(); return (tot==4 && (o.length==16))?(tpcam).Vee(options.camera.Mul(o).Mul(options.camera.Conjugate)).Wedge(tpy):(o.length==2**tot)?Element.sw(options.camera,o):o;};\n      // gl escape.\n        if (options.gl && !(tot==4 && options.conformal)) return Element.graphGL(f,options); if (options.up) return Element.graphGL2(f,options);\n      // if we get an array or function without parameters, we render c2d or p2d SVG points/lines/circles/etc\n        if (!(f instanceof Function) || f.length===0) {\n        // Our current cursor, color, animation state and 2D mapping.\n          var lx,ly,lr,color,res,anim=false,to2d=(tot==3)?[0,1,2,3,4,5,6,7]:[0,7,9,10,13,12,14,15];\n        // Make sure we have an array of elements. (if its an object, convert to array with elements and names.)\n          if (f instanceof Function) f=f(); if (!(f instanceof Array)) f=[].concat.apply([],Object.keys(f).map((k)=>typeof f[k]=='number'?[f[k]]:[f[k],k]));\n        // The build function generates the actual SVG. It will be called everytime the user interacts or the anim flag is set.\n          function build(f,or) {\n          // Make sure we have an aray.\n            if (or && f && f instanceof Function) f=f();\n          // Reset position and color for cursor.\n            lx=-2;ly=-1.85;lr=0;color='#444';\n          // Create the svg element. (master template string till end of function)\n            var svg=new DOMParser().parseFromString(`<SVG onmousedown=\"if(evt.target==this)this.sel=undefined\" viewBox=\"-2 -${2*(hh/ww||1)} 4 ${4*(hh/ww||1)}\" style=\"width:${ww||512}px; height:${hh||512}px; background-color:#eee; -webkit-user-select:none; -moz-user-select:none; -ms-user-select:none; user-select:none\">\n            // Add a grid (option)\n            ${options.grid?(()=>{\n              var n = Math.floor(10 / options.scale);\n              return n>50?'':[...Array(2*n + 1)].map((x,xi)=>`<line x1=\"-10\" y1=\"${((xi-n)/2-(tot<4?2*options.camera.e02:0))*options.scale}\" x2=\"10\" y2=\"${((xi-n)/2-(tot<4?2*options.camera.e02:0))*options.scale}\" stroke-width=\"0.005\" stroke=\"#CCC\"/><line y1=\"-10\" x1=\"${((xi-n)/2-(tot<4?2*options.camera.e01:0))*options.scale}\" y2=\"10\" x2=\"${((xi-n)/2-(tot<4?2*options.camera.e01:0))*options.scale}\"  stroke-width=\"0.005\" stroke=\"#CCC\"/>`);\n            })():''}\n            // Handle conformal 2D elements.\n            ${options.conformal?f.map&&f.map((o,oidx)=>{\n            // Optional animation handling.\n              if((o==Element.graph && or!==false)||(oidx==0&&options.animate&&or!==false)) { anim=true; requestAnimationFrame(()=>{var r=build(origf,(!res)||(document.body.contains(res))).innerHTML; if (res) res.innerHTML=r; }); if (!options.animate) return; }\n            // Resolve expressions passed in.\n              while (o.call) o=o();\n              if (options.ipns && o instanceof Element) o = o.Dual;\n              var sc = options.scale;\n              var lineWidth = options.lineWidth || 1;\n              var pointRadius = options.pointRadius || 1;\n              var dash_for_r2 = (r2, render_r, target_width) => {\n                // imaginary circles are dotted\n                if (r2 >= 0) return 'none';\n                var half_circum = render_r*Math.PI;\n                var width = half_circum / Math.max(Math.round(half_circum / target_width), 1);\n                return `${width} ${width}`;\n              };\n            // Arrays are rendered as segments or polygons. (2 or more elements)\n              if (o instanceof Array)  { lx=ly=lr=0; o=o.map(o=>{ while(o.call)o=o(); return o.Scale(-1/o.Dot(ni).s); }); o.forEach((o)=>{lx+=sc*(o.e1);ly+=sc*(-o.e2)});lx/=o.length;ly/=o.length; return o.length>2?`<POLYGON STYLE=\"pointer-events:none; fill:${color};opacity:0.7\" points=\"${o.map(o=>(sc*o.e1+','+(-o.e2*sc)+' '))}\"/>`:`<LINE style=\"pointer-events:none\" x1=${o[0].e1*sc} y1=${-o[0].e2*sc} x2=${o[1].e1*sc} y2=${-o[1].e2*sc} stroke-width=\"${lineWidth*0.005}\" stroke=\"${color||'#888'}\"/>`; }\n            // Strings are rendered at the current cursor position.\n              if (typeof o =='string') { var res2=(o[0]=='_')?'':`<text x=\"${lx}\" y=\"${ly}\" font-family=\"Verdana\" font-size=\"${options.fontSize*0.1||0.1}\" style=\"pointer-events:none\" fill=\"${color||'#333'}\" transform=\"rotate(${lr},${lx},${ly})\">&nbsp;${o}&nbsp;</text>`; ly+=0.14; return res2; }\n            // Numbers change the current color.\n              if (typeof o =='number') { color='#'+(o+(1<<25)).toString(16).slice(-6); return ''; };\n            // All other elements are rendered ..\n              var ni_part = o.Dot(no.Scale(-1));  // O_i + n_o O_oi\n              var no_part = ni.Scale(-1).Dot(o);  // O_o + O_oi n_i\n              if (ni_part.VLength * 1e-6 > no_part.VLength) {\n                // direction or dual - nothing to render\n                return \"\";\n              }\n              var no_ni_part = no_part.Dot(no.Scale(-1));  // O_oi\n              var no_only_part = ni.Wedge(no_part).Dot(no.Scale(-1));  // O_o\n\n              /* Note: making 1e-6 smaller increases the maximum circle radius before they are drawn as lines */\n              if (no_ni_part.VLength * 1e-6 > no_only_part.VLength) {\n                var is_flat = true;\n                var direction = no_ni_part;\n              }\n              else {\n                var is_flat = false;\n                var direction = no_only_part;\n              }\n              // normalize to make the direction unitary\n              var dl = direction.Length;\n              o = o.Scale(1/dl);\n              direction = direction.Scale(1/dl)\n\n              var b0=direction.Grade(0).VLength>0.001,b1=direction.Grade(1).VLength>0.001,b2=direction.Grade(2).VLength>0.001;\n              if (!is_flat && b0 && !b1 && !b2) {\n                // Points\n                if (direction.s < 0) { o = Element.Sub(o); }\n                lx=sc*(o.e1); ly=sc*(-o.e2); lr=0; return res2=`<CIRCLE onmousedown=\"this.parentElement.sel=${oidx}\" cx=\"${lx}\" cy=\"${ly}\" r=\"${pointRadius*0.03}\" fill=\"${color||'green'}\"/>`;\n              } else if (is_flat && !b0 && b1 && !b2) {\n                // Lines.\n                var loc=minus_no.LDot(o).Div(o), att=ni.Dot(o);\n                lx=sc*(-loc.e1); ly=sc*(loc.e2); lr=Math.atan2(-o[14],o[13])/Math.PI*180; return `<LINE style=\"pointer-events:none\" x1=${lx-10} y1=${ly} x2=${lx+10} y2=${ly} stroke-width=\"${lineWidth*0.005}\" stroke=\"${color||'#888'}\" transform=\"rotate(${lr},${lx},${ly})\"/>`;\n              } else if (!is_flat && !b0 && !b1 && b2) {\n                // Circles\n                var loc=o.Div(ni.LDot(o)); lx=sc*(-loc.e1); ly=sc*(loc.e2);\n                var r2=o.Mul(o.Conjugate).s;\n                var r = Math.sqrt(Math.abs(r2))*sc;\n                return `<CIRCLE onmousedown=\"this.parentElement.sel=${oidx}\" cx=\"${lx}\" cy=\"${ly}\" r=\"${r}\" stroke-width=\"${lineWidth*0.005}\" fill=\"none\" stroke=\"${color||'green'}\" stroke-dasharray=\"${dash_for_r2(r2, r, lineWidth*0.020)}\"/>`;\n              } else if (!is_flat && !b0 && b1 && !b2) {\n                // Point Pairs.\n                lr=0; var ei=ni,eo=no, nix=o.Wedge(ei), sqr=o.LDot(o).s/nix.LDot(nix).s, r=Math.sqrt(Math.abs(sqr)), attitude=((ei.Wedge(eo)).LDot(nix)).Normalized.Mul(Element.Scalar(r)), pos=o.Div(nix); pos=pos.Div( pos.LDot(Element.Sub(ei)));\n                if (nix==0) { pos = o.Dot(Element.Coeff(4,-1)); sqr=-1; }\n                lx=sc*(pos.e1); ly=sc*(-pos.e2);\n                if (sqr==0) return `<CIRCLE onmousedown=\"this.parentElement.sel=${oidx}\" cx=\"${lx}\" cy=\"${ly}\" r=\"${pointRadius*0.03}\" stroke-width=\"${lineWidth*0.01}\" fill=\"none\" stroke=\"${color||'green'}\"/>`;\n                // Draw imaginary pairs hollow\n                if (sqr > 0) var fill = color||'green', stroke = 'none', dash_array = 'none';\n                else var fill = 'none', stroke = color||'green';\n                lx=sc*(pos.e1+attitude.e1); ly=sc*(-pos.e2-attitude.e2);\n                var res2=`<CIRCLE onmousedown=\"this.parentElement.sel=${oidx}\" cx=\"${lx}\" cy=\"${ly}\" r=\"${pointRadius*0.03}\" fill=\"${fill}\" stroke-width=\"${lineWidth*0.01}\" stroke=\"${stroke}\" stroke-dasharray=\"${dash_for_r2(sqr, pointRadius*0.03, lineWidth*0.020)}\" />`;\n                lx=sc*(pos.e1-attitude.e1); ly=sc*(-pos.e2+attitude.e2);\n                return res2+`<CIRCLE onmousedown=\"this.parentElement.sel=${oidx}\" cx=\"${lx}\" cy=\"${ly}\" r=\"${pointRadius*0.03}\" fill=\"${fill}\" stroke-width=\"${lineWidth*0.01}\" stroke=\"${stroke}\" stroke-dasharray=\"${dash_for_r2(sqr, pointRadius*0.03, lineWidth*0.020)}\" />`;\n              } else {\n                /* Unrecognized */\n                return \"\";\n              }\n            // Handle projective 2D and 3D elements.\n            }):f.map&&f.map((o,oidx)=>{  if((o==Element.graph && or!==false)||(oidx==0&&options.animate&&or!==false)) { anim=true; requestAnimationFrame(()=>{var r=build(origf,(!res)||(document.body.contains(res))).innerHTML; if (res) res.innerHTML=r; }); if (!options.animate) return; } while (o instanceof Function) o=o(); o=(o instanceof Array)?o.map(project):project(o); if (o===undefined) return;\n            // dual option dualizes before render\n              if (options.dual && o instanceof Element) o = o.Dual;\n            // line segments and polygons\n              if (o instanceof Array && o.length)  { lx=ly=lr=0; o.forEach((o)=>{while (o.call) o=o(); lx+=options.scale*((drm[1]==6||drm[1]==14)?-1:1)*o[drm[2]]/o[drm[1]];ly+=options.scale*o[drm[3]]/o[drm[1]]});lx/=o.length;ly/=o.length; return o.length>2?`<POLYGON STYLE=\"pointer-events:none; fill:${color};opacity:0.7\" points=\"${o.map(o=>((drm[1]==6||drm[1]==14)?-1:1)*options.scale*o[drm[2]]/o[drm[1]]+','+options.scale*o[drm[3]]/o[drm[1]]+' ')}\"/>`:`<LINE style=\"pointer-events:none\" x1=${options.scale*((drm[1]==6||drm[1]==14)?-1:1)*o[0][drm[2]]/o[0][drm[1]]} y1=${options.scale*o[0][drm[3]]/o[0][drm[1]]} x2=${options.scale*((drm[1]==6||drm[1]==14)?-1:1)*o[1][drm[2]]/o[1][drm[1]]} y2=${options.scale*o[1][drm[3]]/o[1][drm[1]]} stroke-width=\"${options.lineWidth*0.005||0.005}\" stroke=\"${color||'#888'}\"/>`; }\n            // svg\n              if (typeof o =='string' && o[0]=='<') { return o; }\n            // Labels\n              if (typeof o =='string') { var res2=(o[0]=='_')?'':`<text x=\"${lx}\" y=\"${ly}\" font-family=\"Verdana\" font-size=\"${options.fontSize*0.1||0.1}\" style=\"pointer-events:none\" fill=\"${color||'#333'}\" transform=\"rotate(${lr},0,0)\">&nbsp;${o}&nbsp;</text>`; ly+=0.14; return res2; }\n            // Colors\n              if (typeof o =='number') { color='#'+(o+(1<<25)).toString(16).slice(-6); return ''; };\n            // Points\n              if (o[to2d[6]]**2        >0.0001) { lx=options.scale*o[drm[2]]/o[drm[1]]; if (drm[1]==6||drm[1]==14) lx*=-1; ly=options.scale*o[drm[3]]/o[drm[1]]; lr=0;  var res2=`<CIRCLE onmousedown=\"this.parentElement.sel=${oidx}\" cx=\"${lx}\" cy=\"${ly}\" r=\"${options.pointRadius*0.03||0.03}\" fill=\"${color||'green'}\"/>`; ly-=0.05; lx-=0.1; return res2; }\n            // Lines\n              if (o[to2d[2]]**2+o[to2d[3]]**2>0.0001) { var l=Math.sqrt(o[to2d[2]]**2+o[to2d[3]]**2); o[to2d[2]]/=l; o[to2d[3]]/=l; o[to2d[1]]/=l; lx=0.5; ly=options.scale*((drm[1]==6)?-1:-1)*o[to2d[1]]; lr=-Math.atan2(o[to2d[2]],o[to2d[3]])/Math.PI*180; var res2=`<LINE style=\"pointer-events:none\" x1=-10 y1=${ly} x2=10 y2=${ly} stroke-width=\"${options.lineWidth*0.005||0.005}\" stroke=\"${color||'#888'}\" transform=\"rotate(${lr},0,0)\"/>`; ly-=0.05; return res2; }\n            // Vectors\n              if (o[to2d[4]]**2+o[to2d[5]]**2>0.0001) { lr=0; ly+=0.05; lx+=0.1; var res2=`<LINE style=\"pointer-events:none\" x1=${lx} y1=${ly} x2=${lx-o.e02} y2=${ly+o.e01} stroke-width=\"0.005\" stroke=\"${color||'#888'}\"/>`; ly=ly+o.e01/4*3-0.05; lx=lx-o.e02/4*3; return res2; }\n            }).join()}`,'text/html').body;\n          // return the inside of the created svg element.\n            return svg.removeChild(svg.firstChild);\n          };\n        // Create the initial svg and install the mousehandlers.\n          res=build(f); res.value=f; res.options=options;\n          res.onmousemove=(e)=>{ if (res.sel===undefined || !e.buttons) return;var resx=res.getBoundingClientRect().width,resy=res.getBoundingClientRect().height,x=((e.clientX-res.getBoundingClientRect().left)/(resx/4||128)-2)*(resx>resy?resx/resy:1),y=((e.clientY-res.getBoundingClientRect().top)/(resy/4||128)-2)*(resy>resx?resy/resx:1);x/=options.scale;y/=options.scale; if (options.conformal) { f[res.sel].set(this.Coeff(1,x,2,-y).Add(no).Add(ni.Scale(0.5*(x*x+y*y))) ) } else {f[res.sel][drm[2]]=((drm[1]==6)?-x:x)-((tot<4)?2*options.camera.e01:0); f[res.sel][drm[3]]=y+((tot<4)?2*options.camera.e02:0); f[res.sel][drm[1]]=1; f[res.sel].set(f[res.sel].Normalized)} if (!anim) {var r=build(origf,(!res)||(document.body.contains(res))).innerHTML; if (res) res.innerHTML=r; } res.dispatchEvent(new CustomEvent('input')) };\n          return res;\n        }\n      // 1d and 2d functions are rendered on a canvas.\n        cvs=cvs||document.createElement('canvas'); if(ww)cvs.width=ww; if(hh)cvs.height=hh; var w=cvs.width,h=cvs.height,context=cvs.getContext('2d'), data=context.getImageData(0,0,w,h);\n      // two parameter functions .. evaluate for both and set resulting color.\n        if (f.length==2) for (var px=0; px<w; px++) for (var py=0; py<h; py++) { var res=f(px/w*2-1, py/h*2-1); res=res.buffer?[].slice.call(res):res.slice?res:[res,res,res]; data.data.set(res.map(x=>x*255).concat([255]),py*w*4+px*4); }\n      // one parameter function.. go over x range, use result as y.\n        else if (f.length==1) for (var px=0; px<w; px++) { var res=f(px/w*2-1); res=Math.round((res/2+0.5)*h); if (res > 0 && res < h-1) data.data.set([0,0,0,255],res*w*4+px*4); }\n        return context.putImageData(data,0,0),cvs;\n      }\n\n    // webGL2 Graphing function. (for OPNS/IPNS implicit 2D and 1D surfaces in 3D space).\n      static graphGL2(f,options) {\n      // Create canvas, get webGL2 context.\n        var canvas=document.createElement('canvas'); canvas.style.width=options.width||''; canvas.style.height=options.height||''; canvas.style.backgroundColor='#EEE';\n        if (options.width && options.width.match && options.width.match(/px/i)) canvas.width = parseFloat(options.width)*(options.devicePixelRatio||1); if (options.height && options.height.match && options.height.match(/px/i)) canvas.height = parseFloat(options.height)*(options.devicePixelRatio||1);\n        var gl=canvas.getContext('webgl2',{alpha:options.alpha||false,preserveDrawingBuffer:true,antialias:true,powerPreference:'high-performance'});\n        var gl2=!!gl; if (!gl) gl=canvas.getContext('webgl',{alpha:options.alpha||false,preserveDrawingBuffer:true,antialias:true,powerPreference:'high-performance'});\n        gl.clearColor(240/255,240/255,240/255,1.0); gl.enable(gl.DEPTH_TEST); if (!gl2) { gl.getExtension(\"EXT_frag_depth\"); gl.va = gl.getExtension('OES_vertex_array_object'); }\n        else gl.va = { createVertexArrayOES : gl.createVertexArray.bind(gl), bindVertexArrayOES : gl.bindVertexArray.bind(gl), deleteVertexArrayOES : gl.deleteVertexArray.bind(gl)  }\n      // Compile vertex and fragment shader, return program.\n        var compile=(vs,fs)=>{\n          var s=[gl.VERTEX_SHADER,gl.FRAGMENT_SHADER].map((t,i)=>{\n            var r=gl.createShader(t); gl.shaderSource(r,[vs,fs][i]); gl.compileShader(r);\n            return gl.getShaderParameter(r, gl.COMPILE_STATUS)&&r||console.error(gl.getShaderInfoLog(r));\n          });\n          var p = gl.createProgram(); gl.attachShader(p, s[0]); gl.attachShader(p, s[1]); gl.linkProgram(p);\n          gl.getProgramParameter(p, gl.LINK_STATUS)||console.error(gl.getProgramInfoLog(p));\n          return p;\n        };\n      // Create vertex array and buffers, upload vertices and optionally texture coordinates.\n        var createVA=function(vtx) {\n          var r = gl.va.createVertexArrayOES(); gl.va.bindVertexArrayOES(r);\n          var b = gl.createBuffer(); gl.bindBuffer(gl.ARRAY_BUFFER, b);\n          gl.bufferData(gl.ARRAY_BUFFER, new Float32Array(vtx), gl.STATIC_DRAW);\n          gl.vertexAttribPointer(0, 3, gl.FLOAT, false, 0, 0); gl.enableVertexAttribArray(0);\n          return {r,b}\n        },\n      // Destroy Vertex array and delete buffers.\n        destroyVA=function(va) {\n          if (va.b) gl.deleteBuffer(va.b); if (va.r) gl.va.deleteVertexArrayOES(va.r);\n        }\n      // Drawing function\n        var M=[1,0,0,0,0,1,0,0,0,0,1,0,0,0,5,1];\n        var draw=function(p, tp, vtx, color, color2, ratio, texc, va, b,color3,r,g){\n            gl.useProgram(p); gl.uniformMatrix4fv(gl.getUniformLocation(p, \"mv\"),false,M);\n            gl.uniformMatrix4fv(gl.getUniformLocation(p, \"p\"),false, [5,0,0,0,0,5*(ratio||1),0,0,0,0,1,2,0,0,-1,0])\n            gl.uniform3fv(gl.getUniformLocation(p, \"color\"),new Float32Array(color));\n            gl.uniform3fv(gl.getUniformLocation(p, \"color2\"),new Float32Array(color2));\n            if (color3) gl.uniform3fv(gl.getUniformLocation(p, \"color3\"),new Float32Array(color3));\n            if (b) gl.uniform1fv(gl.getUniformLocation(p, \"b\"),(new Float32Array(counts[g])).map((x,i)=>b[g][i]||0));\n            if (texc) gl.uniform1i(gl.getUniformLocation(p, \"texc\"),0);\n            if (r) gl.uniform1f(gl.getUniformLocation(p,\"ratio\"),r);\n            var v; if (!va) v = createVA(vtx); else gl.va.bindVertexArrayOESOES(va.r);\n            gl.drawArrays(tp, 0, (va&&va.tcount)||vtx.length/3);\n            if (v) destroyVA(v);\n        }\n      // Compile the OPNS renderer. (sphere tracing)\n        var programs = [], genprog = grade=>compile(`${gl2?\"#version 300 es\":\"\"}\n             ${gl2?\"in\":\"attribute\"} vec4 position; ${gl2?\"out\":\"varying\"} vec4 Pos; uniform mat4 mv; uniform mat4 p;\n             void main() { Pos=mv*position; gl_Position = p*Pos; }`,\n            `${!gl2?\"#extension GL_EXT_frag_depth : enable\":\"#version 300 es\"}\n             precision highp float;\n             uniform vec3 color; uniform vec3 color2;\n             uniform vec3 color3; uniform float b[${counts[grade]}];\n             uniform float ratio; ${gl2?\"out vec4 col;\":\"\"}\n             ${gl2?\"in\":\"varying\"} vec4 Pos;\n             float dist (in float z, in float y, in float x, in float[${counts[grade]}] b) {\n                ${this.nVector(1,[]).OPNS_GLSL(this.nVector(grade,[]), options.up)}\n                return ${grade!=tot-1?\"sign(sum)*sqrt(abs(sum))\":\"res\"};\n             }\n             vec3 trace_depth (in vec3 start, vec3 dir, in float thresh) {\n                vec3 orig=start; float lastd = 1000.0; const int count=${(options.maxSteps||64)};\n                float s =  sign(dist(start[0],start[1],start[2],b));\n                for (int i=0; i<count; i++) {\n                  float d = s*dist(start[0],start[1],start[2],b);\n                  if (d < thresh) return start - lastd*${(options.stepSize||0.25)}*dir*(thresh-d)/(lastd-d);\n                  lastd = d; start += dir*${(options.stepSize||0.25)}*d;\n                }\n                return orig;\n             }\n             void main() {\n               vec3 p = -5.0*normalize(color2);\n               vec3 dir = normalize((-Pos[0]/5.0)*color + color2 + vec3(0.0,Pos[1]/5.0*ratio,0.0));  p += 1.0*dir;\n               vec3 L = 5.0*normalize( -0.5*color + 0.85*color2 + vec3(0.0,-0.5,0.0) );\n               vec3 d2 = trace_depth( p , dir, ${grade!=tot-1?(options.thresh||0.2):\"0.0075\"} );\n               float dl2 = dot(d2-p,d2-p); const float h=0.1;\n               if (dl2>0.0) {\n                 vec3 n = normalize(vec3(\n                        dist(d2[0]+h,d2[1],d2[2],b)-dist(d2[0]-h,d2[1],d2[2],b),\n                        dist(d2[0],d2[1]+h,d2[2],b)-dist(d2[0],d2[1]-h,d2[2],b),\n                        dist(d2[0],d2[1],d2[2]+h,b)-dist(d2[0],d2[1],d2[2]-h,b)\n                      ));\n                 ${gl2?\"gl_FragDepth\":\"gl_FragDepthEXT\"} = dl2/50.0;\n                 ${gl2?\"col\":\"gl_FragColor\"} = vec4(max(0.2,abs(dot(n,normalize(L-d2))))*color3 + pow(abs(dot(n,normalize(normalize(L-d2)+dir))),100.0),1.0);\n               } else discard;\n             }`),genprog2D = grade=>compile(`${gl2?\"#version 300 es\":\"\"}\n             ${gl2?\"in\":\"attribute\"} vec4 position; ${gl2?\"out\":\"varying\"} vec4 Pos; uniform mat4 mv; uniform mat4 p;\n             void main() { Pos=mv*position; gl_Position = p*Pos; }`,\n            `${!gl2?\"#extension GL_EXT_frag_depth : enable\":\"#version 300 es\"}\n             precision highp float;\n             uniform vec3 color; uniform vec3 color2;\n             uniform vec3 color3; uniform float b[${counts[grade]}];\n             uniform float ratio; ${gl2?\"out vec4 col;\":\"\"}\n             ${gl2?\"in\":\"varying\"} vec4 Pos;\n             float dist (in float z, in float y, in float x, in float[${counts[grade]}] b) {\n                ${this.nVector(1,[]).OPNS_GLSL(this.nVector(grade,[]), options.up)}\n                return ${grade!=tot-1?\"sqrt(abs(sum))\":\"res\"};\n             }\n             float trace_depth (in vec3 start, vec3 dir, in float thresh) {\n                vec3 orig=start; float lastd = 1000.0; const int count=${(options.maxSteps||64)};\n                float s = dist(start[0]*5.0,start[1]*5.0,start[2]*5.0,b);\n                s=s*s;\n                return 1.0-s*150.0;\n             }\n             void main() {\n               vec3 p = -5.0*normalize(color2);\n               vec3 dir = normalize((-Pos[0]/5.0)*color + color2 + vec3(0.0,Pos[1]/5.0*ratio,0.0));  p += 1.0*dir;\n               vec3 L = 5.0*normalize( -0.5*color + 0.85*color2 + vec3(0.0,-0.5,0.0) );\n               float d2 = trace_depth( p , dir, ${grade!=tot-1?(options.thresh||0.2):\"0.0075\"} );\n               if (d2>0.0) {\n                 ${gl2?\"gl_FragDepth\":\"gl_FragDepthEXT\"} = d2/50.0;\n                 ${gl2?\"col\":\"gl_FragColor\"} = vec4(d2*color3,d2);\n               } else discard;\n             }`)\n      // canvas update will (re)render the content.\n        var armed=0;\n        canvas.update = (x)=>{\n        // Start by updating canvas size if needed and viewport.\n          var s = getComputedStyle(canvas); if (s.width) { canvas.width = parseFloat(s.width)*(options.devicePixelRatio||1); canvas.height = parseFloat(s.height)*(options.devicePixelRatio||1); }\n          gl.viewport(0,0, canvas.width|0,canvas.height|0); var r=canvas.width/canvas.height;\n        // Defaults, resolve function input\n          var a,p=[],l=[],t=[],c=[.5,.5,.5],alpha=0,lastpos=[-2,2,0.2]; gl.clear(gl.COLOR_BUFFER_BIT+gl.DEPTH_BUFFER_BIT); while (x.call) x=x();\n        // Loop over all items to render.\n          for (var i=0,ll=x.length;i<ll;i++) {\n            var e=x[i]; while (e&&e.call) e=e(); if (e==undefined) continue;\n            if (typeof e == \"number\") { alpha=((e>>>24)&0xff)/255; c[0]=((e>>>16)&0xff)/255; c[1]=((e>>>8)&0xff)/255; c[2]=(e&0xff)/255; }\n            if (e instanceof Element){\n              var tt = options.spin?-performance.now()*options.spin/1000:-options.h||0; tt+=Math.PI/2; var r = canvas.height/canvas.width;\n              var g=tot-1; while(!e[g]&&g>1) g--;\n              if (!programs[tot-1-g]) programs[tot-1-g] = (options.up.find(x=>x.match&&x.match(\"z\")))?genprog(g):genprog2D(g);\n              gl.enable(gl.BLEND); gl.blendFunc(gl.ONE, gl.ONE_MINUS_SRC_ALPHA);\n              draw(programs[tot-1-g],gl.TRIANGLES,[-2,-2,0,-2,2,0,2,-2,0,-2,2,0,2,-2,0,2,2,0],[Math.cos(tt),0,-Math.sin(tt)],[Math.sin(tt),0,Math.cos(tt)],undefined,undefined,undefined,e,c,r,g);\n              gl.disable(gl.BLEND);\n            }\n          }\n          // if we're no longer in the page .. stop doing the work.\n          armed++; if (document.body.contains(canvas)) armed=0; if (armed==2) return;\n          canvas.value=x; if (options&&!options.animate) canvas.dispatchEvent(new CustomEvent('input'));\n          if (options&&options.animate) { requestAnimationFrame(canvas.update.bind(canvas,f,options)); }\n          if (options&&options.still) { canvas.value=x; canvas.dispatchEvent(new CustomEvent('input')); canvas.im.width=canvas.width; canvas.im.height=canvas.height; canvas.im.src = canvas.toDataURL(); }\n        }\n        // Basic mouse interactivity. needs more love.\n        var sel=-1; canvas.oncontextmenu = canvas.onmousedown = (e)=>{ e.preventDefault(); e.stopPropagation(); sel=-2;\n          var rc = canvas.getBoundingClientRect(), mx=(e.x-rc.left)/(rc.right-rc.left)*2-1, my=((e.y-rc.top)/(rc.bottom-rc.top)*-4+2)*canvas.height/canvas.width;\n          canvas.onwheel=e=>{e.preventDefault(); e.stopPropagation(); options.z = (options.z||5)+e.deltaY/100; if (!options.animate) requestAnimationFrame(canvas.update.bind(canvas,f,options));}\n          canvas.onmouseup=e=>sel=-1; canvas.onmouseleave=e=>sel=-1;\n          canvas.onmousemove=(e)=>{\n            var rc = canvas.getBoundingClientRect();\n            var mx =(e.movementX)/(rc.right-rc.left)*2, my=((e.movementY)/(rc.bottom-rc.top)*-2)*canvas.height/canvas.width;\n            if (sel==-2) { options.h =  (options.h||0)+mx; if (!options.animate) requestAnimationFrame(canvas.update.bind(canvas,f,options)); return; }; if (sel < 0) return;\n          }\n        }\n        canvas.value = f.call?f():f; canvas.options = options;\n        if (options&&options.still) {\n          var i=new Image(); canvas.im = i; return requestAnimationFrame(canvas.update.bind(canvas,f,options)),i;\n        } else return requestAnimationFrame(canvas.update.bind(canvas,f,options)),canvas;\n\n      }\n\n\n    // webGL Graphing function. (for parametric defined objects)\n      static graphGL(f,options) {\n      // Create a canvas, webgl2 context and set some default GL options.\n        var canvas=document.createElement('canvas'); canvas.style.width=options.width||''; canvas.style.height=options.height||''; canvas.style.backgroundColor='#EEE';\n        if (options.width && options.width.match && options.width.match(/px/i)) canvas.width = parseFloat(options.width); if (options.height && options.height.match && options.height.match(/px/i)) canvas.height = parseFloat(options.height);\n        var gl=canvas.getContext('webgl',{alpha:options.alpha||false,antialias:true,preserveDrawingBuffer:options.still||true,powerPreference:'high-performance'});\n        gl.enable(gl.DEPTH_TEST); gl.depthFunc(gl.LEQUAL); if (!options.alpha) gl.clearColor(240/255,240/255,240/255,1.0); gl.getExtension(\"OES_standard_derivatives\"); gl.va=gl.getExtension(\"OES_vertex_array_object\");\n      // Compile vertex and fragment shader, return program.\n        var compile=(vs,fs)=>{\n          var s=[gl.VERTEX_SHADER,gl.FRAGMENT_SHADER].map((t,i)=>{\n            var r=gl.createShader(t); gl.shaderSource(r,[vs,fs][i]); gl.compileShader(r);\n            return gl.getShaderParameter(r, gl.COMPILE_STATUS)&&r||console.error(gl.getShaderInfoLog(r));\n          });\n          var p = gl.createProgram(); gl.attachShader(p, s[0]); gl.attachShader(p, s[1]); gl.linkProgram(p);\n          gl.getProgramParameter(p, gl.LINK_STATUS)||console.error(gl.getProgramInfoLog(p));\n          return p;\n        };\n      // Create vertex array and buffers, upload vertices and optionally texture coordinates.\n        var createVA=function(vtx, texc, idx, clr) {\n              var r = gl.va.createVertexArrayOES(); gl.va.bindVertexArrayOES(r);\n              var b = gl.createBuffer(); gl.bindBuffer(gl.ARRAY_BUFFER, b);\n              gl.bufferData(gl.ARRAY_BUFFER, new Float32Array(vtx), gl.STATIC_DRAW);\n              gl.vertexAttribPointer(0, 3, gl.FLOAT, false, 0, 0); gl.enableVertexAttribArray(0);\n              if (texc){\n                var b2=gl.createBuffer(); gl.bindBuffer(gl.ARRAY_BUFFER, b2);\n                gl.bufferData(gl.ARRAY_BUFFER, new Float32Array(texc), gl.STATIC_DRAW);\n                gl.vertexAttribPointer(1, 2, gl.FLOAT, false, 0, 0); gl.enableVertexAttribArray(1);\n              }\n              if (clr){\n                var b3=gl.createBuffer(); gl.bindBuffer(gl.ARRAY_BUFFER, b3);\n                gl.bufferData(gl.ARRAY_BUFFER, new Float32Array(clr), gl.STATIC_DRAW);\n                gl.vertexAttribPointer(texc?2:1, 3, gl.FLOAT, false, 0, 0); gl.enableVertexAttribArray(texc?2:1);\n              }\n              if (idx) {\n                var b4=gl.createBuffer(); gl.bindBuffer(gl.ELEMENT_ARRAY_BUFFER, b4);\n                gl.bufferData(gl.ELEMENT_ARRAY_BUFFER, new Uint16Array(idx), gl.STATIC_DRAW);\n              }\n              return {r,b,b2,b4,b3}\n            },\n      // Destroy Vertex array and delete buffers.\n            destroyVA=function(va) {\n              [va.b,va.b2,va.b4,va.b3].forEach(x=>{if(x) gl.deleteBuffer(x)}); if (va.r) gl.va.deleteVertexArrayOES(va.r);\n            }\n      // Default modelview matrix, convert camera to matrix (biquaternion->matrix)\n        var M=[1,0,0,0,0,1,0,0,0,0,1,0,0,0,5,1], mtx = x=>{ var t=options.spin?performance.now()*options.spin/1000:options.h||0, t2=options.p||0;\n          var ct = Math.cos(t), st= Math.sin(t), ct2 = Math.cos(t2), st2 = Math.sin(t2), xx=options.posx||0, y=options.posy||0, z=options.posz||0, zoom=options.z||5;\n          if (tot==5) return [ct,st*-st2,st*ct2,0,0,ct2,st2,0,-st,ct*-st2,ct*ct2,0,xx*ct+z*-st,y*ct2+(xx*st+z*ct)*-st2,y*st2+xx*st+z*ct*ct2+zoom,1];\n          x=x.Normalized; var y=x.Mul(x.Dual),X=-x.e23,Y=-x.e13,Z=x.e12,W=x.s,m=Array(16);\n          var xx = X*X, xy = X*Y, xz = X*Z, xw = X*W, yy = Y*Y, yz = Y*Z, yw = Y*W, zz = Z*Z, zw = Z*W;\n          var mtx = [ 1-2*(yy+zz), 2*(xy+zw), 2*(xz-yw), 0, 2*(xy-zw), 1-2*(xx+zz), 2*(yz+xw), 0, 2*(xz+yw), 2*(yz-xw), 1-2*(xx+yy), 0, -2*y.e23, -2*y.e13, 2*y.e12+5, 1];\n          var mtx2 = [ct,st*-st2,st*ct2,0,0,ct2,st2,0,-st,ct*-st2,ct*ct2,0,0,0,0,1], mtx3=[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];\n          for (var i=0; i<4; ++i) for (var j=0; j<4; ++j) for (var k=0; k<4; ++k) mtx3[i+k*4] += mtx[i+j*4]*mtx2[j+k*4]; return mtx3;\n          return mtx;\n        }\n      // Render the given vertices. (autocreates/destroys vertex array if not supplied).\n        var draw=function(p, tp, vtx, color, color2, ratio, texc, va){\n          gl.useProgram(p); gl.uniformMatrix4fv(gl.getUniformLocation(p, \"mv\"),false,M);\n          gl.uniformMatrix4fv(gl.getUniformLocation(p, \"p\"),false, [5,0,0,0,0,5*(ratio||2),0,0,0,0,1,2,0,0,-1,0])\n          gl.uniform3fv(gl.getUniformLocation(p, \"color\"),new Float32Array(color));\n          gl.uniform3fv(gl.getUniformLocation(p, \"color2\"),new Float32Array(color2));\n          if (texc) gl.uniform1i(gl.getUniformLocation(p, \"texc\"),0);\n          var v; if (!va) v = createVA(vtx, texc); else gl.va.bindVertexArrayOES(va.r);\n          if (va && va.b4) {\n            gl.drawElements(tp, va.tcount, gl.UNSIGNED_SHORT, 0);\n          } else {\n            gl.drawArrays(tp, 0, (va&&va.tcount)||vtx.length/3);\n          }\n          if (v) destroyVA(v);\n        }\n      // Program for the geometry. Derivative based normals. Basic lambert shading.\n        var program = compile(`attribute vec4 position; varying vec4 Pos; uniform mat4 mv; uniform mat4 p;\n                 void main() { gl_PointSize=12.0; Pos=mv*position; gl_Position = p*Pos; }`,\n                `#extension GL_OES_standard_derivatives : enable\n                 precision highp float; uniform vec3 color; uniform vec3 color2; varying vec4 Pos;\n                 void main() { vec3 ldir = normalize(Pos.xyz - vec3(2.0,2.0,-4.0));\n                 vec3 normal = normalize(cross(dFdx(Pos.xyz), dFdy(Pos.xyz))); float l=dot(normal,ldir);\n                 vec3 E = normalize(-Pos.xyz); vec3 R = normalize(reflect(ldir,normal));\n                 gl_FragColor = vec4(max(0.0,l)*color+vec3(0.5*pow(max(dot(R,E),0.0),20.0))+color2, 1.0);  }`);\n        var programPoint = compile(`attribute vec4 position; varying vec4 Pos; uniform mat4 mv; uniform mat4 p;\n                 void main() { gl_PointSize=${((options.pointRadius||1)*8.0).toFixed(2)}; Pos=mv*position; gl_Position = p*Pos; }`,\n                `precision highp float; uniform vec3 color; uniform vec3 color2; varying vec4 Pos;\n                 void main() {  float distanceToCenter = length(gl_PointCoord - vec2(0.5)); if (distanceToCenter>0.5) discard; \n                 gl_FragColor = vec4(color+color2, (distanceToCenter<0.5?1.0:0.0));  }`);\n        var programcol = compile(`attribute vec4 position; attribute vec3 col; varying vec3 Col; varying vec4 Pos; uniform mat4 mv; uniform mat4 p;\n                 void main() { gl_PointSize=6.0; Pos=mv*position; gl_Position = p*Pos; Col=col; }`,\n                `#extension GL_OES_standard_derivatives : enable\n                 precision highp float; uniform vec3 color; uniform vec3 color2; varying vec4 Pos; varying vec3 Col;\n                 void main() { vec3 ldir = normalize(Pos.xyz - vec3(1.0,1.0,2.0));\n                 vec3 normal = normalize(cross(dFdx(Pos.xyz), dFdy(Pos.xyz))); float l=dot(normal,ldir);\n                 vec3 E = normalize(-Pos.xyz); vec3 R = normalize(reflect(ldir,normal));\n                 gl_FragColor = vec4(max(0.3,l)*Col+vec3(pow(max(dot(R,E),0.0),20.0))+color2, 1.0);  }`);\n        var programmot = compile(`attribute vec4 position; attribute vec2 texc; attribute vec3 col; varying vec3 Col; varying vec4 Pos; uniform mat4 mv; uniform mat4 p; uniform vec3 color2;\n                 void main() { gl_PointSize=2.0; float blend=fract(color2.x+texc.r)*0.5; Pos=mv*(position*(1.0-blend) + (blend)*vec4(col,1.0)); gl_Position = p*Pos; Col=vec3(length(col-position.xyz)*1.); gl_PointSize = 8.0 -  Col.x; Col.y=sin(blend*2.*3.1415); }`,\n                `precision highp float; uniform vec3 color; uniform vec3 color2; varying vec4 Pos; varying vec3 Col; \n                 void main() {  float distanceToCenter = length(gl_PointCoord - vec2(0.5));gl_FragColor = vec4(1.0-pow(Col.x,2.0),0.0,0.0,(.6-Col.x*0.05)*(distanceToCenter<0.5?1.0:0.0)*Col.y);  }`);\n        gl.lineWidth(options.lineWidth||1); // doesn't work yet (nobody supports it)\n      // Create a font texture, lucida console or otherwise monospaced.\n        var fw=33, font = Object.assign(document.createElement('canvas'),{width:(19+94)*fw,height:48}),\n            ctx = Object.assign(font.getContext('2d'),{font:'bold 48px lucida console, monospace'}),\n            ftx = gl.createTexture(); gl.activeTexture(gl.TEXTURE0); gl.bindTexture(gl.TEXTURE_2D, ftx);\n            for (var i=33; i<127; i++) ctx.fillText(String.fromCharCode(i),(i-33)*fw,40);\n            var specialChars = \"∞≅¹²³₀₁₂₃₄₅₆₇₈₉⋀⋁∆⋅\"; specialChars.split('').forEach((x,i)=>ctx.fillText(x,(i-33+127)*fw,40));\n            // 2.0 gl.texImage2D(gl.TEXTURE_2D,0,gl.RGBA,94*fw,32,0,gl.RGBA,gl.UNSIGNED_BYTE,font);\n            gl.texImage2D(gl.TEXTURE_2D, 0, gl.RGBA, gl.RGBA, gl.UNSIGNED_BYTE, font);\n            gl.texParameteri(gl.TEXTURE_2D, gl.TEXTURE_MAG_FILTER, gl.LINEAR); gl.texParameteri(gl.TEXTURE_2D, gl.TEXTURE_MIN_FILTER, gl.LINEAR);\n            gl.texParameteri(gl.TEXTURE_2D, gl.TEXTURE_WRAP_S, gl.CLAMP_TO_EDGE); gl.texParameteri(gl.TEXTURE_2D, gl.TEXTURE_WRAP_T, gl.CLAMP_TO_EDGE);\n      // Font rendering program. Renders billboarded fonts, transforms offset passed as color2.\n        var program2 = compile(`attribute vec4 position; attribute vec2 texc; varying vec2 tex; varying vec4 Pos; uniform mat4 mv; uniform mat4 p; uniform vec3 color2;\n                 void main() { tex=texc; gl_PointSize=6.0; vec4 o=mv*vec4(color2,0.0); Pos=(-1.0/(o.z-mv[3][2]))*position+vec4(mv[3][0],mv[3][1],mv[3][2],0.0)+o; gl_Position = p*Pos; }`,\n                `precision highp float; uniform vec3 color; varying vec4 Pos; varying vec2 tex;\n                 uniform sampler2D texm; void main() { vec4 c = texture2D(texm,tex); if (c.a<0.01) discard; gl_FragColor = vec4(color,c.a);}`);\n      // Conformal space needs a bit extra magic to extract euclidean parametric representations.\n        if (tot==5 && options.conformal) var ni = Element.Coeff(4,1).Add(Element.Coeff(5,1)), no = Element.Coeff(4,0.5).Sub(Element.Coeff(5,0.5));\n        var interprete = (x)=>{\n          if (!(x instanceof Element)) return { tp:0 };\n          if (options.ipns) x=x.Dual;\n          // tp = { 0:unknown 1:point 2:line, 3:plane, 4:circle, 5:sphere\n          var X2 = (x.Mul(x)).s, tp=0, weight2, opnix = ni.Wedge(x), ipnix = ni.LDot(x),\n              attitude, pos, normal, tg,btg,epsilon = 0.000001/(options.scale||1), I3=Element.Coeff(16,-1);\n          var x2zero = Math.abs(X2) < epsilon, ipnixzero = ipnix.VLength < epsilon, opnixzero = opnix.VLength < epsilon;\n          if (opnixzero && ipnixzero) {                 // free flat\n          } else if (opnixzero && !ipnixzero) {         // bound flat (lines)\n            attitude = no.Wedge(ni).LDot(x);\n            weight2 = Math.abs(attitude.LDot(attitude).s)**.5;\n            pos = attitude.LDot(x.Reverse); //Inverse);\n            pos = [-pos.e15/pos.e45,-pos.e25/pos.e45,-pos.e34/pos.e45];\n            if (x.Grade(3).VLength) {\n              normal = [attitude.e1/weight2,attitude.e2/weight2,attitude.e3/weight2]; tp=2;\n            } else if (x.Grade(2).VLength) { // point pair with ni\n              tp = 1;\n            } else {\n              normal = Element.LDot(Element.Mul(attitude,1/weight2),I3).Normalized;\n              var r=normal.Mul(Element.Coeff(3,1)); if (r[0]==-1) r[0]=1; else {r[0]+=1; r=r.Normalized;}\n              tg = [...r.Mul(Element.Coeff(1,1)).Mul(r.Conjugate)].slice(1,4);\n              btg = [...r.Mul(Element.Coeff(2,1)).Mul(r.Conjugate)].slice(1,4);\n              normal = [...normal.slice(1,4)]; tp=3;\n            }\n          } else if (!opnixzero && ipnixzero) {         // dual bound flat\n          } else if (x2zero) {                          // bound vec,biv,tri (points)\n            if (options.ipns) x=x.Dual;\n            attitude = ni.Wedge(no).LDot(ni.Wedge(x));\n            pos = [...(Element.LDot(1/(ni.LDot(x)).s,x)).slice(1,4)].map(x=>-x);\n            tp=1;\n          } else if (!x2zero) {                          // round (point pair,circle,sphere)\n            tp = x.Grade(3).VLength?4:x.Grade(2).VLength?6:5;\n            var nix  = ni.Wedge(x), nix2 = (nix.Mul(nix)).s;\n            attitude = ni.Wedge(no).LDot(nix);\n            pos = [...(x.Mul(ni).Mul(x)).slice(1,4)].map(x=>-x/(2.0*nix2));\n            weight2 = Math.abs((x.LDot(x)).s / nix2)**.5;\n            if (tp==4) {\n              if (x.LDot(x).s < 0) { weight2 = -weight2; }\n              normal = Element.LDot(Element.Mul(attitude,1/weight2),I3).Normalized;\n              var r=normal.Mul(Element.Coeff(3,1)); if (r[0]==-1) r[0]=1; else {r[0]+=1; r=r.Normalized;}\n              tg = [...r.Mul(Element.Coeff(1,1)).Mul(r.Conjugate)].slice(1,4);\n              btg = [...r.Mul(Element.Coeff(2,1)).Mul(r.Conjugate)].slice(1,4);\n              normal = [...normal.slice(1,4)];\n            } else if (tp==6) {\n              weight2 = (x.LDot(x).s < 0)?-(weight2):weight2;\n              normal = Element.Mul(attitude.Normalized,weight2).slice(1,4);\n            } else {\n              normal = [...((Element.LDot(Element.Mul(attitude,1/weight2),I3)).Normalized).slice(1,4)];\n            }\n          }\n          return {tp,pos:pos?pos.map(x=>x*(options.scale||1)):[0,0,0],normal,tg,btg,weight2:weight2*(options.scale||1)}\n        };\n      // canvas update will (re)render the content.\n        var armed=0,sphere,e14 = Element.Coeff(14,1);\n        canvas.update = (x)=>{\n        // restore from still..\n          if (options && !options.still && canvas.im && canvas.im.parentElement) { canvas.im.parentElement.insertBefore(canvas,canvas.im); canvas.im.parentElement.removeChild(canvas.im); }\n        // Start by updating canvas size if needed and viewport.\n          var s = getComputedStyle(canvas); if (s.width) { canvas.width = parseFloat(s.width)*(options.devicePixelRatio||1); canvas.height = parseFloat(s.height)*(options.devicePixelRatio||1); }\n          gl.viewport(0,0, canvas.width|0,canvas.height|0); var r=canvas.width/canvas.height;\n        // Defaults, resolve function input\n          var a,p=[],l=[],t=[],c=[.5,.5,.5],alpha=0,lastpos=[-2,2,0.2]; gl.clear(gl.COLOR_BUFFER_BIT+gl.DEPTH_BUFFER_BIT); while (x.call) x=x();\n        // Create default camera matrix and initial lastposition (contra-compensated for camera)\n          M = mtx(options.camera); lastpos = options.camera.Normalized.Conjugate.Mul(((a=new this()).set(lastpos,11),a)).Mul(options.camera.Normalized).slice(11,14);\n        // Grid.\n          if (options.grid) {\n            if (!options.gridLines) { options.gridLines=[[],[],[]]; for (var i=-5; i<=5; i++) {\n                options.gridLines[0].push(i,0,5, i,0,-5, 5,0,i, -5,0,i); options.gridLines[1].push(i,5,0, i,-5,0, 5,i,0, -5,i,0); options.gridLines[2].push(0,i,5, 0,i,-5, 0,5,i, 0,-5,i);\n            }}\n            gl.depthMask(false);\n              draw(program,gl.LINES,options.gridLines[0],[0,0,0],[.6,1,.6],r); draw(program,gl.LINES,options.gridLines[1],[0,0,0],[1,.8,.8],r); draw(program,gl.LINES,options.gridLines[2],[0,0,0],[.8,.8,1],r);\n            gl.depthMask(true);\n          }\n        // Z-buffer override.\n          if (options.noZ) gl.depthMask(false);\n        // Loop over all items to render.\n          for (var i=0,ll=x.length;i<ll;i++) {\n            var e=x[i]; while (e&&e.call&&e.length==0) e=e(); if (e==undefined) continue;\n          // CGA\n            if (tot==5 && options.conformal) {\n              if (e instanceof Array && e.length==2) { e.forEach(x=>{ while (x.call) x=x.call(); x=interprete(x);l.push.apply(l,x.pos); });  var d = {tp:-1}; }\n              else if (e instanceof Array && e.length==3) { e.forEach(x=>{ while (x.call) x=x.call(); x=interprete(x);t.push.apply(t,x.pos); });  var d = {tp:-1}; }\n              else var d = interprete(e);\n              if (d.tp) lastpos=d.pos;\n              if (d.tp==1) p.push.apply(p,d.pos);\n              if (d.tp==2) { l.push.apply(l,d.pos.map((x,i)=>x-d.normal[i]*10)); l.push.apply(l,d.pos.map((x,i)=>x+d.normal[i]*10)); }\n              if (d.tp==3) { t.push.apply(t,d.pos.map((x,i)=>x+d.tg[i]+d.btg[i])); t.push.apply(t,d.pos.map((x,i)=>x-d.tg[i]+d.btg[i])); t.push.apply(t,d.pos.map((x,i)=>x+d.tg[i]-d.btg[i]));\n                             t.push.apply(t,d.pos.map((x,i)=>x-d.tg[i]+d.btg[i])); t.push.apply(t,d.pos.map((x,i)=>x+d.tg[i]-d.btg[i])); t.push.apply(t,d.pos.map((x,i)=>x-d.tg[i]-d.btg[i])); }\n              if (d.tp==4) {\n                var ne=0,la=0;\n                if (d.weight2<0) { c[0]=1;c[1]=0;c[2]=0; }\n                for (var j=0; j<65; j++) {\n                  ne = d.pos.map((x,i)=>x+Math.cos(j/32*Math.PI)*d.weight2*d.tg[i]+Math.sin(j/32*Math.PI)*d.weight2*d.btg[i]); if (ne&&la&&(d.weight2>0||j%2==0)) { l.push.apply(l,la); l.push.apply(l,ne); }; la=ne;\n                }\n              }\n              if (d.tp==6) {\n                if (d.weight2<0) { c[0]=1;c[1]=0;c[2]=0; }\n                if (options.useUnnaturalLineDisplayForPointPairs) {\n                  l.push.apply(l,d.pos.map((x,i)=>x-d.normal[i]*(options.scale||1)));\n                  l.push.apply(l,d.pos.map((x,i)=>x+d.normal[i]*(options.scale||1)));\n                }\n                p.push.apply(p,d.pos.map((x,i)=>x-d.normal[i]*(options.scale||1)));\n                p.push.apply(p,d.pos.map((x,i)=>x+d.normal[i]*(options.scale||1)));\n              }\n              if (d.tp==5) {\n                if (!sphere) {\n                  var pnts = [], tris=[], S=Math.sin, C=Math.cos, pi=Math.PI, W=96, H=48;\n                  for (var j=0; j<W+1; j++) for (var k=0; k<H; k++) {\n                    pnts.push( [S(2*pi*j/W)*S(pi*k/(H-1)), C(2*pi*j/W)*S(pi*k/(H-1)), C(pi*k/(H-1))]);\n                    if (j && k) {\n                      tris.push.apply(tris, pnts[(j-1)*H+k-1]);tris.push.apply(tris, pnts[(j-1)*H+k]);tris.push.apply(tris, pnts[j*H+k-1]);\n                      tris.push.apply(tris, pnts[j*H+k-1]); tris.push.apply(tris, pnts[(j-1)*H+k]); tris.push.apply(tris, pnts[j*H+k]);\n                  }}\n                  sphere = { va : createVA(tris,undefined) }; sphere.va.tcount = tris.length/3;\n                }\n                var oldM = M;\n                M=[].concat.apply([],Element.Mul([[d.weight2,0,0,0],[0,d.weight2,0,0],[0,0,d.weight2,0],[d.pos[0],d.pos[1],d.pos[2],1]],[[M[0],M[1],M[2],M[3]],[M[4],M[5],M[6],M[7]],[M[8],M[9],M[10],M[11]],[M[12],M[13],M[14],M[15]]])).map(x=>x.s);\n                gl.enable(gl.BLEND); gl.blendFunc(gl.CONSTANT_ALPHA, gl.ONE_MINUS_CONSTANT_ALPHA); gl.blendColor(1,1,1,0.5); gl.enable(gl.CULL_FACE)\n                draw(program,gl.TRIANGLES,undefined,c,[0,0,0],r,undefined,sphere.va);\n                gl.disable(gl.BLEND); gl.disable(gl.CULL_FACE);\n                M = oldM;\n              }\n              if (i==ll-1 || d.tp==0) {\n              // render triangles, lines, points.\n                if (alpha) { gl.enable(gl.BLEND); gl.blendFunc(gl.CONSTANT_ALPHA, gl.ONE_MINUS_CONSTANT_ALPHA); gl.blendColor(1,1,1,1-alpha); }\n                if (t.length) { draw(program,gl.TRIANGLES,t,c,[0,0,0],r); t.forEach((x,i)=>{ if (i%9==0) lastpos=[0,0,0]; lastpos[i%3]+=x/3; }); t=[];  }\n                if (l.length) { draw(program,gl.LINES,l,[0,0,0],c,r); var l2=l.length-1; lastpos=[(l[l2-2]+l[l2-5])/2,(l[l2-1]+l[l2-4])/2+0.1,(l[l2]+l[l2-3])/2]; l=[]; }\n                if (p.length) { gl.enable(gl.BLEND); gl.blendFunc(gl.SRC_ALPHA, gl.ONE_MINUS_SRC_ALPHA); draw(programPoint,gl.POINTS,p,[0,0,0],c,r); lastpos = p.slice(-3); lastpos[0]-=0.075; lastpos[1]+=0.075; p=[]; gl.disable(gl.BLEND); }\n                // Motor orbits\n                  if ( e.call && e.length==2 && !e.va3) { var countx=e.dx||32,county=e.dy||32;\n                    var temp=new Float32Array(3*countx*county),o=new Float32Array(3),et=[];\n                    for (var pp=0,ii=0; ii<countx; ii++) for (var jj=0; jj<county; jj++,pp+=3)\n                      temp.set(Element.sw(e(ii/(countx-1),jj/(county-1)),no).slice(1,4),pp);\n                    for (ii=0; ii<countx-1; ii++) for (var jj=0; jj<county; jj++)\n                      et.push((ii+0)*county+(jj+0),(ii+0)*county+(jj+1),(ii+1)*county+(jj+1),(ii+0)*county+(jj+0),(ii+1)*county+(jj+1),(ii+1)*county+(jj+0));\n                    e.va3 = createVA(temp,undefined,et.map(x=>x%(countx*county))); e.va3.tcount = (countx-1)*county*2*3;\n                  }\n                  if ( e.call && e.length==1 && !e.va2) { var countx=e.dx||256;\n                    var temp=new Float32Array(3*countx),o=new Float32Array(3),et=[];\n                    for (var ii=0; ii<countx; ii++) { temp.set(Element.sw(e(ii/(countx-1)),no).slice(1,4),ii*3); if (ii) et.push(ii-1,ii); }\n                    e.va2 = createVA(temp,undefined,et); e.va2.tcount = et.length;\n                  }\n                // Experimental display of motors using particle systems.\n                  if (e instanceof Object && e.motor) {\n                    if (!e.va || e.recalc) {\n                       var seed = 1; function random() { var x = Math.sin(seed++) * 10000; return x - Math.floor(x); }\n                       e.xRange = e.xRange === undefined ? 1:e.xRange; e.yRange = e.yRange === undefined ? 1:e.yRange; e.zRange = e.zRange === undefined ? 1:e.zRange;\n                       var vtx=[], tx=[], vtx2=[];\n                       for (var i=0; i<(e.zRange*e.xRange*e.yRange===0?2500:Math.pow(e.zRange*e.xRange*e.yRange,1/3)*6000); i++) {\n                         var [x,y,z] = [random()*(2*e.xRange)-e.xRange,random()*2*e.yRange-e.yRange,random()*2*e.zRange-e.zRange];\n                         var xyz = (x*x+y*y+z*z)*0.5;\n                         var p  = Element.Vector(x,y,z,xyz-0.5,xyz+0.5);\n                         var p2 = Element.sw(e.motor,p);\n                         var d = p2[5]-p2[4]; p2[1]/=d; p2[2]/=d; p2[3]/=d;\n                         tx.push(random(), random());\n                         vtx.push(...p.slice(1,4)); vtx2.push(...p2.slice(1,4));\n                       }  \n                       e.va = createVA(vtx,tx,undefined,vtx2); e.va.tcount = vtx.length/3;\n                       e.recalc = false;\n                    } \n                    var time = performance.now()/1000;\n                    gl.enable(gl.BLEND); gl.blendFunc(gl.SRC_ALPHA, gl.ONE_MINUS_SRC_ALPHA); gl.disable(gl.DEPTH_TEST);\n                    draw(programmot, gl.POINTS,t,c,[time%1,0,0],r,undefined,e.va);\n                    gl.disable(gl.BLEND); gl.enable(gl.DEPTH_TEST);\n                  }\n                // we could also be an object with cached vertex array of triangles ..\n                  else if (e.va || e.va2 || e.va3 || (e instanceof Object && e.data)) {\n                    // Create the vertex array and store it for re-use.\n                    if (!e.va3 && !e.va2) {\n                      var et=[],et2=[],et3=[],lc=0,pc=0,tc=0; e.data.forEach(e=>{\n                        if (e instanceof Array && e.length==3) { tc++; e.forEach(x=>{ while (x.call) x=x.call(); x=interprete(x);et3.push.apply(et3,x.pos); });  var d = {tp:-1}; }\n                        else {\n                          var d = interprete(e);\n                          if (d.tp==1) { pc++; et.push(...d.pos);  }\n                          if (d.tp==2) { lc++; et2.push(...d.pos.map((x,i)=>x-d.normal[i]*10),...d.pos.map((x,i)=>x+d.normal[i]*10)); }\n                        }\n                      });\n                      e.va = createVA(et,undefined); e.va.tcount = pc;\n                      e.va2 = createVA(et2,undefined); e.va2.tcount = lc*2;\n                      e.va3 = createVA(et3,undefined); e.va3.tcount = tc*3;\n                    }\n                    // render the vertex array.\n                    if (e.va  && e.va.tcount) { gl.enable(gl.BLEND); gl.blendFunc(gl.SRC_ALPHA, gl.ONE_MINUS_SRC_ALPHA); draw(programPoint,gl.POINTS,undefined,[0,0,0],c,r,undefined,e.va); gl.disable(gl.BLEND); };\n                    if (e.va2 && e.va2.tcount) draw(program,gl.LINES,undefined,[0,0,0],c,r,undefined,e.va2);\n                    if (e.va3 && e.va3.tcount) draw(program,gl.TRIANGLES,undefined,c,[0,0,0],r,undefined,e.va3);\n                  }\n                if (alpha) gl.disable(gl.BLEND); // no alpha for text printing.\n              // setup a new color\n                if (typeof e == \"number\") { alpha=((e>>>24)&0xff)/255; c[0]=((e>>>16)&0xff)/255; c[1]=((e>>>8)&0xff)/255; c[2]=(e&0xff)/255; }\n                if (typeof(e)=='string') {\n                  if (options.htmlText) {\n                    if (!x['_'+i]) { console.log('creating div'); Object.defineProperty(x,'_'+i, {value: document.body.appendChild(document.createElement('div')), enumerable:false }) };\n                    var rc = canvas.getBoundingClientRect(), div = x['_'+i];\n                    var pos2 = Element.Mul( [[M[0],M[4],M[8],M[12]],[M[1],M[5],M[9],M[13]],[M[2],M[6],M[10],M[14]],[M[3],M[7],M[11],M[15]]], [...lastpos,1]).map(x=>x.s);\n                    pos2 = Element.Mul( [[5,0,0,0],[0,5*(r||2),0,0],[0,0,1,-1],[0,0,2,0]], pos2).map(x=>x.s).map((x,i,a)=>x/a[3]);\n                    Object.assign(div.style,{position:'fixed',pointerEvents:'none',left:rc.left + (rc.right-rc.left)*(pos2[0]/2+0.5),top: rc.top + (rc.bottom-rc.top)*(-pos2[1]/2+0.5) - 20});\n                    if (div.last != e) { div.innerHTML = e; div.last = e; if (self.renderMathInElement) self.renderMathInElement(div); }\n                  } else { \n                    gl.enable(gl.BLEND); gl.blendFunc(gl.SRC_ALPHA,gl.ONE_MINUS_SRC_ALPHA);\n                    var fw = 113, mapChar = (x)=>{ var c = x.charCodeAt(0)-33; if (c>=94) c = 94+specialChars.indexOf(x); return c/fw; }\n                    draw(program2,gl.TRIANGLES,\n                         [...Array(e.length*6*3)].map((x,i)=>{ var x=0,z=-0.2, o=x+(i/18|0)*1.1; return (0.05*(options.z||5))*[o,-1,z,o+1.2,-1,z,o,1,z,o+1.2,-1,z,o+1.2,1,z,o,1,z][i%18]}),c,lastpos,r,\n                         [...Array(e.length*6*2)].map((x,i)=>{ var o=mapChar(e[i/12|0]); return [o,1,o+1/fw,1,o,0,o+1/fw,1,o+1/fw,0,o,0][i%12]})); gl.disable(gl.BLEND); lastpos[1]-=0.18;\n                  }      \n                }\n              }\n              continue;\n            }\n          // PGA\n            if (options.dual && e instanceof Element) e = e.Dual;\n          // Convert planes to polygons.\n            if (e instanceof Element && e.Grade(1).Length) {\n              var m = Element.Add(1, Element.Mul(e.Normalized, Element.Coeff(3,1))).Normalized, e0 = 0;\n              e=Element.sw(m,[[-1,-1],[-1,1],[1,1],[-1,-1],[1,1],[1,-1]].map(([x,z])=>Element.Trivector(x,e0,z,1)));\n            }\n          // Convert lines to line segments.\n            if (e instanceof Element && e.Grade(2).Length)\n               e=[e.LDot(e14).Wedge(e).Add(e.Wedge(Element.Coeff(1,1)).Mul(Element.Coeff(0,-500))),e.LDot(e14).Wedge(e).Add(e.Wedge(Element.Coeff(1,1)).Mul(Element.Coeff(0,500)))];\n          // If euclidean point, store as point, store line segments and triangles.\n            if (e.e123) p.push.apply(p,e.slice(11,14).map((y,i)=>(i==0?1:-1)*y/e[14]).reverse());\n            if (e instanceof Array && e.length==2) l=l.concat.apply(l,e.map(x=>[...x.slice(11,14).map((y,i)=>(i==0?1:-1)*y/x[14]).reverse()]));\n            if (e instanceof Array && e.length%3==0) t=t.concat.apply(t,e.map(x=>[...x.slice(11,14).map((y,i)=>(i==0?1:-1)*y/x[14]).reverse()]));\n          // Render orbits of parametrised motors, as well as lists of points.. \n            function sw_mot_orig(A,R){\n              var a0=A[0],a1=A[5],a2=A[6],a3=A[7],a4=A[8],a5=A[9],a6=A[10],a7=A[15];\n              R[2] = -2*(a0*a3+a4*a7-a6*a2-a5*a1);\n              R[1] = -2*(a4*a1-a0*a2-a6*a3+a5*a7);\n              R[0] =  2*(a0*a1+a4*a2+a5*a3+a6*a7);\n              return R\n            }\n            if ( e.call && e.length==1) { var count=e.dx||64;\n              for (var ismot,xx,o=new Float32Array(3),ii=0; ii<count; ii++) {\n                if (ii>1) l.push(xx[0],xx[1],xx[2]);\n                var m = e(ii/(count-1));\n                if (ii==0) ismot = m[0]||m[5]||m[6]||m[7]||m[8]||m[9]||m[10];\n                xx = ismot?sw_mot_orig(m,o):m.slice(11,14).map((y,i)=>(i==0?1:-1)*y).reverse(); //Element.sw(e(ii/(count-1)),o);\n                l.push(xx[0],xx[1],xx[2]);\n              }\n            }\n            if ( e.call && e.length==2 && !e.va) { var countx=e.dx||64,county=e.dy||32;\n              var temp=new Float32Array(3*countx*county),o=new Float32Array(3),et=[];\n              for (var pp=0,ii=0; ii<countx; ii++) for (var jj=0; jj<county; jj++,pp+=3) temp.set(sw_mot_orig(e(ii/(countx-1),jj/(county-1)),o),pp);\n              for (ii=0; ii<countx-1; ii++) for (var jj=0; jj<county; jj++) et.push((ii+0)*county+(jj+0),(ii+0)*county+(jj+1),(ii+1)*county+(jj+1),(ii+0)*county+(jj+0),(ii+1)*county+(jj+1),(ii+1)*county+(jj+0));\n              e.va = createVA(temp,undefined,et.map(x=>x%(countx*county))); e.va.tcount = (countx-1)*county*2*3;\n            }\n          // Experimental display of motors using particle systems.\n            if (e instanceof Object && e.motor) {\n              if (!e.va || e.recalc) {\n                 var seed = 1; function random() { var x = Math.sin(seed++) * 10000; return x - Math.floor(x); }\n                 e.xRange = e.xRange === undefined ? 1:e.xRange; e.yRange = e.yRange === undefined ? 1:e.yRange; e.zRange = e.zRange === undefined ? 1:e.zRange;\n                 var vtx=[], tx=[], vtx2=[];\n                 for (var i=0; i<(e.zRange===0?5000:60000); i++) {\n                   var p  = Element.Trivector(random()*(2*e.xRange)-e.xRange,random()*2*e.yRange-e.yRange,random()*2*e.zRange-e.zRange,1);\n//                   var p2 = Element.sw(e.motor,p);\n                   var p2 = e.motor.Mul(p).Mul(e.motor.Inverse);\n                   tx.push(random(), random());\n                   vtx.push(...p.slice(11,14).reverse()); vtx2.push(...p2.slice(11,14).reverse());\n                 }  \n                 e.va = createVA(vtx,tx,undefined,vtx2); e.va.tcount = vtx.length/3;\n                 e.recalc = false;\n              } \n              var time = performance.now()/1000;\n              gl.enable(gl.BLEND); gl.blendFunc(gl.SRC_ALPHA, gl.ONE_MINUS_SRC_ALPHA); gl.disable(gl.DEPTH_TEST);\n              draw(programmot, gl.POINTS,t,c,[time%1,0,0],r,undefined,e.va);\n              gl.disable(gl.BLEND); gl.enable(gl.DEPTH_TEST);\n            }\n          // we could also be an object with cached vertex array of triangles ..\n            else if (e.va || (e instanceof Object && e.data)) {\n              // Create the vertex array and store it for re-use.\n              if (!e.va) {\n                if (e.idx) {\n                  var et = e.data.map(x=>[...x.slice(11,14).map((y,i)=>(i==0?1:-1)*y/x[14]).reverse()]).flat();\n                } else {\n                  var et=[]; e.data.forEach(e=>{if (e instanceof Array && e.length==3) et=et.concat.apply(et,e.map(x=>[...x.slice(11,14).map((y,i)=>(i==0?1:-1)*y/x[14]).reverse()]));});\n                }\n                e.va = createVA(et,undefined,e.idx,e.color?new Float32Array(e.color):undefined); e.va.tcount = (e.idx && e.idx.length)?e.idx.length:e.data.length*3;\n              }\n              // render the vertex array.\n              if (e.transform) { M=mtx(options.camera.Mul(e.transform)); }\n              if (alpha) { gl.enable(gl.BLEND); gl.blendFunc(gl.CONSTANT_ALPHA, gl.ONE_MINUS_CONSTANT_ALPHA); gl.blendColor(1,1,1,1-alpha); }\n              draw(e.color?programcol:program,gl.TRIANGLES,t,c,[0,0,0],r,undefined,e.va);\n              if (alpha) gl.disable(gl.BLEND);\n              if (e.transform) { M=mtx(options.camera); }\n            }\n          // if we're a number (color), label or the last item, we output the collected items.\n            else if (typeof e=='number' || i==ll-1 || typeof e == 'string') {\n            // render triangles, lines, points.\n              if (alpha) { gl.enable(gl.BLEND); gl.blendFunc(gl.CONSTANT_ALPHA, gl.ONE_MINUS_CONSTANT_ALPHA); gl.blendColor(1,1,1,1-alpha); }\n              if (t.length) { draw(program,gl.TRIANGLES,t,c,[0,0,0],r); t.forEach((x,i)=>{ if (i%9==0) lastpos=[0,0,0]; lastpos[i%3]+=x/3; }); t=[];  }\n              if (l.length) { draw(program,gl.LINES,l,[0,0,0],c,r); var l2=l.length-1; lastpos=[(l[l2-2]+l[l2-5])/2,(l[l2-1]+l[l2-4])/2+0.1,(l[l2]+l[l2-3])/2]; l=[]; }\n              if (p.length) { gl.enable(gl.BLEND); gl.blendFunc(gl.SRC_ALPHA, gl.ONE_MINUS_SRC_ALPHA); draw(programPoint,gl.POINTS,p,[0,0,0],c,r); lastpos = p.slice(-3); lastpos[0]-=0.075; lastpos[1]+=0.075; p=[];gl.disable(gl.BLEND); }\n              if (alpha) gl.disable(gl.BLEND);\n            // setup a new color\n              if (typeof e == \"number\") { alpha=((e>>>24)&0xff)/255; c[0]=((e>>>16)&0xff)/255; c[1]=((e>>>8)&0xff)/255; c[2]=(e&0xff)/255; }\n            // render a label\n              if (typeof(e)=='string') {\n                if (options.htmlText) { \n                  if (!canvas['_'+i]) { console.log('creating div'); Object.defineProperty(canvas,'_'+i, {value: document.body.appendChild(document.createElement('div')), enumerable:false }) };\n                  var rc = canvas.getBoundingClientRect(), div = canvas['_'+i];\n                  var pos2 = Element.Mul( [[M[0],M[4],M[8],M[12]],[M[1],M[5],M[9],M[13]],[M[2],M[6],M[10],M[14]],[M[3],M[7],M[11],M[15]]], [...lastpos,1]).map(x=>x.s);\n                  pos2 = Element.Mul( [[5,0,0,0],[0,5*(r||2),0,0],[0,0,1,-1],[0,0,2,0]], pos2).map(x=>x.s).map((x,i,a)=>x/a[3]);\n                  Object.assign(div.style,{position:'fixed',pointerEvents:'none',left:rc.left + (rc.right-rc.left)*(pos2[0]/2+0.5),top: rc.top + (rc.bottom-rc.top)*(-pos2[1]/2+0.5) - 20});\n                  if (div.last != e) { div.innerHTML = e; div.last = e; if (self.renderMathInElement) self.renderMathInElement(div,{output:'html'});  }\n                } else { \n                  gl.enable(gl.BLEND); gl.blendFunc(gl.SRC_ALPHA,gl.ONE_MINUS_SRC_ALPHA);\n                  var fw = 113, mapChar = (x)=>{ var c = x.charCodeAt(0)-33; if (c>=94) c = 94+specialChars.indexOf(x); return c/fw; }\n                  draw(program2,gl.TRIANGLES,\n                       [...Array(e.length*6*3)].map((x,i)=>{ var x=0,z=-0.2, o=x+(i/18|0)*1.1; return 0.25*[o,-1,z,o+1.2,-1,z,o,1,z,o+1.2,-1,z,o+1.2,1,z,o,1,z][i%18]}),c,lastpos,r,\n                       [...Array(e.length*6*2)].map((x,i)=>{ var o=mapChar(e[i/12|0]); return [o,1,o+1/fw,1,o,0,o+1/fw,1,o+1/fw,0,o,0][i%12]})); gl.disable(gl.BLEND); lastpos[1]-=0.18;\n                }\n              }\n            }\n          };\n          // if we're no longer in the page .. stop doing the work.\n          armed++; if (document.body.contains(canvas)) armed=0; if (armed==2) return;\n          canvas.value=x; if (options&&!options.animate) canvas.dispatchEvent(new CustomEvent('input')); canvas.options=options;\n          if (options&&options.animate) { requestAnimationFrame(canvas.update.bind(canvas,f,options)); }\n          if (options&&options.still) { canvas.value=x; canvas.dispatchEvent(new CustomEvent('input')); canvas.im.style.width=canvas.style.width; canvas.im.style.height=canvas.style.height; canvas.im.src = canvas.toDataURL();\n            var p=canvas.parentElement;  if (p) { p.insertBefore(canvas.im,canvas); p.removeChild(canvas); }\n          }\n        }\n        // Basic mouse interactivity. needs more love.\n        var sel=-1; canvas.oncontextmenu = canvas.onmousedown = (e)=>{e.preventDefault(); e.stopPropagation();  if (e.detail===0) return;\n          var rc = canvas.getBoundingClientRect(), mx=(e.x-rc.left)/(rc.right-rc.left)*2-1, my=((e.y-rc.top)/(rc.bottom-rc.top)*-4+2)*canvas.height/canvas.width;\n          sel = (e.button==2)?-3:-2; canvas.value.forEach((x,i)=>{\n            if (tot != 5) { if (x[14]) {\n              var pos2 = Element.Mul( [[M[0],M[4],M[8],M[12]],[M[1],M[5],M[9],M[13]],[M[2],M[6],M[10],M[14]],[M[3],M[7],M[11],M[15]]], [-x[13]/x[14],-x[12]/x[14],x[11]/x[14],1]).map(x=>x.s);\n              pos2 = Element.Mul( [[5,0,0,0],[0,5*(2),0,0],[0,0,1,-1],[0,0,2,0]], pos2).map(x=>x.s).map((x,i,a)=>x/a[3]);\n              if ((mx-pos2[0])**2 + ((my)-pos2[1])**2 < 0.01) sel=i;\n            }} else {\n              x = interprete(x); if (x.tp==1) {\n                var pos2 = Element.Mul( [[M[0],M[4],M[8],M[12]],[M[1],M[5],M[9],M[13]],[M[2],M[6],M[10],M[14]],[M[3],M[7],M[11],M[15]]], [...x.pos,1]).map(x=>x.s);\n                pos2 = Element.Mul( [[5,0,0,0],[0,5*(r||2),0,0],[0,0,1,-1],[0,0,2,0]], pos2).map(x=>x.s).map((x,i,a)=>x/a[3]);\n                if ((mx-pos2[0])**2 + (my-pos2[1])**2 < 0.01) sel=i;\n              }\n            }\n          });\n          canvas.onwheel=e=>{e.preventDefault(); e.stopPropagation(); options.z = (options.z||5)+e.deltaY/100; if (!options.animate) requestAnimationFrame(canvas.update.bind(canvas,f,options));}\n          canvas.onmouseup=e=>sel=-1; canvas.onmouseleave=e=>sel=-1;\n          var tx,ty; canvas.ontouchstart = (e)=>{e.preventDefault();  canvas.focus(); var x = e.changedTouches[0].pageX, y = e.changedTouches[0].pageY; tx=x; ty=y; }\n          canvas.ontouchmove = function (e) { e.preventDefault();\n             var x = e.changedTouches[0].pageX, y = e.changedTouches[0].pageY, mx = (x-(tx||x))/1000, my = -(y-(ty||y))/1000; tx=x; ty=y;\n             options.h = (options.h||0)+mx; options.p = Math.max(-Math.PI/2,Math.min(Math.PI/2, (options.p||0)+my)); if (!options.animate) requestAnimationFrame(canvas.update.bind(canvas,f,options)); return;\n          };\n          canvas.onmousemove=(e)=>{\n            var rc = canvas.getBoundingClientRect(),x; if (sel>=0) { if (tot==5) x=interprete(canvas.value[sel]); else { x=canvas.value[sel]; x={pos:[-x[13]/x[14],-x[12]/x[14],x[11]/x[14]]};  }}\n            var mx =(e.movementX)/(rc.right-rc.left)*2, my=((e.movementY)/(rc.bottom-rc.top)*-2)*canvas.height/canvas.width;\n            if (sel==-2) { options.h =  (options.h||0)+mx; options.p = Math.max(-Math.PI/2,Math.min(Math.PI/2, (options.p||0)+my)); if (!options.animate) requestAnimationFrame(canvas.update.bind(canvas,f,options)); return; };\n            if (sel==-3) { var ct = Math.cos(options.h||0), st= Math.sin(options.h||0), ct2 = Math.cos(options.p||0), st2 = Math.sin(options.p||0);\n              if (e.shiftKey) { options.posy = (options.posy||0)+my; } else { options.posx = (options.posx||0)+mx*ct+my*st; options.posz = (options.posz||0)+mx*-st+my*ct*ct2; } if (!options.animate) requestAnimationFrame(canvas.update.bind(canvas,f,options));return; }; if (sel < 0) return;\n            if (tot==5) { \n               x.pos[0] += (e.buttons!=2)?Math.cos(-(options.h||0))*mx:Math.sin((options.h||0))*my; x.pos[1]+=(e.buttons!=2)?my:0; x.pos[2]+=(e.buttons!=2)?Math.sin(-(options.h||0))*mx:Math.cos((options.h||0))*my;\n               canvas.value[sel].set(Element.Mul(ni,(x.pos[0]**2+x.pos[1]**2+x.pos[2]**2)*0.5).Sub(no)); canvas.value[sel].set(x.pos,1); }\n            else if (x) { \n               x.pos[2] += (e.buttons!=2)?Math.sin(-(options.h||0))*mx:Math.cos((options.h||0))*my; x.pos[1]+=(e.buttons!=2)?my:0; x.pos[0]+=(e.buttons!=2)?Math.cos(-(options.h||0))*mx:Math.sin((options.h||0))*my;\n               canvas.value[sel].set([x.pos[2],-x.pos[1],-x.pos[0],1],11);\n            }\n            if (!options.animate) requestAnimationFrame(canvas.update.bind(canvas,f,options));\n          }\n        }\n        canvas.value = f.call?f():f; canvas.options=options;\n        if (options&&options.still) {\n          var i=new Image(); canvas.im = i; return requestAnimationFrame(canvas.update.bind(canvas,f,options)),canvas;\n        } else return requestAnimationFrame(canvas.update.bind(canvas,f,options)),canvas;\n      }\n\n    // The inline function is a js to js translator that adds operator overloading and algebraic literals.\n    // It can be called with a function, a string, or used as a template function.\n      static inline(intxt) {\n      // If we are called as a template function.\n        if (arguments.length>1 || intxt instanceof Array) {\n          var args=[].slice.call(arguments,1);\n          return res.inline(new Function(args.map((x,i)=>'_template_'+i).join(),'return ('+intxt.map((x,i)=>(x||'')+(args[i]&&('_template_'+i)||'')).join('')+')')).apply(res,args);\n        }\n      // Get the source input text.\n        var txt = (intxt instanceof Function)?intxt.toString():`function(){return (${intxt})}`;\n      // Our tokenizer reads the text token by token and stores it in the tok array (as type/token tuples).\n        var tok = [], resi=[], t, tokens = [/^[\\s\\uFFFF]|^[\\u000A\\u000D\\u2028\\u2029]|^\\/\\/[^\\n]*\\n|^\\/\\*[\\s\\S]*?\\*\\//g,                 // 0: whitespace/comments\n          /^\\\"\\\"|^\\'\\'|^\\\".*?[^\\\\]\\\"|^\\'.*?[^\\\\]\\'|^\\`[\\s\\S]*?[^\\\\]\\`/g,                                                                // 1: literal strings\n          /^\\d+[.]{0,1}\\d*[ei][\\+\\-_]{0,1}\\d*|^\\.\\d+[ei][\\+\\-_]{0,1}\\d*|^e_\\d*/g,                                                       // 2: literal numbers in scientific notation (with small hack for i and e_ asciimath)\n          /^\\d+[.]{0,1}\\d*[E][+-]{0,1}\\d*|^\\.\\d+[E][+-]{0,1}\\d*|^0x\\d+|^\\d+[.]{0,1}\\d*|^\\.\\d+|^\\(\\/.*[^\\\\]\\/\\)/g,                       // 3: literal hex, nonsci numbers and regex (surround regex with extra brackets!)\n          /^(\\.Normalized|\\.Length|\\.\\.\\.|>>>=|===|!==|>>>|<<=|>>=|=>|\\|\\||[<>\\+\\-\\*%&|^\\/!\\=]=|\\*\\*|\\+\\+|\\-\\-|<<|>>|\\&\\&|\\^\\^|^[{}()\\[\\];.,<>\\+\\-\\*%|&^!~?:=\\/]{1})/g,   // 4: punctuator\n          /^[A-Za-z0-9_]*/g]                                                                                                            // 5: identifier\n        while (txt.length) for(t in tokens) if(resi=txt.match(tokens[t])){ tok.push([t|0,resi[0]]); txt=txt.slice(resi[0].length); break;} // tokenise\n      // Translate algebraic literals. (scientific e-notation to \"this.Coeff\"\n        tok=tok.map(t=>(t[0]==2)?[2,'Element.Coeff('+basis.indexOf((!options.Cayley?simplify:(x)=>x)('e'+t[1].split(/e_|e|i/)[1]||1).replace('-',''))+','+(simplify(t[1].split(/e_|e|i/)[1]||1).match('-')?\"-1*\":\"\")+parseFloat(t[1][0]=='e'?1:t[1].split(/e_|e|i/)[0])+')']:t);\n      // String templates (limited support - needs fundamental changes.).\n        tok=tok.map(t=>(t[0]==1 && t[1][0]=='`')?[1,t[1].replace(/\\$\\{(.*?)\\}/g,a=>\"${\"+Element.inline(a.slice(2,-1)).toString().match(/return \\((.*)\\)/)[1]+\"}\")]:t);  \n      // We support two syntaxes, standard js or if you pass in a text, asciimath.\n        var syntax = (intxt instanceof Function)?[[['.Normalized','Normalize',2],['.Length','Length',2]],[['~','Conjugate',1],['!','Dual',1]],[['**','Pow',0,1]],[['^','Wedge'],['&','Vee'],['<<','LDot']],[['*','Mul'],['/','Div']],[['|','Dot']],[['>>>','sw',0,1]],[['-','Sub'],['+','Add']],[['%','%']],[['==','eq'],['!=','neq'],['<','lt'],['>','gt'],['<=','lte'],['>=','gte']]]\n                                                :[[['pi','Math.PI'],['sin','Math.sin']],[['ddot','this.Reverse'],['tilde','this.Involute'],['hat','this.Conjugate'],['bar','this.Dual']],[['^','Pow',0,1]],[['^^','Wedge'],['*','LDot']],[['**','Mul'],['/','Div']],[['-','Sub'],['+','Add']],[['<','lt'],['>','gt'],['<=','lte'],['>=','gte']]];\n      // For asciimath, some fixed translations apply (like pi->Math.PI) etc ..\n        tok=tok.map(t=>(t[0]!=5)?t:[].concat.apply([],syntax).filter(x=>x[0]==t[1]).length?[5,[].concat.apply([],syntax).filter(x=>x[0]==t[1])[0][1]]:t);\n      // Now the token-stream is translated recursively.\n        function translate(tokens) {\n           // helpers : first token to the left of x that is not of a type in the skip list.\n           var left = (x=ti-1,skip=[0])=>{ while(x>=0&&~skip.indexOf(tokens[x][0])) x--; return x; },\n           // first token to the right of x that is not of a type in the skip list.\n               right= (x=ti+1,skip=[0])=>{ while(x<tokens.length&&~skip.indexOf(tokens[x][0])) x++; return x; },\n           // glue from x to y as new type, optionally replace the substring with sub.\n               glue = (x,y,tp=5,sub)=>{tokens.splice(x,y-x+1,[tp,...(sub||tokens.slice(x,y+1))])},\n           // match O-C pairs. returns the 'matching bracket' position\n               match = (O=\"(\",C=\")\")=>{var o=1,x=ti+1; while(o){if(tokens[x][1]==O)o++;if(tokens[x][1]==C)o--; x++;}; return x-1;};\n           // grouping (resolving brackets).\n           for (var ti=0,t,si;t=tokens[ti];ti++) if (t[1]==\"(\") glue(ti,si=match(),6,[[4,\"(\"],...translate(tokens.slice(ti+1,si)),[4,\")\"]]);\n           // [] dot call and new\n           for (var ti=0,t,si; t=tokens[ti];ti++) {\n             if (t[1]==\"[\") { glue(ti,si=match(\"[\",\"]\"),6,[[4,\"[\"],...translate(tokens.slice(ti+1,si)),[4,\"]\"]]); if (ti)ti--;}    // matching []\n             else if (t[1]==\".\") { glue(left(),right()); ti--; }                                                                   // dot operator\n             else if (t[0]==6 && ti && left()>=0 && tokens[left()][0]>=5 && tokens[left()][1]!=\"return\") { glue(left(),ti--) }     // collate ( and [\n             else if (t[1]=='new') { glue(ti,right()) };                                                                           // collate new keyword\n           }\n           // ++ and --\n           for (var ti=0,t; t=tokens[ti];ti++) if (t[1]==\"++\" || t[1]==\"--\") glue(left(),ti);\n           // unary - and + are handled seperately from syntax ..\n           for (var ti=0,t,si; t=tokens[ti];ti++)\n             if (t[1]==\"-\" && (left()<0 || (tokens[left()]||[4])[0]==4)) glue(ti,right(),5,[\"Element.Sub(\",tokens[right()],\")\"]);   // unary minus works on all types.\n             else if (t[1]==\"+\" && (tokens[left()]||[0])[0]==4 && (tokens[left()]||[0])[1][0]!=\".\") glue(ti,ti+1);                   // unary plus is glued, only on scalars.\n           // now process all operators in the syntax list ..\n           for (var si=0,s; s=syntax[si]; si++) for (var ti=s[0][3]?tokens.length-1:0,t; t=tokens[ti];s[0][3]?ti--:ti++) for (var opi=0,op; op=s[opi]; opi++) if (t[1]==op[0]) {\n             // exception case .. \".Normalized\" and \".Length\" properties are re-routed (so they work on scalars etc ..)\n                    if (op[2]==2) { var arg=tokens[left()]; glue(ti-1,ti,5,[\"Element.\"+op[1],\"(\",arg,\")\"]); }\n             // unary operators (all are to the left)\n               else if (op[2])    { var arg=tokens[right()]; glue(ti, right(), 5, [\"Element.\"+op[1],\"(\",arg,\")\"]); }\n             // binary operators\n                             else { var l=left(),r=right(),a1=tokens[l],a2=tokens[r]; if (op[0]==op[1]) glue(l,r,5,[a1,op[1],a2]); else glue(l,r,5,[\"Element.\"+op[1],\"(\",a1,\",\",a2,\")\"]); ti--; }\n           }\n           return tokens;\n        }\n      // Glue all back together and return as bound function.\n        return eval( ('('+(function f(t){return t.map(t=>t instanceof Array?f(t):typeof t == \"string\"?t:\"\").join('');})(translate(tok))+')') );\n      }\n    }\n\n    if (options.dual) {\n      Object.defineProperty(res.prototype, 'Inverse', {configurable:true, get(){ var s = 1/this.s**2; return this.map((x,i)=>i?-x*s:1/x ); }});\n    } else {\n    // Matrix-free inverses up to 5D. Should translate this to an inline call for readability.\n    // http://repository.essex.ac.uk/17282/1/TechReport_CES-534.pdf\n      Object.defineProperty(res.prototype, 'Inverse', {configurable: true, get(){\n        return (tot==0)?new this.constructor.Scalar([1/this[0]]):\n               (tot==1)?this.Involute.Mul(this.constructor.Scalar(1/this.Mul(this.Involute)[0])):\n               (tot==2)?this.Conjugate.Mul(this.constructor.Scalar(1/this.Mul(this.Conjugate)[0])):\n               (tot==3)?this.Reverse.Mul(this.Involute).Mul(this.Conjugate).Mul( this.constructor.Scalar(1/this.Mul(this.Conjugate).Mul(this.Involute).Mul(this.Reverse)[0])):\n               (tot==4)?this.Conjugate.Mul(this.Mul(this.Conjugate).Map(3,4)).Mul( this.constructor.Scalar(1/this.Mul(this.Conjugate).Mul(this.Mul(this.Conjugate).Map(3,4))[0])):\n                        this.Conjugate.Mul(this.Involute).Mul(this.Reverse).Mul(this.Mul(this.Conjugate).Mul(this.Involute).Mul(this.Reverse).Map(1,4)).Mul(this.constructor.Scalar(1/this.Mul(this.Conjugate).Mul(this.Involute).Mul(this.Reverse).Mul(this.Mul(this.Conjugate).Mul(this.Involute).Mul(this.Reverse).Map(1,4))[0]));\n      }});\n    }\n\n    if (options.over) {\n     // experimental. do not use.\n      res.over = options.over;\n      [\"Mul\",\"Add\",\"Sub\",\"Scale\",\"Dot\",\"Wedge\",\"LDot\",\"Vee\"].forEach(x=>res.prototype[x] = options.over.inline(res.prototype[x]));\n      res.prototype.Coeff   = function() { for (var i=0,l=arguments.length; i<l; i+=2) this[arguments[i]]=(arguments[i+1] instanceof options.over)?arguments[i+1]:options.over.Scalar(arguments[i+1]); return this; }\n      res.prototype.upgrade = function () { for (var i=0; i<this.length; i++) this[i] = options.over.Scalar(0); }\n      Object.defineProperty(res.prototype, 'Conjugate', {configurable:true,get(){var res = new this.constructor(); for (var i=0; i<this.length; i++) res[i]= this[i].slice().Scale([1,-1,-1,1][grades[i]%4]); return res; }});\n      Object.defineProperty(res.prototype, 'Reverse',   {configurable:true,get(){var res = new this.constructor(); for (var i=0; i<this.length; i++) res[i]= this[i].slice().Scale([1,1,-1,-1][grades[i]%4]); return res; }});\n      Object.defineProperty(res.prototype, 'Involute',  {configurable:true,get(){var res = new this.constructor(); for (var i=0; i<this.length; i++) res[i]= this[i].slice().Scale([1,-1,1,-1][grades[i]%4]); return res; }});\n      Object.defineProperty(res.prototype, 'Inverse', {configurable: true, get(){\n        return (tot==0)?new this.constructor.Scalar([this[0].Inverse]):\n               (tot==1)?this.Involute.Mul(this.constructor.Scalar(this.Mul(this.Involute)[0].Inverse)):\n               (tot==2)?this.Conjugate.Mul(this.constructor.Scalar(this.Mul(this.Conjugate)[0].Inverse)):\n               (tot==3)?this.Reverse.Mul(this.Involute).Mul(this.Conjugate).Mul( this.constructor.Scalar(this.Mul(this.Conjugate).Mul(this.Involute).Mul(this.Reverse)[0].Inverse)):\n               (tot==4)?this.Conjugate.Mul(this.Mul(this.Conjugate).Map(3,4)).Mul( this.constructor.Scalar(this.Mul(this.Conjugate).Mul(this.Mul(this.Conjugate).Map(3,4))[0].Inverse)):\n                        this.Conjugate.Mul(this.Involute).Mul(this.Reverse).Mul(this.Mul(this.Conjugate).Mul(this.Involute).Mul(this.Reverse).Map(1,4)).Mul(this.constructor.Scalar(this.Mul(this.Conjugate).Mul(this.Involute).Mul(this.Reverse).Mul(this.Mul(this.Conjugate).Mul(this.Involute).Mul(this.Reverse).Map(1,4))[0].Inverse));\n      }});\n      res.prototype.toString = function() { return [...this].map((x,i)=>x==0?undefined:(i?'('+x+')'+basis[i]:x.toString())).filter(x=>x).join(' + '); }\n    }\n\n  // Experimental differential operator.\n    var _D, _DT, _DA, totd = basis.length;\n    function makeD(transpose=false){\n      _DA = _DA || Algebra({ p:p,q:q,r:r,basis:options.basis,even:options.even,over:Algebra({dual:totd})}); // same algebra, but over dual numbers.\n      return (func)=>{\n        var dfunc = _DA.inline(func);                                                                       // convert input function to dual algebra\n        return (val,...args)=>{                                                                             // return a new function (the derivative w.r.t 1st param)\n          if (!(val instanceof res)) val = res.Scalar(val);                                                 // allow to be called with scalars.\n          args = args.map(x=>{ var r = _DA.Scalar(0); for (var i=0; i<totd; i++) r[i][0]=x[i]; return r;}); // upcast args.\n          for (var dval=_DA.Scalar(0),i=0; i<totd; i++) { dval[i][0] = val[i]; dval[i][1+i] = 1; };         // fill in coefficients and dual components\n          var rval = dfunc(dval,...args); var r = [...Array(totd)].map(x=>val.slice());                     // call the function in the dual algebra.\n          if (transpose) for (var i=0; i<totd; i++) for (var j=0; j<totd; j++) { r[i][j] = rval[i][j+1]; }  // downcast transpose from dual algebra to Jacobian vector.\n          else for (var i=0; i<totd; i++) for (var j=0; j<totd; j++) { r[j][i] = rval[i][j+1]; }            // downcast from dual algebra to Jacobian vector.\n          return r.length<=2?r[0]:r;                                                                        // return derivative or jacobian.\n        }\n      }\n    }\n    Object.defineProperty(res, 'D',  {configurable:true,get(){ if (_D) return _D; _D = makeD(false); return _D }});\n    Object.defineProperty(res, 'Dt', {configurable:true,get(){ if (_DT) return _DT; _DT = makeD(true); return _DT }});\n\n  // If a function was passed in, translate, call and return its result. Else just return the Algebra.\n    if (fu instanceof Function) return res.inline(fu)(); else return res;\n  }\n}));\n\n    // take a closure on element before the next cell replaces it\n    (function(element) {\n        (requirejs||require)(['Algebra'], function(Algebra) {\n            var opts = {\"p\": 2, \"q\": 0, \"r\": 1, \"mv_length\": \"8\", \"graph\": {\"conformal\": false, \"gl\": false, \"useUnnaturalLineDisplayForPointPairs\": true, \"grid\": true, \"labels\": true}};  // injected from python\n            var output = Algebra({p: opts.p, q: opts.q, r: opts.r, baseType: Float64Array}).inline((opts)=>{\n                var data = [13696993, [[0, 0, 0, 0, 1, -1, 1, 0], [0, 0, 0, 0, 1, 1, 1, 0], [0, 0, 0, 0, -1, 1, 1, 0]].map(x=>x.length==8?new Element(x):x), 8921736, [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0], \"project B onto AC\", 8921736, [0.0, 4.0, 2.0, -2.0, 0.0, 0.0, 0.0, 0.0], \"project AC onto B\", 34884, [0, 0, 2, 2, 0, 0, 0, 0], \"reject AC from B\", 43656, [0, 0, 2, -2, 0, 0, 0, 0], \"AC\", 2245768, [0, 0, 0, 0, 1, -1, 1, 0], \"A\", 2245768, [0, 0, 0, 0, 1, 1, 1, 0], \"B\", 2245768, [0, 0, 0, 0, -1, 1, 1, 0], \"C\"];  // injected from python\n                // When we get a file, we load and display.\n                var canvas;\n                var h=0, p=0;\n                // convert arrays of floats back to CGA elements.\n                data = data.map(x=>x.length==opts.mv_length?new Element(x):x);\n                // add the graph to the page.\n                canvas = this.graph(data, opts.graph);\n                canvas.options.h = h;\n                canvas.options.p = p;\n                // make it big.\n                canvas.style.width = '100%';\n                canvas.style.height = '50vh';\n                return canvas;\n            })(opts);\n            element.append(output);\n            var a = document.createElement(\"button\");\n            var t = document.createTextNode(\"💾 Save\");\n            a.appendChild(t);\n            function screenshot(){\n                //output.width = 1920;  output.height = 1080;\n                output.update(output.value);\n                output.toBlob(function(blob) {\n                    var url = URL.createObjectURL(blob);\n                    window.open(url, '_blank');\n                });\n            }\n            window.addEventListener('resize', function() {\n                output.update(output.value);\n            });\n            a.onclick = screenshot\n            var butnelem = element.append(a);\n        });\n    })(element);\n    "},"metadata":{}}]},{"metadata":{"trusted":true},"cell_type":"code","source":"","execution_count":null,"outputs":[]}],"metadata":{"kernelspec":{"name":"python3","display_name":"Python 3","language":"python"},"language_info":{"name":"python","version":"3.7.8","mimetype":"text/x-python","codemirror_mode":{"name":"ipython","version":3},"pygments_lexer":"ipython3","nbconvert_exporter":"python","file_extension":".py"}},"nbformat":4,"nbformat_minor":1}

Here is what the visualisation outputs: