Symbolic Geometric Algebra/Calculus package for SymPy.

## Project description

# GAlgebra

Symbolic Geometric Algebra/Calculus package for SymPy.

## Development Status

brombo/galgebra was originally written by Alan Bromborsky, but was no longer actively maintained, and as of 2019-11-25 no longer exists.

pygae/galgebra is a community fork, maintained by Pythonic Geometric Algebra Enthusiasts.

The fork supports Python 3, increases test coverage, sets up CI and linters, maintains releases to PyPI, improves docs and has many bug fixes, see Changelog.

## Features

### Geometric Algebra

- Arbitrary Vector Basis and Metric
- Scalar, Vector, Bivector, Multivector, Pseudoscalar, Spinor, Blade
- Basic Geometic Algebra Operations
- Sum Difference
- Geometric Product
- Outer and Inner Products
- Left and Right Contractions
- Reverse, Dual, Exponential
- Commutator
- Projection, Reflection, Rotation
- Reciprocal Frames

- Inspecting Base/Blade Representation
- Symbolic Manipulations
`expand`

,`factor`

,`simplify`

,`subs`

,`trigsimp`

etc.

Overloaded Python operators for basic GA operations:

```
\begin{split}\begin{aligned}
A+B &= \texttt{A+B} \\
A-B &= \texttt{A-B} \\
AB &= \texttt{A*B} \\
A \wedge B &= \mathtt{A \verb!^! B} \\
A \cdot B &= \texttt{A|B} \\
A \rfloor B &= \texttt{A<B} \\
A \lfloor B &= \texttt{A>B} \\
A/B &= \texttt{A/B} \\
\end{aligned}\end{split}
```

### Geometric Calculus

- Geometric Derivative
- Submanifolds
- Linear Transformations
- Differential Operators

The various derivatives of a multivector function is accomplished by multiplying the gradient operator vector with the function:

```
\begin{aligned}
\nabla F &= \texttt{grad*F} \\
F \bar{\nabla} &= \texttt{F*rgrad} \\
\nabla {\wedge}F &= \mathtt{grad \verb!^! F} \\
F {\wedge}\bar{\nabla} &= \mathtt{F \verb!^! rgrad} \\
\nabla \cdot F &= \texttt{grad|F} \\
F \cdot \bar{\nabla} &= \texttt{F|rgrad} \\
\nabla \rfloor F &= \texttt{grad<F} \\
F \rfloor \bar{\nabla} &= \texttt{F<rgrad} \\
\nabla \lfloor F &= \texttt{grad>F} \\
F \lfloor \bar{\nabla} &= \texttt{F>rgrad}
\end{aligned}
```

```
\begin{aligned}
F \nabla &= \texttt{F*grad} \\
\bar{\nabla} F &= \texttt{rgrad*F} \\
F {\wedge}\nabla &= \mathtt{F \verb!^! grad} \\
\bar{\nabla} {\wedge}F &= \mathtt{rgrad \verb!^! F} \\
F \cdot \nabla &= \texttt{F|grad} \\
\bar{\nabla}\cdot F &= \texttt{rgrad|F} \\
F \rfloor \nabla &= \texttt{F<grad} \\
\bar{\nabla} \rfloor F &= \texttt{rgrad<F} \\
F \lfloor \nabla &= \texttt{F>grad} \\
\bar{\nabla} \lfloor F &= \texttt{rgrad>F}
\end{aligned}
```

Tip: an example for getting `grad`

and `rgrad`

of a 3-d Euclidean geometric algebra in rectangular coordinates:

```
from sympy import symbols
from galgebra.ga import Ga
o3d = Ga('e', g=[1,1,1], coords=symbols('x,y,z',real=True))
(grad,rgrad) = o3d.grads()
```

### Printing

- Enhanced Console Printing
- Latex Printing
- out-of-the-box support for Jupyter Notebook
- PDF generation and croping support if you have
`pdflatex`

/`pdfcrop`

installed

## Getting Started

After installing GAlgebra (see section Installing GAlgebra below), in a Jupyter Notebook:

```
from sympy import symbols
from galgebra.ga import Ga
from galgebra.printer import Format
Format(Fmode = False, Dmode = True)
st4coords = (t,x,y,z) = symbols('t x y z', real=True)
st4 = Ga('e',
g=[1,-1,-1,-1],
coords=st4coords)
M = st4.mv('M','mv',f = True)
M.grade(3).Fmt(3,r'\langle \mathbf{M} \rangle _3')
```

You will see:

```
\begin{aligned} \langle \mathbf{M} \rangle _3 =& M^{txy} \boldsymbol{e}_{t}\wedge \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} \\ & + M^{txz} \boldsymbol{e}_{t}\wedge \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} \\ & + M^{tyz} \boldsymbol{e}_{t}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \\ & + M^{xyz} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{aligned}
```

You may also check out more examples here.

For detailed documentation, please visit https://galgebra.readthedocs.io/ .

**NOTE:** If you are coming from sympy.galgebra or brombo/galgebra, please check out section Migration Guide below.

## Installing GAlgebra

### Prerequisites

- Works on Linux, Windows, Mac OSX
- Python >= 3.8
- 0.5.0 was the last supported release for Python 3.5-3.7
- 0.4.x was the last supported release series for Python 2.7

- SymPy >= 1.3
- Only SymPy 1.12 is tested via CI, see
`.github/workflows/ci.yml`

for more details - 0.5.0 was the last supported release for SymPy 1.7

- Only SymPy 1.12 is tested via CI, see

### Installing GAlgebra From PyPI (Recommended for users)

```
pip install galgebra
```

Then you are all set!

### Installing GAlgebra From Source (Recommended for developers)

To install from the latest source code of GAlgebra:

```
git clone https://github.com/pygae/galgebra.git
cd galgebra
pip install -e .
```

Note that the optional `-e`

argument is used here for a developer install so modifying the source will take effect immediately without the need of reinstallation.

Now you may run tests to verify the installation, run from the root of the repository:

```
pip install pytest
pytest test
```

Further, to run the complete test suite including the ones using nbval, just run:

```
pip install nbval
pytest --nbval examples/ipython/ test --nbval-current-env --nbval-sanitize-with test/.nbval_sanitize.cfg
```

This could take more than 10 minutes, please be patient.

## Migration Guide

Note: The APIs have changed since the era of

`sympy.galgebra`

and`brombo/galgebra`

, some properties and methods are deprecated, the supported versions of Python and SymPy have also changed, please check Changelog and further update your scripts accordingly besides the following. If you encounter any problems, feel free to open an issue!

### Migrating from sympy.galgebra

GAlgebra is no longer part of SymPy since 1.0.0, if you have an import like this in your source:

```
from sympy.galgebra.ga import *
```

Simply remove the `sympy.`

prefix before `galgebra`

then you are good to go:

```
from galgebra.ga import *
```

### Migrating from brombo/galgebra

The `setgapth.py`

way to install is now deprecated by `pip install galgebra`

and all modules in GAlgebra should be imported from `galgebra`

, for example:

```
from galgebra.printer import Format, Eprint, latex, GaPrinter
from galgebra.ga import Ga
from galgebra.mv import Mv, Nga
```

## Bundled Resources

Note that in the doc/books directory there are:

`BookGA.pdf`

which is a collection of notes on Geometric Algebra and Calculus based of "Geometric Algebra for Physicists" by Doran and Lasenby and on some papers by Lasenby and Hestenes.`galgebra.pdf`

which is the original main doc of GAlgebra in PDF format, while the math part is still valid, the part describing the installation and usage of GAlgebra is outdated, please read with caution or visit https://galgebra.readthedocs.io/ instead.`Macdonald`

which contains bundled supplementary materials for Linear and Geometric Algebra and Vector and Geometric Calculus by Alan Macdonald, see here and here for more information.

## Star History

## Contributors

Made with contrib.rocks.

## Citing This Library

For citation information, see our `CITATION.md`

file.

## Project details

## Release history Release notifications | RSS feed

## Download files

Download the file for your platform. If you're not sure which to choose, learn more about installing packages.