GeometricAlgebra.jl

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GeometricAlgebra defines a collection of basic types and operations that support numerical geometric algebra computations. Our aim is to simplify the process of implementing numerical algorithms for geometric operations expressed algebraically in the language of geometric algebra.

The GeometricAlgebra package features:

• an easy-to-use interface for constructing geometric algebra objects,

• intuitive notation for operations on and between geometric algebra objects, and

• behavior (including limitations) analogous to standard numerical linear algebra libraries.

Overview

Geometric algebra extends classical linear algebra (of inner product spaces over the real numbers $\mathbb{R}$) by including the higher-dimensional siblings of the vector (a 1-dimensional object) and defining algebraic operations that augment the basic operations on inner product spaces (i.e., vector addition, scalar multiplication, and inner product). With these extensions, geometric algebra makes it possible to use simple algebraic expressions to represent geometric objects (e.g., subspaces) and operations (e.g., projection, computing the orthogonal complement).

The key mathematical object introduced in geometric algebra is the blade. Geometrically, blades represent two concepts (1) subspaces of $\mathbb{R}^n$ and (2) $k$-dimensional hypervolumes (with dimension less than or equal to $n$ and arbitrary shape). The exterior product ($\wedge$), which combines lower-grade blades into higher-grade blades (or zero), is an example of an algebraic operation with geometric meaning. For instance, it enables any blade $B$ to be expressed algebraically in terms of a basis $b_1, \ldots b_k$ for the subspace represented by the blade:

$$ B = b_1 \wedge \cdots \wedge b_k. $$

The GeometricAlgebra package implements types and functions that make it straightforward to do computational geometric algebra calculations.

Getting Started

• Add the Velexi Julia package registry.

  julia>  # Press ']' to enter the Pkg REPL mode.
  pkg> registry add https://github.com/velexi-research/JuliaRegistry.git

  Notes

  • Only needed once. This step only needs to be performed once per Julia installation.

  • GeometricAlgebra is registered with a local Julia package registry. The Velexi registry needs to be added to your Julia installation because GeometricAlgebra is currently registered with Velexi Julia package registry (not the General Julia package registry).

• Install the GeometricAlgebra package via the Pkg REPL. That's it!

  julia>  # Press ']' to enter the Pkg REPL mode.
  pkg> add GeometricAlgebra

Examples

• Create a blade.

  julia> vectors = Matrix([3 3; -4 -4; 0 1]);
  
  julia> B = Blade(vectors)
  Blade{Float64}(3, 2, [-0.6 0.0; 0.8 0.0; 0.0 -1.0], 5.0)
  
  julia> dim(B)
  3
  
  julia> grade(B)
  2
  
  julia> norm(B)
  5.0

• Compute the outer product of vectors (1-blades).

  julia> u = [3, -4, 0, 0];  # a vector is a 1-blade
  
  julia> v = [3, -4, 1, 0];  # a vector is a 1-blade
  
  julia> B = u ∧ v
  Blade{Float64}(3, 2, [-0.6 0.0; 0.8 0.0; 0.0 -1.0], 5.0)
  
  julia> grade(Blade(u))
  1
  
  julia> grade(Blade(v))
  1
  
  julia> grade(B)
  2

• Compute the outer product of blades.

  julia> A = Blade([0, 0, 0, 2]);
  
  julia> B = Blade(Matrix([3 3; -4 -4; 0 1; 0 0]));
  
  julia> C = A ∧ B
  Blade{Float64}(4, 3, [0.0 0.6 0.0; 0.0 -0.8 0.0; 0.0 0.0 1.0; -1.0 0.0 0.0], -10.0)
  
  julia> grade(A)
  1
  
  julia> grade(B)
  2
  
  julia> grade(C)
  3

• Compute the additive and multiplicative inverses of a vector (1-blade).

  julia> v = Blade([3, 4, 0, 0])
  Blade{Float64}(4, 1, [0.6; 0.8; 0.0; 0.0;], 5.0)
  
  julia> -v
  Blade{Float64}(4, 1, [0.6; 0.8; 0.0; 0.0;], -5.0)
  
  julia> inv(v)
  Blade{Float64}(4, 1, [0.6; 0.8; 0.0; 0.0;], 0.2)
  
  julia> 1 / v
  Blade{Float64}(4, 1, [0.6; 0.8; 0.0; 0.0;], 0.2)
  
  julia> v^-1
  Blade{Float64}(4, 1, [0.6; 0.8; 0.0; 0.0;], 0.2)

• Create a pseudoscalar for $\mathbb{G}^n$.

  julia> n = 10;
  julia> value = -2;
  julia> B = Pseudoscalar(n, value)
  Pseudoscalar{Float64}(10, -2.0)
  
  julia> grade(B)
  10
  
  julia> norm(B)
  2.0
  
  julia> volume(B)
  -2.0

• Compute the dual of a blade.

  julia> vectors = Matrix([3 3; -4 -4; 0 1]);
  
  julia> B = Blade(vectors);
  
  julia> dual(B)
  Blade{Float64}(3, 1, [0.8; 0.6; 0.0;], -5.0)
  
  julia> isapprox(dot(B.basis[:, 1], dual(B).basis), 0; atol=1e-15)
  true
  
  julia> isapprox(dot(B.basis[:, 2], dual(B).basis), 0; atol=1e-15)
  true
  
  julia> B ∧ dual(B)
  Pseudoscalar{Float64}(3, 25.0)

  We can confirm that is the dual of blade $B$ is the same as division by the unit pseudoscalar for $\mathbb{G}^n$.

  julia> dual(B) ≈ B / Pseudoscalar(3, 1)
  true

• Compute the geometric product of two vectors. Note that the geometric product of two vectors is a multivector (a linear combination of blades).

  julia> u = Blade([1, 0, 0]);
  
  julia> v = Blade([sqrt(3), 1, 0]);
  
  julia> M = u * v
  Multivector{Float64}(3, DataStructures.SortedDict{Int64, Vector{AbstractBlade}, Base.Order.ForwardOrdering}(0 => [Scalar{Float64}(1.7320508075688776)], 2 => [Blade{Float64}(3, 2, [1.0 0.0; 0.0 1.0; 0.0 0.0], 1.0)]), 2.8284271247461903)
  
  julia> M[0]
  1-element Vector{AbstractBlade}:
   Scalar{Float64}(1.7320508075688776)
  
  julia> M[1]
  AbstractBlade[]
  
  julia> M[3]
  1-element Vector{AbstractBlade}:
   Blade{Float64}(3, 2, [1.0 0.0; 0.0 1.0; 0.0 0.0], 1.0)

  Right multiplication of M by the multiplicative inverse of v yields u.

  julia> M * inv(v) ≈ u
  true
  
  julia> M * (1 / v) ≈ u
  true

  Left multiplication of M by the multiplicative inverse of u yields v.

  julia> inv(u) * M ≈ v
  true
  
  julia> (1 / u) * M ≈ v
  true

Related Packages

There are several active geometric algebra packages in the Julia ecosystem. Most of the available packages are implemented using an additive blade representation and emphasize symbolic computations. To the best of our knowledge, GeometricAlgebra.jl is the only package currently uses a multiplicative blade representation and focuses on numerical computations.

• Grassmann.jl

• Multivectors.jl

• Jollywatt/GeometricAlgebra.jl

• serenity4/GeometricAlgebra.jl

• GAlgebra.jl