ganja.js - Geometric Algebra for javascript.

Geometric Algebra - Not Just Algebra

Ganja.js is a Geometric Algebra code generator for javascript. It generates Clifford algebras and sub-algebras of any signature and implements operator overloading and algebraic constants.

(Mathematically, an algebra generated by ganja.js is a graded exterior (Grassmann) algebra (or one of its subalgebras) with a non-metric outer product, extended (Clifford) with geometric and contraction inner products, a Poincare duality operator and the main involutions and automorphisms.)

(Technically, ganja.js is a code generator producing classes that reificate algebraic literals and expressions by using reflection, a built-in tokenizer and a simple AST translator to rewrite functions containing algebraic constructs to their procedural counterparts.)

(Seriously, look at the examples, run some quick numbers using the GAlculator or play the wedge game first.)

Contents

1. Reasons to use ganja
2. Using ganja for the first time
3. Getting free ganja samples
4. Ganja for experienced users
5. Ganja ingredients and syntax
6. Ganja starterkit : PGA2D P(R*_2,0,1)
7. Ganja starterkit : PGA3D P(R*_3,0,1)

Reasons to use ganja

Ganja.js makes doing Geometric Algebra in your browser easy and fun. Its inline syntax and graphing makes math in the browser feel like .. math.

Operator overloading
Algebraic constants
Supports any metric (positive,negative,zero)
smallish (190 lines)
matrix-free inverses up to 5D.
geometric, inner (contraction), outer (wedge) and vee product
conjugate, Reverse, Involute, Dual (Poincare), Negative
4 API's (inline, asciimath, object oriented, functional)
Easy graph function for 1D and 2D functions, Projective 2D, 3D and conformal 2D elements.
There's a game that teaches you how to use ganja.js !

Using ganja for the first time

Start off by including the ganja.js script. (ganja.js has no dependencies - just 7.9kb on the wire)

<SCRIPT SRC="https://raw.githubusercontent.com/enkimute/ganja.js/master/ganja.js"></SCRIPT>

The Algebra Function

To create an Algebra, call the Algebra function specifying the metric signature (number of positive,negative and zero dimensions). The result is an ES6 class implementing the requested clifford algebra.

function Algebra( 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)

An extended syntax is also available that allows you to further tweak the created Algebra.

function Algebra( 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.

Here are some examples :

// Basic 
var Hyper   = Algebra(1);       // Hyperbolic numbers. 
var Complex = Algebra(0,1);     // Complex numbers.
var Dual    = Algebra(0,0,1);   // Dual numbers.
var H = Algebra(0,2);           // Quaternions.

// Clifford
var Cl2 = Algebra(2);           // Clifford algebra for 2D vector space.
var Cl3 = Algebra(3);           // Clifford algebra for 3D vector space.
var timeSpace = Algebra(1,3);   // Clifford algebra for timespace vectors.

// SubAlgebras
var Complex = Algebra({p:3,basis:['1','e123']});        // Complex Numbers as subalgebra of Cl3
var H = Algebra({p:3,basis:['1','e12','e13','e23']});   // Quaternions as even subalgebra of Cl3

// Geometric
var PGA2D = Algebra(2,0,1);     // Projective Euclidean 2D plane. (dual)
var PGA3D = Algebra(3,0,1);     // Projective Euclidean 3D space. (dual)
var CGA2D = Algebra(3,1);       // conformal 2D space. 
var CGA3D = Algebra(4,1);       // Conformal 3D space.

You can now use these classes to generate algebraic elements. Those elements will have all of the expected properties. (Length, blade access, Dot, Wedge, Mul, Dual, Inverse, etc ...)

And while not advised you could use them in a 'classic' programming style syntax like the example below.

var Complex = Algebra(0,1);     // Complex numbers.
var a = new Complex([3,2]);     // 3+2i
var b = new Complex([1,4]);     // 1+4i
return a.Mul(b);                // returns [-5, 14]

This however, is not very pretty. It's not that much fun either. Luckily, ganja.js provides an alternate way to write algebraic functions, literals and expressions.

The inline function

Your Algebra class exposes this interface through the inline function. It accepts a javascript function, and translates it to use the Algebra of your choice. Using the inline function, the above example is written :

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

Note that if you are immediately executing the function, you can add it as a last parameter to your Algebra constructor call.

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

The inline syntax is powerful and flexible. It offers full operator overloading, overloads scientific e-notation to allow you to directly specify basis blades and allows using lambda expressions without the need for calling brackets in algebraic expressions.

Algebra(2,0,1,()={

  // Direct specification of basis blades using e-notation.
  var xy_bivector = 1e12,
      pseudoscalar = 1e012;

  // Operator overloading .. * = geometric product, ^ = wedge, & = vee, << = dot, >>> = sandwich ... 
  var xy_bivector_from_product = 1e1 * 1e2;

  // Directly specified point.
  var some_point = 1e12 + 0.4e01 + 0.5e02;  

  // Function that returns point.
  var function_that_returns_point = ()=>some_point + 0.5e01;

  // Join of point and function .. notice no calling brackets .. 
  var join_between_point_and_function = some_point & function_that_returns_point;

  // Same line as above.. but as function.. (so will update if the point changes)
  var function_that_returns_join = ()=>some_point & function_that_returns_point;

  // All elements and functions can be rendered directly. (again, no calling brackets). 
  var canvas = this.graph([ some_point, function_that_returns_point, function_that_returns_join ]);

});

Under the hood, ganja.js will translate these functions.

// 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))))

In the example above, functions a and b do the same thing, but it should be clear that a-b=headeache. Because I'm out of aspirin, I'll leave the proof of that to the reader.

See the coffeeshop for more examples of how to use the inline syntax.

The graph function.

Your Algebra also exposes a static graph function that allows you to easily graph 1D or 2D functions as well as 2D and 3D PGA elements.

canvas = Algebra(0).graph(x=>Math.sin(x*5));                   // Graph a 1D function in R

canvas = Algebra(0).graph((x,y)=>x+y);                         // Graph a 2D function in R

svg = Algebra(2,0,1,()=>this.graph([1e12,1e1,1e2]));           // Graph the origin and x and y-axis in 2D

svg = Algebra(3,0,1,()=>this.graph([1e123,1e23,1e13,1e12],{camera:1+.5e01-.5e02})); // and in 3D

Again, many more examples can be found at the coffeeshop.

The describe function.

To display the basis blade names, metric, Cayley table and more, use the static describe function.

Algebra(0,1).describe();

sample output :

Basis
1,e1
Metric
-1
Cayley
  1, e1
 e1, -1
Matrix Form:
 A,-B
 B, A

Getting free ganja samples.

Please visit the coffeeshop and play around with the examples. They are interactive and you can easily change the code online. No need to download or install anything !

Complex Mandelbrot	Mandelbrot with Quaternion Hue colorisation
Quaternion color rotations	PGA3D Points and Lines
PGA3D Distances and Angles	PGA3D Rotors and Translators
PGA3D Slicing	PGA3D Icosahedron
PGA3D Sampling	PGA3D Automatic Differentiation
PGA2D Points and Lines	PGA2D Distances and Angles
PGA2D Project and Reject	PGA2D Rotors and Translators
PGA2D Isometries	PGA2D Inverse Kinematics
PGA2D Separating Axis	PGA2D Pose Estimation
PGA2D Euler Line	PGA2D Desargues Theorem
CGA2D Euler Line	CGA2D Basics

ganja.js is also the engine behind the GAlculator - try it online or get it on the play store

Or - get some hands on experience with eucledian plane PGA by playing the wedge game.

Ganja for experienced users.

Ganja.js allows you to further customise the algebra class it generates, allowing you to generate subalgebras (who's elements use less storage), or algebra's where you decide on the order and name of the basis blades. (the name should always be e_xyz but you can pick e.g. e₂₀ instead of the default e₀₂ and expect ganja.js to make appropriate sign changes)

The advanced options are available by passing in an options object as the first parameter to the Algebra call.

Custom subalgebra's

// The complex numbers as the even subalgebra of R2 
C = Algebra({p:2,basis:['1','e12']});

// The Quaternions as the even subalgebra of R3
var H = Algebra({p:3,basis:['1','e12','e13','e23']});

Custom basis names.

When not specified, ganja.js will generate basis names that are grouped by rank and numerically sorted. By default, a single zero dimension will get generator name e₀. Zero dimensions come first.

signature	default basis names
2,0,0 and 1,1,0	1,e₁,e₂,e₁₂
1,0,1	1,e₀,e₁,e₀₁
3,0,0 and 2,1,0	1,e₁,e₂,e₃,e₁₂,e₁₃,e₂₃,e₁₂₃
2,0,1	1,e₀,e₁,e₂,e₀₁,e₀₂,e₁₂,e₀₁₂
4,0,0 and 3,1,0	1,e₁,e₂,e₃,e₄,e₁₂,e₁₃,e₁₄,e₂₃,e₂₄,e₃₄,e₁₂₃,e₁₂₄,e₁₃₄,e₂₃₄,e₁₂₃₄
3,0,1	1,e₀,e₁,e₂,e₃,e₀₁,e₀₂,e₀₃,e₁₂,e₁₃,e₂₃,e₀₁₂,e₀₁₃,e₀₂₃,e₁₂₃,e₀₁₂₃

note the scalar part of a multivector "mv" can be addressed with "mv.s", other basis blades follow the expected pattern. e.g. "mv.e12" or "mv.e012".

By default, your algebra elements will inherit from Float32Array. You can change the underlying datatype used by ganja.js to any of the typed array basis types :

var R3_32 = Algebra(3);

var R3_64 = Algebra({p:3,baseType:Float64Array});

Custom Cayley Table

Or take things a bit further and simply specify a Cayley table to your liking. The example below shows automatic numerical differentiation and calculates the value, 1st, 2nd and 3rd derivative of any polynomial.

var basis=['1','e1','e2','e3'];
var Cayley=[['1', 'e1','e2','e3'],
            ['e1','e2','e3', '0'],
            ['e2','e3', '0', '0'],
            ['e3', '0', '0', '0']];

Algebra({basis,Cayley},()=>{
  var f = (x)=>0.25*x*x*x*x-0.5; 
  for (var i=-5; i<5; i++) console.log( i, f(i+1e1) );
});

outputs : x [f(x),f'(x),f''(x)/2!,f'''(x)/3!]

-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.

For storage and performance reasons it can be interesting to combine elements of various sub-algebras of a given Algebra. Ganja.js supports this by setting options.mix to true when you create your algebra.

// Create R2 - Clifford algebra of 2D vectors - indicate you want mix mode.
var R2 = Algebra({p:2,q:0,r:0,mix:true}); 

// Create the complex numbers as the even subalgebra of R2.
var C  = Algebra({p:2,q:0,r:0,basis:['1','e12'],mix:true});

// Elements of R2 have four components.
// Create the complex number 1+4i
var a=new R2([1,0,0,4])

// Elements of C have two components.
// Create the complex number 3+2i
var b=new C([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]

With the mix mode enabled, all operations generated by ganja.js will use basis name access instead of array indexing. (and all operations are protected to substitute missing blades with 0).

The Inline syntax can still be used, keep in mind that in most cases you would want that to be the inline function of the 'parent' algebra. (In the example above, use R2.Inline and not C.Inline as the latter will reduce all your operations to the field of C).

Ganja ingredients and syntax.

Here's a list of the supported operators in all syntax flavors :

Please note that operator precedence is as always in javaScript, except for Wedge, Vee, Dot and Sandwich which have higher precedence than * and /, resulting in less brackets in many common GA expressions.

Precedence	Inline JS	AsciiMath	Object Oriented	Functional
5	x.Involute	tilde(x)	x.Involute	A.Involute(x)
5	x.Reverse	ddot(x)	x.Reverse	A.Reverse(x)
5	~x	hat(x)	x.Conjugate	A.Conjugate(x)
5	!x	bar(x)	x.Dual	A.Dual(x)
4 rtl	x**-1	x^-1	x.Inverse	A.Inverse(x)
4 rtl	x**y	x^y	x.Pow(y)	A.Pow(x,y)
3	x^y	x^^y	x.Wedge(y)	A.Wedge(x,y)
3	x<<y	x*y	x.Dot(y)	A.Dot(x,y)
3	x&y	bar(bar(x)^^bar(y))	x.Vee(y)	A.Vee(x,y)
3	x>>>y	x y hat(x)	x.Mul(y).Mul(x.Conjugate)	A.sw(x,y)
2	x*y	x**y	x.Mul(y)	A.Mul(x,y)
2	x/y	x/y	x.Div(y)	A.Div(x,y)
1	x-y	x-y	x.Sub(y)	A.Sub(x,y)
1	x+y	x+y	x.Add(y)	A.Add(x,y)
	1e1	1e_1	new A([0,1])	A.Vector(1)
	2e2	2e_2	new A([0,0,2,0])	A.Vector(0,2)
	2e12	2e_12	new A([0,0,0,2])	A.Bivector(2)

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

Want to get started quickly with 2D Projective Geometric Algebra ? The boiler plate below gets you going with a bunch of usefull identities. (and the coffeeshop has plenty of examples).

We start off with a clifford algebra with signature (2,0,1).

var PGA2D = Algebra(2,0,1);

Next, we upgrade it to a geometric algebra by extending it with geometric operators. (this is where we decide our bivectors will be points, effectively making this P(R*_2,0,1).

PGA2D.inline(function(){ 
  this.point           = (X,Y)=>1e12-X*1e02+Y*1e01;
  this.line            = (a,b,c)=>a*1e1+b*1e2+c*1e0;
  this.join            = (x,y)=>x&y;
  this.meet            = (x,y)=>x^y;
  this.dist_points     = (P1,P2)=>(P1.Normalized&P2.Normalized).Length;
  this.dist_point_line = (P,l)=>((P.Normalized)^(l.Normalized)).e012;
  this.angle_lines     = (l1,l2)=>(l1.Normalized<<l2.Normalized).s;
  this.project         = (P,l)=>(P<<l)*l;
  this.parallel        = (P,l)=>(P<<l)*P;
  this.ortho           = (P,l)=>P<<l;
  this.rotor           = (a,P)=>Math.cos(a*0.5)+Math.sin(a*0.5)*P;
  this.translator      = (x,y)=>1+0.5*(x*1e20-y*1e01);
})();

We can now use our 2D Projective Algebra. We'll use the graph function to visualize some algebraic objects (i.e. lines and points).

PGA2D.inline(function(){ 

   var O = PGA2D.point(-1,-1), 
       Y = PGA2D.point( 1,-1), 
       X = PGA2D.point(-1, 1),
       z = PGA2D.join(Y,X),
       o = -1*PGA2D.ortho(O,z),
       rot = this.rotor(0.3,O);

   document.body.appendChild(this.graph({
     O,X,Y,
     "color1"        : 0xff0000,
     "O˅X"           : PGA2D.join(O,X),
     "O˅Y"           : PGA2D.join(Y,O),
     "z=X˅Y"         : z,
     "color2"        : 0x008800,
     "o=ortho(O,z)"  : o,
     "parallel(O,z)" : PGA2D.parallel(O,z),
     "rot*o*~rot"    : rot>>>o,
     "color3"        : 0x8888ff,
     "proj(O,z)"     : PGA2D.project(O,z),
     "rot*X*~rot"    : rot>>>X,
   }))

})();

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

This example implements the table on page 15 of Gunn's Geometric Algebra for Computer Graphics. We apply the same strategy from above and start from a Clifford Algebra in R_3,0,1.

var P3 = Algebra({
           metric:[0,1,1,1],
           basis:['1','e0','e1','e2','e3','e01','e02','e03','e12','e31','e23','e123','e012','e031','e023','e0123']
         });

Extending it with geometric operators to form P(R*_3,0,1).

P3.inline(function(){ 
  // We extend the basic (clifford) algebra with PGA specific items.
  // Easier names for triVectors.  
    var E0=1e123, E1=1e012, E2=1e031, E3=1e023;
  // Homogenous points.
    this.point    = (X,Y,Z)=>E0+X*E1+Y*E2+(Z||0)*E3,
    this.to_point = (p)=>p.e123?[p.e012/p.e123,p.e023/p.e123,p.e031/p.e123]
                        :[p.e012*Infinity||0,p.e023*Infinity||0,p.e031*Infinity||0],
  // Join and Meet
    this.join = (x,y)=>x&y;
    this.meet = (x,y)=>x^y;
  // Table from "Geometric Algebra for Computer Graphics" p.15
    this.LineFromPoints            = (P,Q)=>P3.join(P,Q);
    this.LineFromPlanes            = (a,b)=>a^b;
    this.PointFromPlanes           = (a,b,c)=>a^b^c;
    this.PlaneFromPoints           = (P,Q,R)=>P3.join(P3.join(P,Q),R);
    this.DistPointToPlane          = (a,P)=>P3.join(a,P);
    this.DistPoints                = (P,Q)=>P3.join(P,Q).Length;
    this.AnglePlanes               = (a,b)=>Math.acos((a<<b).Length);
    this.LineThroughPointPerpPlane = (P,a)=>P<<a;
    this.OrthProjPointToPlane      = (P,a)=>(P<<a)*a;
    this.PlaneThroughPointParPlane = (P,a)=>(P<<a)*P;
    this.IntersectLinePlane        = (PI,a)=>PI^a;
    this.PlaneThroughPointPerpLine = (PI,P)=>P<<PI;
    this.OrthProjPointToLine       = (PI,P)=>(P<<PI)*PI;
    this.LineThroughPointParLine   = (PI,P)=>(P<<PI)*P;
    this.LineThroughPointPerpLine  = (PI,P)=>P3.join((P<<PI)*P,P);
    this.DistLines                 = (PI,EP)=>P3.join(PI,EP);
    this.AngleLines                = (PI,EP)=>Math.acos((PI<<EP).Length);
    this.ReflectionInPlane         = (a,X)=>a*X*a;
    this.Rotor                     = (PI,alpha)=>Math.cos(alpha/2) + Math.sin(alpha/2)*PI;
    this.RotationAroundLine        = (X,PI,alpha)=>this.rotor(PI,alpha)*X*~this.rotor(PI,alpha);
    this.Translator                = (x,y,z)=>1+0.5*(x*1e03+y*1e01+z*1e02);
})();

Resulting in P3, the geometric algebra for projective euclidean space.

P3.inline(function(){ 
// points
  var o=P3.point(0,0,0), x=P3.point(1,0,0), y=P3.point(0,1,0), z=P3.point(0,0,1);
// lines
  var ox=P3.LineFromPoints(o,x), oy=P3.LineFromPoints(o,y);
// planes
  var oxy=P3.PlaneFromPoints(o,x,y), oyz=P3.PlaneFromPoints(o,y,z), xyz=P3.PlaneFromPoints(x,y,z);
// distance
  console.log('distances |o,x| and |x,y| = ',P3.DistPoints(o,x),',', P3.DistPoints(x,y));
// angles
  console.log('angles lines |ox,oy| and |ox,ox| = ',P3.AngleLines(ox,oy),',',P3.AngleLines(ox,ox));
  console.log('angles planes |oxy,oyz| and |oxy,oxy| =',P3.AnglePlanes(oxy,oyz),',',P3.AnglePlanes(oxy,oxy));
// projection
  console.log('project o onto xyz = ',P3.to_point(P3.OrthProjPointToPlane(o,xyz)));
// rotation and translation
  var tran = P3.Translator(2,3,4), rot = P3.Rotor(x,Math.PI);
  console.log('x translated 2,3,4 = ',P3.to_point(tran*x*~tran));
  console.log('o rotated PI around x = ',P3.to_point(rot*o*~rot));
})();

License

unao/ganja.js

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