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This contains series expansions of trigonometric, exponential, and hyperbolic functions.

This also tweaks the implementation of the hitzer inverse to behave slightly better on degenerate algebras, and fixes a crash when computing an inverse in `Cl(0, 0, 0)`.

`cf.general_exp(mv)` has been deprecated in favor of `cf.taylor_expansions.exp(mv)`, or the more convenient spelling `mv.exp()`. The new function is jitted, unlike the old one.

Finally, this overloads pow in numba to allow it to be used in jitted code.

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clifford: Geometric Algebra for Python

PyPI DOI Binder Documentation Status Build Status Build Status codecov

clifford is a numerical Geometric Algebra (a.k.a. Clifford algebra) package for python.

Geometric Algebra (GA) is a universal algebra which among several other independent mathematical systems, subsumes:

  • Complex numbers
  • Quaternions
  • Linear algebra

Scalars, vectors, and higher-grade entities can be mixed freely and consistently in the form of mixed-grade multivectors. Like this,

Visual explanation of blades

Quick Installation

Requires Python version >=3.5

Install using conda:

conda install clifford -c conda-forge

Install using pip:

pip3 install clifford

Detailed instructions


Try out a notebook in binder

Or have a go on your own pc:

from clifford.g3 import *  # import GA for 3D space
from math import e, pi
a = e1 + 2*e2 + 3*e3 # vector 
R = e**(pi/4*e12)    # rotor 
R*a*~R    # rotate the vector  

Syntax Summary

Syntax Operation
| Symmetric inner product
<< Left contraction
^ Outer product
* Geometric product
X(i) Return the section of the multivector X of grade i
X(ei) Return the section of the multivector X for which ei is the pseudo scalar
X[i] Return the i'th coefficient from the multivector X
X.normal() Return the normalised multivector so that X*~X is +- 1

For installation instructions, api documention, and tutorials, head over to our documentation!

Citing This Library

As linked at the top of this page, clifford is published to zenodo. DOI 10.5281/zenodo.1453978 refers to all versions of clifford.

To obtain BibTex citation information for a specific release (recommended):

  • Run python -m pip show clifford to determine which version you are using (or print clifford.__version__ from python)
  • Click on the corresponding version from this list of versions
  • Scroll down to the bottom of the page, and click the "BibTex" link in the "Export" sidebar

If you want to cite all releases, use:

  author       = {Hugo Hadfield and
                  Eric Wieser and
                  Alex Arsenovic and
                  Robert Kern and
                  {The Pygae Team}},
  title        = {pygae/clifford},
  publisher    = {Zenodo},
  doi          = {10.5281/zenodo.1453978},
  url          = {}