@misc{ganja.js,doi ={10.5281/ZENODO.3635774},url ={https://zenodo.org/record/3635774},author ={De Keninck,  Steven},title ={ganja.js},howpublished ={Zenodo. https://doi.org/10.5281/ZENODO.3635774},year ={2020}}

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Contents

Reasons to use ganja

Using ganja for the first time

npm install ganja.js

varAlgebra=require('ganja.js');

<SCRIPTSRC="https://unpkg.com/ganja.js"></SCRIPT>

The Algebra Function

functionAlgebra(p,q,r,func);// p    = number of positive dimensions.// q    = optional number of negative dimensions.// r    = optional number of zero dimensions.// func = optional function. (shorthand .. it is passed to .inline and executed)

functionAlgebra(options,func);// options = object containing subset of//           {//             p,            integer number of positive dimensions.//             q,            integer number of negative dimensions.//             r,            integer number of zero dimensions.//             metric,       [a,b,..] array with metric per generating dimensions. (e.g. [0,1,1] for PGA2D)//             basis,        ["1","e1","e2"] basis that overrules the standard cannonical basis.//             Cayley,       [["1","e1"],["e1","-1"]] Cayley table to overrule standard GA tables.//             baseType,     float32Array (default), float64Array, .. baseType to be used for the Elements.//             mix           Set to true to enable interoperable sub-algebras. (defaults to false).//           }// returns : algebra class if no func supplied, function result if func supplied.

// BasicvarHyper=Algebra(1);// Hyperbolic numbers.varComplex=Algebra(0,1);// Complex numbers.varDual=Algebra(0,0,1);// Dual numbers.varH=Algebra(0,2);// Quaternions.// CliffordvarCl2=Algebra(2);// Clifford algebra for 2D vector space.varCl3=Algebra(3);// Clifford algebra for 3D vector space.vartimeSpace=Algebra(1,3);// Clifford algebra for timespace vectors.// SubAlgebrasvarComplex=Algebra({p:3,basis:['1','e123']});// Complex Numbers as subalgebra of Cl3varH=Algebra({p:3,basis:['1','e12','e13','e23']});// Quaternions as even subalgebra of Cl3// GeometricvarPGA2D=Algebra(2,0,1);// Projective Euclidean 2D plane. (dual)varPGA3D=Algebra(3,0,1);// Projective Euclidean 3D space. (dual)varCGA2D=Algebra(3,1);// conformal 2D space.varCGA3D=Algebra(4,1);// Conformal 3D space.// High-Dimensional GAvarDCGA3D=Algebra(6,2);// Double Conformal 3D Space.varTCGA3D=Algebra(9,3);// Triple Conformal 3D Space.varDCGSTA=Algebra(4,8);// Double Conformal Geometric Space Time Algebra.varQCGA=Algebra(9,6);// Quadric Conformal Geometric Algebra.

varComplex=Algebra(0,1);// Complex numbers.vara=newComplex([3,2]);// 3+2ivarb=newComplex([1,4]);// 1+4ireturna.Mul(b);// returns [-5, 14]

The inline function

Algebra(0,1).inline(()=>(3+2e1)*(1+4e1))();// return [-5,14]

Algebra(0,1,()=>(3+2e1)*(1+4e1));// return [-5,14]

Algebra(2,0,1,()=>{// Direct specification of basis blades using e-notation.varxy_bivector=1e12,pseudoscalar=1e012;// Operator overloading .. * = geometric product, ^ = wedge, & = vee, << = dot, >>> = sandwich ...varxy_bivector_from_product=1e1*1e2;// Directly specified point.varsome_point=1e12+0.4e01+0.5e02;// Function that returns point.varfunction_that_returns_point=()=>some_point+0.5e01;// Join of point and function .. notice no calling brackets ..varjoin_between_point_and_function=some_point&function_that_returns_point;// Same line as above.. but as function.. (so will update if the point changes)varfunction_that_returns_join=()=>some_point&function_that_returns_point;// Binary operations on arrays also work as expected.vareven=[1,2,3,4,5]*2;// Even if those contain multivectors or other arrays :varfunky=[1,1e01+0.5e02,[3,4]]*3+[1,2,3];// All elements and functions can be rendered directly. (again, no calling brackets).varcanvas=this.graph([some_point,function_that_returns_point,function_that_returns_join]);});

// the pretty mathematical expression (!=dual, ^=wedge)a=()=>!(!a^!b)*(c*1e23)// gets translated to ..b=()=>this.Mul(this.Dual((this.Wedge(this.Dual(a),this.Dual(b)))),(this.Mul(c,this.Coeff(6,1))))

The graph function.

canvas=Algebra(0).graph(x=>Math.sin(x*5));// Graph a 1D function in Rcanvas=Algebra(0).graph((x,y)=>x+y);// Graph a 2D function in Rsvg=Algebra(2,0,1,()=>this.graph([1e12,1e1,1e2]));// Graph the origin and x and y-axis in 2D PGAsvg=Algebra(3,0,1,()=>this.graph([1e123,1e23,1e13,1e12],{camera:1+.5e01-.5e02}));// and in 3D PGAcanvas=Algebra(4,1,()=>this.graph([.5e4-.5e5],{conformal:true,gl:true});// The origin in 3D CGA

The describe function.

Algebra(0,1).describe();

Basis1,e1Metric-1Cayley  1, e1 e1, -1Matrix Form: A,-B B, A

Getting free ganja samples.

Ganja for experienced users.

Custom subalgebra's

// The complex numbers as the even subalgebra of R2C=Algebra({p:2,basis:['1','e12']});// The Quaternions as the even subalgebra of R3varH=Algebra({p:3,basis:['1','e12','e13','e23']});

Custom basis names.

varR3_32=Algebra(3);varR3_64=Algebra({p:3,baseType:Float64Array});

Custom Cayley Table

varbasis=['1','e1','e2','e3'];varCayley=[['1','e1','e2','e3'],['e1','e2','e3','0'],['e2','e3','0','0'],['e3','0','0','0']];Algebra({basis,Cayley},()=>{varf=(x)=>0.25*x*x*x*x-0.5;for(vari=-5;i<5;i++)console.log(i,f(i+1e1));});

-5 [155.75, -125, 37.5, -5]-4 [  63.5,  -64,   24, -4]-3 [ 19.75,  -27, 13.5, -3]-2 [   3.5,   -8,    6, -2]-1 [ -0.25,   -1,  1.5, -1]0  [  -0.5,    0,    0,  0]1  [ -0.25,    1,  1.5,  1]2  [   3.5,    8,    6,  2]3  [ 19.75,   27, 13.5,  3]4  [  63.5,   64,   24,  4]

Mixed mode.

// Create R2 - Clifford algebra of 2D vectors - indicate you want mix mode.varR2=Algebra({p:2,q:0,r:0,mix:true});// Create the complex numbers as the even subalgebra of R2.varC=Algebra({p:2,q:0,r:0,basis:['1','e12'],mix:true});// Elements of R2 have four components.// Create the complex number 1+4ivara=newR2([1,0,0,4])// Elements of C have two components.// Create the complex number 3+2ivarb=newC([3,2]);// They inter-operate ..a.Mul(b);// returns an element of R2 : [-5,0,0,14]b.Mul(a);// returns an element of C  : [-5,14]

Ganja ingredients and syntax.

Duality

Dot Product

Vee product

a&b = !(!a^!b)

Ganja starterkit : PGA2D P(R*_2,0,1)

// Create a Clifford Algebra with 2 positive and one zero generator.Algebra(2,0,1,()=>{// Output algebra info to the console.this.describe();// The default basis is s,e0,e1,e2,e01,e02,e12,e012// The metric for vectors is    0,  1,  1 - Vectors will represent lines.// The metric for bivectors is  0,  0, -1 - Bivectors will represent points.// The pseudoscalar is degenerate, so use the built-in duality operator instead.// The bivectors consist of two motor elements and one rotation element -// exactly what is needed to represent translations and rotations in the plane.// In dual projectivized space, the origin is represented by the e12 bivector.varorigin=1e12,EX=-1e02,EY=1e01;// Points and lines can be specified directly. (note : -e02 = e20)varpoint=(x,y)=>origin+x*EX+y*EY;varline=(a,b,c)=>a*1e1+b*1e2+c*1e0;// Or through join and meet operations. (dual so wedge is meet and vee is join.)varjoin=(p1,p2)=>p1&p2;varmeet=(l1,l2)=>l1^l2;// Distances and anglesvardist_points=(P1,P2)=>(P1.Normalized&P2.Normalized).Length;vardist_point_line=(P,l)=>((P.Normalized)^(l.Normalized)).e012;varangle_lines=(l1,l2)=>(l1.Normalized<<l2.Normalized).s;// Points and lines can be projected and rejected.varproject=(P,l)=>P<<l*l;varparallel=(P,l)=>P<<l*P;varortho=(P,l)=>P<<l;// translations and rotations.varrotor=(a,P)=>Math.cos(a*0.5)+Math.sin(a*0.5)*P;vartranslator=(x,y)=>1+0.5*(x*1e02-y*1e01);// To demonstrate graphing, we create some points and lines.// Users can drag points in the graph, lambda expressions can be// used to create dynamic updating items.varA=point(-1,-1),B=point(1,-1),C=point(-1,1),l=line(-1,1,0.5);// Ganja.js can directly graph 2D PGA elements. Pass in an array of// items to render. (numbers are colors, strings are labels, PGA points// and lines are rendered automatically and arrays can be used for line// segments and polygons). The graph function returns a HTML SVG element.document.body.appendChild(this.graph([// use numbers to set the current color.0x444444,// strings label the items they follow, first string is a title."title",// render points (user can drag these)A,B,C,"Label for point",// render linesl,"Label for line",// line segments()=>[A,B],"Label for segment",// polygons0xffeeee,()=>[A,B,C],0xff7777,"Label for polygon"],{grid:true,animate:false}));// When using the animation mode, all lambda's will be evaluated every frame.// Use Date.now() or similar. (many examples in the coffeeshop.)});

Ganja starterkit : PGA3D P(R*_3,0,1)

// Create a Clifford Algebra with 3 positive and one zero generator.Algebra(3,0,1,()=>{// Output algebra info to the console.this.describe();// The default basis is 1,e0,e1,e2,e3,e01,e02,e03,e12,e13,e23,e012,e013,e023,e123,e0123// The metric for vectors is     0, 1, 1,  1          - Vectors will represent planes.// The metric for bivectors is   0, 0, 0, -1, -1, -1  - Bivectors will represent lines.// The metric for trivectors is  0, 0, 0, -1          - Trivectors will represent points.// The pseudoscalar is degenerate, so use the built-in duality operator instead.// The bivectors consist of three motor elements and three rotation element -// exactly what is needed to represent translations and rotations in euclidean space.// In dual projectivized space, the origin is represented by the e12 bivector.varorigin=1e123,EX=-1e023,EY=1e013,EZ=-1e012;// Points and planes can be specified directly.varpoint=(x,y,z)=>origin+x*EX+y*EY+z*EZ,plane=(a,b,c,d)=>a*1e1+b*1e2+c*1e3+d*1e0;// Table from "Geometric Algebra for Computer Graphics" p.15varLineFromPoints=(P,Q)=>P&Q,LineFromPlanes=(a,b)=>a^b,PointFromPlanes=(a,b,c)=>a^b^c,PlaneFromPoints=(P,Q,R)=>P&Q&R,DistPointToPlane=(a,P)=>a&P,DistPoints=(P,Q)=>(P&Q).Length,AnglePlanes=(a,b)=>Math.acos((a<<b).Length),LineThroughPointPerpPlane=(P,a)=>P<<a,OrthProjPointToPlane=(P,a)=>P<<a*a,PlaneThroughPointParPlane=(P,a)=>P<<a*P,IntersectLinePlane=(PI,a)=>PI^a,PlaneThroughPointPerpLine=(PI,P)=>P<<PI,OrthProjPointToLine=(PI,P)=>P<<PI*PI,LineThroughPointParLine=(PI,P)=>P<<PI*P,LineThroughPointPerpLine=(PI,P)=>(P<<PI*P)&P,DistLines=(PI,EP)=>PI&EP,AngleLines=(PI,EP)=>Math.acos((PI<<EP).Length),ReflectionInPlane=(a,X)=>a*X*a,Rotor=(PI,alpha)=>Math.cos(alpha/2)+Math.sin(alpha/2)*PI,RotationAroundLine=(X,PI,alpha)=>Rotor(PI,alpha)*X*~Rotor(PI,alpha),Translator=(x,y,z)=>1+0.5*(x*EX+y*EY+z*EZ);// To demonstrate graphing, we create some points and lines.// Users can drag points in the graph, lambda expressions can be// used to create dynamic updating items.varA=point(-1,-1,0),B=point(1,-1,0),C=point(-1,1,0);// Ganja.js can directly graph 3D PGA elements. Pass in an array of// items to render. (numbers are colors, strings are labels, PGA points// and lines are rendered automatically and arrays can be used for line// segments and polygons). The graph function returns a HTML SVG element.document.body.appendChild(this.graph([// use numbers to set the current color.0x444444,// strings label the items they follow, first string is a title."title",// render points (user can drag these)A,B,C,"Label for point",// render lines()=>(B&C&A)<<(B&C)<<-A,"Label for line",// line segments()=>[A,B],"Label for segment",// polygons0xffeeee,()=>[A,B,C],0xff7777,"Label for polygon"],{grid:false,animate:false}));// When using the animation mode, all lambda's will be evaluated every frame.// Use Date.now() or similar. (many examples in the coffeeshop.)});