GeometricAlgebra.jl

GeometricAlgebra.jl 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 that is consistent with expectations for 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, such as the exterior product and geometric 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) $k$-dimensional subspaces of $\mathbb{R}^n$ and (2) $k$-dimensional hypervolumes (of arbitrary shape). The exterior product ($\wedge$), which combines two blades and yields a blade, 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
```
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

Summary of Types

GeometricAlgebra defines concrete types for the following hierarchy of mathematical objects.

multivector: Multivector
- blade: Blade
  - scalar ($0$-blade) : Scalar
    - multiplicative identity: One
    - additive identity: Zero
  - pseudoscalar ($n$-blade): Pseudoscalar

Summary of Operations

Unary Operations

additive inverse: -
multiplicative inverse: inv
reverse: reverse
dual: dual

Binary Operations

addition/subtraction: +, -
multiplication/division (geometric product): *, /
exterior product: ∧ (wedge)
inner product (left contraction): < (contract_left), dot

Geometric Operations

projection: project

Supported Numeric Types

All GeometricAlgebra types are parametrized by the numeric type used to represent the real numbers. Only subtypes of AbstractFloat are supported (i.e., Float16, Float32, Float64, BigFloat). Other subtypes of Real (e.g., Rational, Irrational) are not allowed as parameters to GeometricAlgebra types.

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.