Clifford Geometric Algebra Cl(p, q, r)¶

The CliffordSemiring in algebrax.clifford implements Clifford Geometric Algebra \(Cl(p, q, r)\) over QuotientMonoidAlgebraSemiring. Multivectors unify scalars, vectors, bivectors, and pseudoscalars into a single sparse dictionary representation {blade_tuple: coeff}.

Geometric Product & Blade Reduction¶

• Geometric Product: \(A B = A \cdot B + A \wedge B\)
• Blade Sign Flips: \(\mathbf{e}_i \mathbf{e}_j = -\mathbf{e}_j \mathbf{e}_i\) for \(i \ne j\).
• Metric Signatures: \(\mathbf{e}_i^2 = +1\) (\(i \le p\)), \(-1\) (\(p < i \le p+q\)), \(0\) (\(i > p+q\)).

Python Example: 3D Spatial Vector Rotor Rotation¶

import math
import algebrax as ax

# Instantiate Cl(3, 0) ax.semiring.Semiring
cs = ax.clifford.CliffordSemiring(p=3, q=0, r=0)

# Define 3D Vector v = 3 e1 + 4 e2
v = {(1,): 3.0, (2,): 4.0}

# Geometric Vector Squared v^2 = |v|^2 = 25.0
v_sq = cs.mul(v, v)
print("Vector Squared v^2:", v_sq)
# Output: {(): 25.0}

# Rotate v in e12 bivector plane by 90 degrees (pi/2)
v_rot = ax.clifford.rotor_rotation(v, bivector=(1, 2), angle_rad=math.pi / 2.0, p=3, q=0, r=0)
print(f"Rotated Vector v': e1={v_rot.get((1,), 0.0):.2f}, e2={v_rot.get((2,), 0.0):.2f}")
# Output: Rotated Vector v': e1=-4.00, e2=3.00