repository·master·Indexed 21 days ago
https://github.com/pygae/cliffordA numerical Geometric Algebra (Clifford algebra) package for Python that allows for the seamless mixing of scalars, vectors, and higher-grade entities as multivectors. It provides tools for defining algebraic structures via Cl(), managing multivectors with the MultiVector class, and performing operations like 3D rotations using rotors. The library includes predefined algebras such as g2, g3, and pga, as well as utilities like conformalize() for mapping to Conformal Geometric Algebra.
The clifford module implements Geometric Algebras (Clifford algebras). It is a universal algebra that subsumes several mathematical systems, including:
In this library, scalars, vectors, and higher-grade entities can be mixed freely and consistently in the form of mixed-grade multivectors.
The clifford module does not prioritize preserving array dtype or optimizing storage.
MultiVectors will have integer coefficients if instantiated that way. Dividing them by Python integers follows standard Python integer division rules.To use Geometric Algebra for 3D space, import from clifford.g3. You can define vectors using basis vectors (e.g., e1, e2, e3) and perform rotations using rotors. A rotor $R$ can rotate a vector $a$ using the sandwich product $R a \tilde{R}$, where ~ denotes the reverse of the multivector.
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 vectorTo begin using Geometric Algebra for 3D space, import the g3 module from clifford. This provides the basis vectors e1, e2, and e3. You can then create vectors, rotors, and perform geometric operations like rotations using multivector arithmetic.
Key concepts:
e1, e2, e3 represent the orthogonal axes.math.e**(math.pi/4*e12)).a with a rotor R is performed via R * a * ~R, where ~R is the inverse of the rotor.from clifford.g3 import *
import math
a = e1 + 2*e2 + 3*e3 # vector
R = math.e**(math.pi/4*e12) # rotor
# rotate the vector
result = R * a * ~R
# Output: (2.0^e1) - (1.0^e2) + (3.0^e3)You can supplement your use of the clifford library with the following educational resources:
clifford.To install the latest development version from the master branch on GitHub, use the following command:
pip3 install git+https://github.com/pygae/clifford.git@mastermpl_toolkits.clifford integrates with matplotlib, allowing you to use standard plotting workflows. This is useful when you need to combine geometric visualizations with other scientific plots.
Key usage details:
plot(ax, [list_of_objects], ...) where ax is a matplotlib Axes object.plt.subplots().plt.subplots(..., subplot_kw=dict(projection='3d')).color, label, marker, linestyle, linewidth, and markeredgewidth.mpl_toolkits.clifford.clifford.tools.classify, you may need to call .normal() on objects to ensure they render correctly.from matplotlib import pyplot as plt
from mpl_toolkits.clifford import plot
fig, ax = plt.subplots()
plot(ax, [point, line, circle], color='tab:blue', label='objects')
plt.show()You can install the clifford package using pip3. If your Python installation is system-wide and you are not running as sudo, append the --user flag to the command.
pip3 install cliffordInstead of manually constructing a Clifford algebra using clifford.Cl, you can use predefined algebra modules. These modules are optimized for faster startup and provide convenient access to basis blades and layout information.
Available predefined algebras include:
clifford.g2: 2D Euclidean, Cl(2)clifford.g3: 3D Euclidean, Cl(3)clifford.g4: 4D Euclidean, Cl(4)clifford.g2c: Conformal space for G2, Cl(3, 1)clifford.g3c: Conformal space for G3, Cl(4, 1)clifford.pga: Projective space for G3, Cl(3, 0, 1)clifford.pga2d: Projective space for G2, Cl(2, 0, 1)clifford.gac: Geometric Algebra for Conics, Cl(5, 3)clifford.dpga: Double PGA (Mother Algebra), Cl(4, 4)clifford.dg3c: Double Conformal Geometric Algebra, Cl(8, 2)from clifford import g2
# Access basis blades directly from the module
blade = g2.e1 * g2.e2Install the clifford package using either conda or pip. Requires Python version >=3.5.
# Using conda
conda install clifford -c conda-forge
# Using pip
pip3 install cliffordYou can install the two primary visualization tools for clifford using pip:
pyganja: A Python interface to the ganja.js library.mpl_toolkits.clifford: A toolkit for visualizing geometric objects within matplotlib.pip install pyganja
pip install mpl_toolkits.cliffordFor a deeper theoretical understanding of Geometric Algebra beyond the clifford library, consider these resources: