30:T4fd,// Create a Clifford Algebra with 3,0,1 metric. Algebra(3,0,1,()=>{ // Specify a point directly (trivectors specified with overloaded e-notation.) var point = (x,y,z)=>1e123-x*1e012+y*1e013+z*1e023; var rotor = (P,a)=>Math.cos(a/2)+Math.sin(a/2)*P; // Our camera position and orientation var camera=0e0; // We construct faces, edges and vertices of an icosahedron. var r = rotor(1e13,Math.PI/2.5); var A = point(0,1,0); var B = point((1-Math.atan(0.5)**2)**0.5,Math.atan(0.5),0); var C = rotor(1e13,Math.PI/5)>>>(1e2>>>B); var items=[A,"A",B,"B",C,"C"]; // faces items.push(0xFFCCCC); for (var i=0;i<5;i++) items.push([A,B,r>>>B],[B,B=r>>>B,C],[C,B,r>>>C],[C,1e2>>>A,C=r>>>C]); // edges items.push(0x444444); for (var i=0;i<5;i++) items.push([A,B],[B,C],[B,B=r>>>B],[B,C],[C,C=r>>>C],[1e2>>>A,C]); // vertices items.push(0x4444FF); for (var i=0;i<5;i++) items.push(A,B=r>>>B,C=r>>>C,1e2>>>A); // Graph the 3D items return this.graph(()=>{ var time=performance.now()/4000; camera.set(rotor(1e13,time)*rotor(1e12,time*1.23131)); // animate camera return items.slice(0,1+((Math.floor(time*50))%(items.length+20))); // show more and more elements },{animate:true,camera}); })31:T8077,Algebra = { // I ever so slightly modified the library, // by changing the single occurrence of: // new DOMParser().parseFromString(`...`, 'text/html') // to the Observable-specific: // html`...` // For some reason, the former already injected the html // object into the DOM or something, for Observable // gave me the `SVGSVGElement` message, instead of just // showing the element itself. /** Ganja.js - Geometric Algebra - not just algebra. (The worlds smallest Geometric Algebra Implementation Generator) * @author Enki * @link https://github.com/enkimute/ganja.js */ return function Algebra(p,q,r) { // p can be options object. var fu=arguments[arguments.length-1],options=p; if (options instanceof Object) { q = (p.q || (p.metric && p.metric.filter(x=>x==-1).length))| 0; r = (p.r || (p.metric && p.metric.filter(x=>x==0).length)) | 0; p = p.p === undefined ? (p.metric && p.metric.filter(x=>x==1).length) : p.p || 0; } else { options={}; p=p|0; r=r|0; q=q|0; }; // Initialise basis names and multiplication table. var tot = options.tot||(p||0)+(q||0)+(r||0), // p = #dimensions that square to 1, q to -1, r to 0. basis=options.basis||Array.apply([],{length:2**tot}) // => [undefined, undefined, undefined, undefined, undefined, undefined, undefined, undefined] .map((x,xi)=>(((1<<20)+xi).toString(2)).slice(-tot||-1) // => ["000", "001", "010", "011", "100", "101", "110", "111"] (index of array in base 2) .replace(/./g,(a,ai)=>a=='0'?'':ai+1-(r==0?0:1))) // => ["", "3", "2", "23", "1", "13", "12", "123"] (1 bits replaced with their positions, 0's removed) .sort((a,b)=>(a.toString().length==b.toString().length)?(a|0)-(b|0):a.toString().length-b.toString().length) // => ["", "1", "2", "3", "12", "13", "23", "123"] (sorted numerically) .map(x=>x&&'e'+x||'1'), // => ["1", "e1", "e2", "e3", "e12", "e13", "e23", "e123"] (converted to commonly used basis names) low=basis.join('').split('').filter(x=>x.match(/\d/)).sort()[0]*1, // lowest x in ex basis names. grades=basis.map(x=>x.length-1), // => [0, 1, 1, 1, 2, 2, 2, 3] (grade per basis blade) grade_start=grades.map((a,b,c)=>c[b-1]!=a?b:-1).filter(x=>x+1).concat([basis.length]), // => [0, 1, 4, 7, 8] (first blade of given grade, with extra stop element) simplify = (s,p,q,r)=>{ // implement ex*ey=-ey*ex (for x!=y) and ex*ex=metric[x] var sign=1,c,l,t=[],f=true;s=[].slice.call(s.replace(/e/g,''));l=s.length; while (f) { f=false; for (var i=0; i=(p+r)) sign*=-1; else if ((s[i]-low)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; } } var ret=(sign==0)?'0':((sign==1)?'':'-')+(t.length?'e'+t.join(''):'1'); return (brm&&brm[ret])||(brm&&brm['-'+ret]&&'-'+brm['-'+ret])||ret; }, 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), drm=basis.map((a,i)=>{ return {a:a,i:i} }).sort((a,b)=>a.a.length>b.a.length?1:a.a.lengthx.i).reverse(), mulTable = options.Cayley||basis.map(x=>basis.map(y=>(x==1)?y:(y==1)?x:simplify(x+y,p,q,r))), // Generate Cayley multiplication table. mulTable2 = options.Cayley||basis.map(x=>basis.map(y=>(x==1)?y:(y==1)?x:simplify(x+y,p+q+r,0,0))), // Generate Cayley multiplication table. metric = basis.map((x,xi)=>mulTable[xi][xi]|0), // extended metric .. per basis blade (not just for the 1D subspaces) .. diagonal of cayley table. gp=basis.map(x=>basis.map(x=>'0')), cp=gp.map(x=>gp.map(x=>'0')), op=gp.map(x=>gp.map(x=>'0')); // Storage for our product tables. // Convert Caeyley table to product matrices. basis.forEach((output,i)=>basis.forEach((x,xi)=>basis.forEach((y,yi)=>{ if (mulTable2[xi][yi].replace('-','') == output) { // For each output component scan Cayley table. gp[i][xi] = (mulTable2[xi][yi]=='0')?'0':((mulTable2[xi][yi][0]!='-')?'':'-')+'b['+yi+']*this['+xi+']'; // Fill in the geometric table op[i][xi] = ( grades[i] == grades[xi]+grades[yi] ) ? gp[i][xi]:'0'; // strict sum of grades in outer if ((mulTable[xi][yi].replace('-','') == output)||(mulTable[xi][yi]=='0')) { gp[i][xi] = (mulTable[xi][yi]=='0')?'0':((mulTable[xi][yi][0]!='-')?'':'-')+'b['+yi+']*this['+xi+']'; // Fill in the geometric table cp[i][xi] = ( grades[i] == grades[yi]-grades[xi] ) ? gp[i][xi]:'0'; // strict diff of grades in inner } }}))); // Generate a new class for our algebra. var res = class Element extends (options.baseType||Float32Array) { // Our elements will be Float32Arrays. // constructor - create a floating point array with the correct number of coefficients. constructor(a) { super(a||basis.length); return this; } // Construct with correct default lenght. Blade(grade,res) { var res=res||new Element(); for (var i=0,l=res.length; iElement.Add(x,b)):b.map(x=>Element.Add(a,x)); if ((a instanceof Array)&&(b instanceof Array)&&a.length==b.length) return a.map((x,xi)=>Element.Add(x,b[xi])); if ((typeof a=='string')||(typeof b=='string')) return a.toString()+b.toString(); if (!(a instanceof Element || b instanceof Element)) return a+b; res=res||new Element(); a=Element.toEl(a); b=Element.toEl(b); for(var i=0;iElement.Sub(x,b)):b.map(x=>Element.Sub(a,x)); if (b&&(a instanceof Array)&&(b instanceof Array)&&a.length==b.length) return a.map((x,xi)=>Element.Sub(x,b[xi])); if (arguments.length==1) return Element.Mul(a,-1); if (!(a instanceof Element || b instanceof Element)) return a-b; res=res||new Element(); a=Element.toEl(a); b=Element.toEl(b); for(var i=0;iElement.Mul(x,b)):b.map(x=>Element.Mul(a,x));var r=a*b; if (!isNaN(r)) return r; a=Element.toEl(a); b=Element.toEl(b); return a.Mul(b,res); } static Div(a,b,res) { a=a.call?a():a; b=b.call?b():b; if (!(a instanceof Element || b instanceof Element)) return a/b; a=Element.toEl(a);b=Element.toEl(b); return a.Div(b,res); } static Pow(a,b,res) { a=a.call?a():a; b=b.call?b():b; if (!(a instanceof Element || b instanceof Element)) return a**b; if (a==Math.E) return b.Exp(); a=Element.toEl(a); if (b==-1) return a.Inverse; if (b==2) return a.Mul(a,res); return a.Pow(b); } static Dot(a,b,res) { a=a.call?a():a; b=b.call?b():b; if (!(a instanceof Element || b instanceof Element)) return a*b; a=Element.toEl(a);b=Element.toEl(b); return a.Dot(b,res); } static Wedge(a,b,res) { a=a.call?a():a; b=b.call?b():b; if (!(a instanceof Element || b instanceof Element)) return a*b; a=Element.toEl(a);b=Element.toEl(b); return a.Wedge(b,res); } static Vee(a,b,res) { a=a.call?a():a; b=b.call?b():b; if (!(a instanceof Element || b instanceof Element)) return a*b; a=Element.toEl(a);b=Element.toEl(b); return a.Vee(b,res); } static Dual(a) { if (r) return Element.toEl(a).map((x,i,a)=>a[drm[i]]); return Element.toEl(a).Dual; }; static Involute(a) { return Element.toEl(a).Involute; }; static Reverse(a) { return Element.toEl(a).Reverse; }; static Conjugate(a) { return Element.toEl(a).Conjugate; } static Normalize(a) { return Element.toEl(a).Normalized; }; static Length(a) { return Element.toEl(a).Length }; static lt(a,b) { a=a.call?a():a; b=b.call?b():b; if (!(a instanceof Element || b instanceof Element)) return ab; a=Element.toEl(a);b=Element.toEl(b); a=(a instanceof Element)?a.Length:a; b=(b instanceof Element)?b.Length:b; return a>b; } static lte(a,b) { a=a.call?a():a; b=b.call?b():b; if (!(a instanceof Element || b instanceof Element)) return a<=b; a=Element.toEl(a);b=Element.toEl(b); a=(a instanceof Element)?a.Length:a; b=(b instanceof Element)?b.Length:b; return a<=b; } static gte(a,b) { a=a.call?a():a; b=b.call?b():b; if (!(a instanceof Element || b instanceof Element)) return a>=b; a=Element.toEl(a);b=Element.toEl(b); a=(a instanceof Element)?a.Length:a; b=(b instanceof Element)?b.Length:b; return a>=b; } static sw(a,b) { a=Element.toEl(a); b=Element.toEl(b); return a.Mul(b).Mul(a.Conjugate); } static exp(a) { a=a.call?a():a; return a.Exp(); } // Debug static describe() { 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')); } toString() { var res=[]; for (var i=0; i1e-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'; } // Parse expressions, translate functions and render graphs. static graph(f,options) { if (!f) return; var origf=f; options=options||{}; options.scale=options.scale||1; options.camera=options.camera||new Element([0.7071067690849304, 0, 0, 0, 0, 0, 0, 0, 0, 0.7071067690849304, 0, 0, 0, 0, 0, 0]); 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(), 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}; // p2d SVG points/lines/rotations/translations/labels . if (!(f instanceof Function) || f.length===0) { 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]; 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])); function build(f,or) { if (or && f && f instanceof Function) f=f(); lx=-2;ly=-1.85;lr=0;color='#444'; return html`