Dual Operations Tutorial¶

This tutorial explains the concept of dual in geometric algebra and its applications.

Table of Contents¶

What is the Dual?
Mathematical Definition
Dual and Cross Product
Inverse Dual
Applications

1. What is the Dual?¶

The dual is a fundamental operation in geometric algebra that maps between different grades of multivectors. In 3D Euclidean space:

Original	Dual
Scalar	Trivector (pseudoscalar)
Vector	Bivector
Bivector	Vector
Trivector	Scalar

The dual essentially gives us the "orthogonal complement" of a multivector.

Intuitive Understanding¶

In 3D: - The dual of a vector is a bivector representing the plane perpendicular to that vector - The dual of a bivector is a vector perpendicular to that plane - The dual of a scalar is the volume element (pseudoscalar)

2. Mathematical Definition¶

Pseudoscalar¶

The pseudoscalar I is the highest-grade element in an algebra:

import nblade

alg = nblade.Algebra.euclidean(3)
e1, e2, e3 = alg.basis_vectors()

# Pseudoscalar I = e1 ∧ e2 ∧ e3
I = e1 ^ e2 ^ e3
print(f"I = {I}")  # e1∧e2∧e3
print(f"I² = {(I * I).scalar_part()}")  # -1 in 3D Euclidean

Dual Operation¶

The dual of a multivector A is defined as:

A* = A · I⁻¹  (right contraction)

Or equivalently:

A* = A I⁻¹   (geometric product)

Using Dual in nblade¶

import nblade

alg = nblade.Algebra.euclidean(3)
e1, e2, e3 = alg.basis_vectors()

# Dual of a vector
v = e1
v_dual = v.dual()
print(f"Dual of e1: {v_dual}")  # e2∧e3

# Dual of a bivector
B = e1 ^ e2
B_dual = B.dual()
print(f"Dual of e1∧e2: {B_dual}")  # e3

3. Dual and Cross Product¶

In 3D, the cross product can be expressed using the dual:

a × b = (a ∧ b)*

This is why the cross product only works in 3D - it requires the dual to map a bivector back to a vector.

Example: Cross Product via Dual¶

import nblade

alg = nblade.Algebra.euclidean(3)
e1, e2, e3 = alg.basis_vectors()

a = e1
b = e2

# Cross product via dual: a × b = (a ∧ b)*
wedge = a ^ b        # a ∧ b = e1∧e2
cross = wedge.dual() # dual = e3

print(f"a ∧ b = {wedge}")  # e1∧e2
print(f"(a ∧ b)* = {cross}")  # e3

# This is equivalent to the cross product!
# e1 × e2 = e3

Advantages of the GA Approach¶

Cross Product	Geometric Algebra Dual
Only works in 3D	Works in any dimension
Returns a vector	Returns a bivector (more natural)
Non-associative	Wedge product is associative

4. Inverse Dual¶

The inverse dual reverses the dual operation:

(A*)* = A (up to sign in some signatures)

Using Inverse Dual in nblade¶

import nblade

alg = nblade.Algebra.euclidean(3)
e1, e2, e3 = alg.basis_vectors()

v = e1 + 2*e2 + 3*e3

# Dual
v_dual = v.dual()

# Inverse dual (should recover original)
v_recovered = v_dual.inverse_dual()

print(f"Original: {v}")
print(f"Dual: {v_dual}")
print(f"Inverse dual: {v_recovered}")

5. Applications¶

Normal Vectors from Planes¶

A plane defined by a bivector B has a normal vector n = B*:

import nblade

alg = nblade.Algebra.euclidean(3)
e1, e2, e3 = alg.basis_vectors()

# Plane defined by e1∧e2 (xy-plane)
plane = e1 ^ e2

# Normal vector (points in z direction)
normal = plane.dual()
print(f"Normal to xy-plane: {normal}")  # e3

Area and Volume¶

The dual naturally relates areas and volumes:

import nblade

alg = nblade.Algebra.euclidean(3)
e1, e2, e3 = alg.basis_vectors()

# Area element
area = e1 ^ e2  # Bivector representing area

# "Volume" of the area (magnitude)
# |B*| = |B| in 3D Euclidean
area_magnitude = area.norm()
print(f"Area magnitude: {area_magnitude}")  # 1.0

Electromagnetic Field¶

In physics, the electromagnetic field F = E + iB uses the dual to relate E and B fields:

import nblade

alg = nblade.Algebra.euclidean(3)
e1, e2, e3 = alg.basis_vectors()
I = e1 ^ e2 ^ e3  # Pseudoscalar

# Electric field
E = e1 + e2

# Magnetic field as bivector via dual
# B_bivector = I * B_vector = dual(B_vector)
B_vector = e3
B_bivector = B_vector.dual()  # e1∧e2

print(f"Magnetic field bivector: {B_bivector}")

Summary¶

Concept	Formula	nblade Method
Dual	A* = AI⁻¹	`A.dual()`
Inverse Dual	(A)	`A.inverse_dual()`
Cross Product (3D)	a × b = (a ∧ b)*	`(a ^ b).dual()`
Pseudoscalar	I = e₁∧e₂∧...∧eₙ	`alg.config.volume_element()`