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.
cite as [](https://zenodo.org/badge/latestdoi/98999731)
```bibtex
@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}
}
```
(**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 morphisms.)
(**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.)
(**Practically**, ganja.js enables real math syntax inside javascript, with element, vector and matrix
operations over **reals**, **complex numbers**, **dual numbers**, **hyperbolic numbers**, **vectors**, **spacetime events**, **quaternions**, **dual quaternions**, **biquaternions** or **any other Clifford Algebra**.)
(**Seriously**, look at the [examples](https://enkimute.github.io/ganja.js/examples/coffeeshop.html),
run some quick numbers using the [GAlculator](https://enkimute.github.io/ganja.js/examples/galculator.html)
or play [the wedge game](https://enkimute.github.io/ganja.js/examples/example_game_wedge.html) first.)
## Discourse and Discord
Visit [bivector.net](https://bivector.net) for our forum and chat - the perfect
place for questions and support.
## New !
ganja.js now has a nodejs based templated source generator that allows the creation of arbitrary algebras
for **C++**, **C#**, **python** and **rust**. The generated code provides in a flat multivector format and operator overloading.
Check the 'codegen' folder for the source, several algebras are available in pregenerated versions.
* [**C++ pre-generated sources**](https://github.com/enkimute/ganja.js/tree/master/codegen/cpp)
* [**C# pre-generated sources**](https://github.com/enkimute/ganja.js/tree/master/codegen/csharp)
* [**rust pre-generated sources**](https://github.com/enkimute/ganja.js/tree/master/codegen/rust)
* [**python pre-generated sources**](https://github.com/enkimute/ganja.js/tree/master/codegen/python)
## Contents
[1. Reasons to use ganja](#Features)[2. Using ganja for the first time](#Started)
[3. Getting free ganja samples](#samples)
[4. Ganja for experienced users](#custom)
[5. Ganja ingredients and syntax](#syntax)
[6. Ganja starterkit : PGA2D P(R*2,0,1)](#P2)
[7. Ganja starterkit : PGA3D P(R*3,0,1)](#P3)
## 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) and dimensionality (also +10) * smallish (20kb on the wire) * matrix-free inverses up to 5D. * geometric, inner (left contraction), outer (wedge) and regressive (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 and 3D elements. (SVG/webGL/OPNS) * Supports vectors and matrices in all its algebras. * There's a [game](https://enkimute.github.io/ganja.js/examples/example_game_wedge.html) that teaches you how to use ganja.js ! ## Using ganja for the first time Install ganja.js using npm : ``` npm install ganja.js ``` And require it in your script : ```javascript var Algebra=require('ganja.js'); ``` Or in the browser, just include the ganja.js script. (ganja.js has no dependencies)  ```html ``` ### 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. ```javascript 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. ``` javascript 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 : ```javascript // 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. // High-Dimensional GA var DCGA3D = Algebra(6,2); // Double Conformal 3D Space. var TCGA3D = Algebra(9,3); // Triple Conformal 3D Space. var DCGSTA = Algebra(4,8); // Double Conformal Geometric Space Time Algebra. var QCGA = Algebra(9,6); // Quadric Conformal Geometric Algebra. ``` 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. ```javascript 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 : ```javascript 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. ```javascript 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 arrays or lambda expressions without the need for calling brackets in algebraic expressions. ```javascript 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; // Binary operations on arrays also work as expected. var even = [1,2,3,4,5]*2; // Even if those contain multivectors or other arrays : var funky = [1, 1e01+0.5e02, [3,4]] * 3 + [1,2,3]; // 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. ```javascript // 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](https://enkimute.github.io/ganja.js/examples/coffeeshop.html) 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 and CGA elements. * canvas output is available for 1D and 2D functions. * SVG output is available for 2D PGA, 3D PGA and 2D CGA. * webGL output is available for 3D PGA and 3D CGA. * webGL2 implicit OPNS rendering is available for all other spaces. ```javascript 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 PGA svg = Algebra(3,0,1,()=>this.graph([1e123,1e23,1e13,1e12],{camera:1+.5e01-.5e02})); // and in 3D PGA canvas = Algebra(4,1,()=>this.graph([.5e4-.5e5],{conformal:true,gl:true}); // The origin in 3D CGA ``` Again, many more examples can be found at [the coffeeshop](https://enkimute.github.io/ganja.js/examples/coffeeshop.html). ### The describe function. To display the basis blade names, metric, Cayley table and more, use the static **_describe_** function. ```javascript 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](https://enkimute.github.io/ganja.js/examples/coffeeshop.html) 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
|
complex least squares
|
dual differentiation
|
dual backpropagation
|
quaternion hue
|
quaternion mandelbrot
|
timespace lorentz
|
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
|
pga2d differentiation
|
pga2d physics moon
|
pga2d origami
|
pga2d poncelet
|
pga3d points and lines
|
pga3d distances and angles
|
pga3d rotors and translators
|
pga3d icosahedron
|
pga3d sampling
|
pga3d slicing
|
pga3d differentiation
|
pga3d skinning
|
pga3d physics planets
|
pga3d origami
|
pga3d physics symmetric top
|
pga3d physics free top
|
pga3d objects
|
cga2d points and circles
|
cga2d project and reject
|
cga2d rotors and translators
|
cga2d euler line
|
cga3d points circles lines
|
cga3d points spheres planes
|
cga3d dual spheres planes
|
cga3d intersections
|
cga3d project reject
|
cga3d opns visualizer
|
cga3d opns line circle
|
cga3d json
|
mga3d points and lines
|
ccga3d points quadrics
|
qcga3d points and more
|
game wedge
|
Or - get some hands on experience with euclidian plane PGA by playing the [wedge game](https://enkimute.github.io/ganja.js/examples/example_game_wedge.html).
## 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 exyz 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
```javascript
// 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 :
```javascript
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.
```javascript
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.
```javascript
// 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<