# GAlgebra.jl

Julia interface to GAlgebra, a symbolic Geometric Algebra/Calculus package for SymPy.

## Development Status

Very early. But it already works and has many tests.

Build Status

## Getting Started

GAlgebra.jl itself doesn't depend on SymPy.jl, but is designed to work with it.

After installing SymPy.jl and GAlgebra.jl (see below for instructions), you may experiment with GAlgebra.jl just like in the Python version of GAlgebra (though there're some syntax differences between Python and Julia like True/true, '/" etc.).

import SymPy: symbols, sympy, Sym
using GAlgebra

# In console, uncomment to enable colored printing with ANSI escape sequences
# galgebra.printer.Eprint()
# In Jupyter, uncomment to enable LaTeX printing with MathJax
# galgebra.printer.Format()

(x, y, z) = xyz = symbols("x,y,z",real=true)
o3d = galgebra.ga.Ga("e", g=[1, 1, 1], coords=xyz)

u = o3d.mv("u", "vector")
v = o3d.mv("v", "vector")
A = o3d.mv("A", "mv")
R = o3d.mv("R", "spinor")
# Volume element
I = o3d.I()

# Wedge product: ∧ \wedge
v ∧ A
# Hestenes' inner product: ⋅ \cdot
v ⋅ A
# Left contraction: ⨼ \intprod
v ⨼ A
# Right contraction: ⨽ \intprodr
v ⨽ A
# Scalar product: ⊛ \circledast
# A ⊛ B = <A B†>
v ⊛ A
# Commutator product: ⊠ \boxtimes
# A⊠B = (AB-BA)/2
v ⊠ A
# Anti-commutator product: ⊙ \odot
# A⊙B = (AB+BA)/2
v ⊙ A

# Norm: norm(A) = A.norm() := ||A||
norm(v)

# Inverse: postfix ⁻¹ \^-\^1
# (A)⁻¹ = A^-1 = inv(A) = A.inv()
(R)⁻¹
R^-1
inv(R)

# Reversion: ~A = rev(A) = A.rev()
# A^† is usually used in literature
~A
rev(A)

# Dual: postfix '
# orthogonal complement, Λ^p -> Λ^(n-p)
# note: Ga.dual_mode_value is default to "I+", so A' = A * I
# change Ga.dual_mode_value to get a different definition
A'
dual(A)

# Grade involution: postfix ˣ \^x
# (A)ˣ = A[:*] = involute(A) := A+ - A- = A.even() - A.odd()
# A^* is usually used in literature
(A)ˣ
involute(A)

# Clifford conjugate: postfix ǂ \doublepipe
# (A)ǂ = conj(A) := ((A)^*)^†
(A)ǂ
conj(A)

# Projection: proj(B, A) = A.project_in_blade(B)
proj(u, v)

# Reflection: refl(B, A) = A.reflect_in_blade(B)
refl(u, v)

# Rotation: rot(itheta, A) = A.rotate_multivector(itheta)
# rotate the multivector A by the 2-blade itheta
rot(u ∧ v, A)

# Natural base exponential of x: e^x
exp(u ∧ v)

A[2]

# Scalar (grade-0) part: scalar(A) = A.scalar() := <A> = <A>_0
# note: it returns a SymPy expression unlike A[0] which returns a Mv object
scalar(A)

# Even-grade part: A[:+] = (A)₊ = even(A) = A.even() := A+
A[:+]
even(A)

# Odd-grade part: A[:-] = (A)₋ = odd(A) = A.odd() := A-
A[:-]
odd(A)

Note: enter unicode symbols like ∧ with corresponding LaTeX commands like \wedge by Tab completion which are provided in the comments.

So far only galgebra.ga.Ga and galgebra.mv.Mv have been verified to work in Julia, see tests. The tests verified many identities in Linear Algebra and Geometric Algebra.

See examples of GAlgebra for more examples in Python. With some small changes, they can be ported to Julia.

## Installation

The package can be installed with the Julia package manager. From the Julia REPL, type ] to enter the Pkg REPL mode (you'll see a prompt like (v1.1) pkg>) and run:

dev https://github.com/pygae/GAlgebra.jl.git

The installation process will take a while, because it will install PyCall.jl and its dependencies and it will also use pip to install GAlgebra if you haven't. If you wish to use a version other than the latest released version of GAlgebra that will be installed by deps/build.jl, you can install that via pip before/after the installation of GAlgebra.jl, e.g. pip install -e <your local path to GAlgebra> or pip install https://github.com/pygae/galgebra/archive/master.zip.

Now you may run GAlgebra.jl tests in Julia Pkg REPL mode:

test GAlgebra

At the first time it will take a while, because SymPy.jl is specified as a test dependency so it'll be installed.

Then you'll see something like:

   Testing GAlgebra
Resolving package versions...
Test Summary: | Pass  Broken  Total
GAlgebra.jl   | 1289       1   1290
Testing GAlgebra tests passed

Hint: To get back to the Julia REPL please press backspace, see Pkg doc to learn more.