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Quick Start (G2)

This notebook gives a terse introduction to using the clifford module, using a two-dimensional geometric algebra as the context.

Setup

First, import clifford and instantiate a two-dimensional algebra (G2),

from numpy import e,pi
import clifford as cf

layout, blades = cf.Cl(2) # creates a 2-dimensional clifford algebra

Inspect blades.

blades

{'': 1, 'e1': (1^e1), 'e2': (1^e2), 'e12': (1^e12)}

Assign blades to variables

e1 = blades['e1']
e2 = blades['e2']
e12 = blades['e12']

Basics

e1*e2 # geometric product

(1^e12)

e1|e2 # inner product

0

e1^e2 # outer product

(1^e12)

Reflection

a = e1+e2    # the vector
n = e1       # the reflector
-n*a*n.inv() # reflect `a` in hyperplane normal to `n`

-(1.0^e1) + (1.0^e2)

Rotation

from numpy  import pi

R = e**(pi/4*e12) # enacts rotation by pi/2
R

0.70711 + (0.70711^e12)

R*e1*~R    # rotate e1 by pi/2 in the e12-plane

-(1.0^e2)