Geometric (Clifford) algebra

import clifford as cf
layout, blades = cf.Cl(2) # create a 2D clifford algebra

# blades is a dict {'': 1, 'e1': (1^e1), 'e2': (1^e2), 'e12': (1^e12)}
# we can assign the variables e1 = blades['e1'], e2 = blades['e2'] and so on automatically by:
locals().update(blades)
v1 = 3*e1
v2 = e1 + e2
print(v1 ^ v2)

'''
Output
(3^e12)
'''

print(v1^v1)
'''
Output
0
'''
    

layout, blades = cf.Cl(3) # create a 3D clifford algebra
locals().update(blades)
a = 2*e1 + e2 + e3
b = e1 + e2 + 2*e3
print(a * b)    # sum of a scalar and a bivector

'''
Output:
5 + (1^e12) + (3^e13) + (1^e23)
'''

print(a ^ b)    # a bivector expressed as a sum of projections on the basis planes

'''
Output:
(1^e12) + (3^e13) + (1^e23)
'''

import clifford as cf
from math import pi, sin, cos
layout, blades = cf.Cl(2) # create a 2D clifford algebra
locals().update(blades)
x = 1^e1        # the vector to be rotated
theta = pi/4    # the angle of rotation

# pick any two vectors with an angle of theta between them
random_angle = 1.4376
p = (cos(random_angle)^e1) + (sin(random_angle)^e2)
q = (cos(random_angle + theta)^e1) + (sin(random_angle + theta)^e2)
pq = p*q    # the bivector R = pq which is also a rotor since the vectors are unit vectors
print(pq)

'''
Output:
0.70711 + (0.70711^e12)
This is equivalent to cos(theta) + i*sin(theta) in the complex plane
'''

print(pq * x)   # Rv rotates the vector x by -theta
'''
Output:
(0.70711^e1) - (0.70711^e2)
'''

print(x * pq)   # vR rotates the vector x by theta
'''
Output:
(0.70711^e1) + (0.70711^e2)
'''

import clifford as cf
from math import sqrt
layout, blades = cf.Cl(3) # create a 3D clifford algebra
locals().update(blades)
x = (0^e1) + (1^e2) + (-4^e3)        # the vector to be rotated
SQRT14 = sqrt(14)
SQRT3 = sqrt(3)
# define coefficients of unit vectors p and q
p1, p2, p3 = 1/SQRT14, 2/SQRT14, -3/SQRT14
q1, q2, q3 = 1/SQRT3, 1/SQRT3, 1/SQRT3
# define the unit vectors p and q
'''
note that the dot product of p and q is zero i.e p|q = 0
thus, the angle between p and q is 90 degrees
'''
p = (p1^e1) + (p2^e2) + (p3^e3)
q = (q1^e1) + (q2^e2) + (q3^e3)
'''
define the rotor R = p^q
the rotor R will rotate any vector x by an angle of ±90 degrees
'''
pq = p*q
print(f'pq is {pq}')

# now let's rotate the vector 
x_rotated_1 = pq * x
x_rotated_2 = x * pq
print(x_rotated_1)
print(x_rotated_2)
print(f'Dot product between x and x_rotated_1 is {x_rotated_1|x}')

'''
pq is -(0.1543^e12) + (0.61721^e13) + (0.77152^e23)
-(2.62316^e1) - (3.08607^e2) - (0.77152^e3)
(2.62316^e1) + (3.08607^e2) + (0.77152^e3)
Dot product between x and x_rotated_1 is 0
'''

x_not_in_plane = (0^e1) + (1^e2) + (2^e3)
print(pq * x_not_in_plane)

'''
Output:
(1.08012^e1) + (1.54303^e2) - (0.77152^e3) - (0.92582^e123)
'''

'''
Two rotate the vector x_not_in_plane by 90 degrees, we find a vector which forms an angle of 45 degrees with p
'''
r1, r2, r3 = (p1+q1)/2, (p2+q2)/2, (p3+q3)/2
r_norm = sqrt(r1**2 + r2**2 + r3**2)
r1, r2, r3 = r1/r_norm, r2/r_norm, r3/r_norm
r = (r1^e1) + (r2^e2) + (r3^e3)
pr = p*r
y_not_in_plane = pr * x_not_in_plane * pr.inv()
print(y_not_in_plane)

'''
Output:
(0.36584^e1) + (2.11446^e2) - (0.62866^e3)
'''