Galixir (galixir v0.22.0)

Galixir is a library for working with Geometric Algebra (GA) in Elixir.

Geometric Algebra provides a unified mathematical language for describing geometric objects such as points, vectors, lines, planes, circles, spheres, and transformations. Instead of representing each object type separately, GA represents them as multivectors and uses algebraic operations to construct and manipulate them.

Galixir provides tools for defining geometric algebras with arbitrary signatures and automatically generating the operations required to work with them.

Geometric Algebras

A geometric algebra is defined by its metric signature. For example:

defmodule PGA3 do

 use Galixir.GeometricAlgebra,
   signature: {1, 1, 1, 0},
   bases: {1, 2, 3, 0}

end

This creates a projective geometric algebra for 3D Euclidean geometry containing basis vectors, multivectors, and operations such as:

geometric product (gp/2)
outer product (wedge/2)
inner product (dot/2)
reverse
dual
inverse
normalization

Multivectors

Multivectors are combinations of basis blades. For example:

iex> import Galixir.Algebras.PGA3, only: [sigil_G: 2]
iex> Galixir.Algebras.PGA3.new(e1: 1, e2: 2, e3: 3)
~G"1.0e1 + 2.0e2 + 3.0e3"

A multivector can contain components of different grades:

iex> import Galixir.Algebras.PGA3, only: [sigil_G: 2]
iex> Galixir.Algebras.PGA3.new(e1: 1, e23: 2, scalar: 3)
~G"3.0 + 1.0e1 + 2.0e23"

Geometry

Higher-level geometry modules build on top of the generated algebra.

For example, projective geometric algebra can represent:

points
directions
lines
planes
motors (rotations and translations)

Conformal geometric algebra extends this further with representations for:

points
circles
spheres
intersections and joins

Design Goals

Galixir aims to provide:

a native Elixir implementation of geometric algebra
compile-time generation of efficient algebra operations
explicit and inspectable multivector representations
a foundation for computational geometry, robotics, graphics, and simulation

Mathematical Background

Geometric algebra is based on the Clifford algebra product:

ab = a · b + a ∧ b

In Elixir written as:

iex> alias Galixir.Algebras.PGA2
iex> {a, b} = {PGA2.vector(1, 2), PGA2.vector(2, 3)}
iex> ab = PGA2.add(PGA2.inner(a, b), PGA2.wedge(a, b))
iex> assert ab == PGA2.gp(a,b)

where the geometric product combines the metric-dependent inner product with the antisymmetric outer product.

Unlike traditional approaches that require different data structures for different geometric primitives, GA represents many geometric entities in a single algebraic framework.