2025-08-09

geometric algebra

notes, definitions, and references for geometric algebra (GA), including conformal geometric algebra (CGA). see also the software implementation sph-ga.

overview

geometric algebra unifies inner, exterior, and geometric products into a single algebraic system. it supports scalars, vectors, and higher-grade elements as first-class entities. applications: geometry, physics, computer graphics, robotics, computational algebra.

resources

introductory

tools

reference lists

terminology

  • scalars: grade-0 elements (real numbers)
  • vectors: grade-1 elements (basis: e₁, e₂, ...)
  • basis vectors: linearly independent set spanning the space
  • null vectors: vectors with self-inner product zero
  • blades: simple multivectors from exterior product of independent vectors
  • bivectors: grade-2 blades (oriented planes)
  • trivectors: grade-3 blades (oriented volumes)
  • multivectors: general linear combination of grades
  • pseudoscalar: highest-grade blade
  • spinors: multivectors representing transformations (rotations, reflections, dilations)
  • dual: multivector's complement w.r.t. pseudoscalar
  • grades: homogeneous components by dimension
  • conformal points: null vectors embedding Euclidean points
  • directions: null vectors for points at infinity
  • rounds: spheres/circles in CGA
  • flats: lines/hyperplanes in CGA
  • tangent vectors: null vectors tangent to rounds

products

  • geometric product: associative combination of inner and exterior products
  • exterior product (wedge): antisymmetric, raises grade
  • inner product (dot): reduces grade, encodes projections
  • commutator product: [A, B] = AB - BA
  • anticommutator product: {A, B} = AB + BA

main operations

  • reversion: reverse vector order in product
  • grade involution: negate odd-grade parts
  • dualization: multiply by pseudoscalar
  • grade projection: extract specified grade(s)
  • norm: magnitude from geometric product
  • sandwich product: A -> R A R⁻¹
  • meet: intersection via dualization
  • join: union via exterior product
  • reflection: v -> -R v R⁻¹
  • rotation: spinor from bivector exponentials
  • inversion: inversion in sphere/plane
  • translation: spinor with e∞
  • dilation: spinor scaling transformation
  • exponential map: bivector -> rotation/dilation
  • logarithm map: spinor -> bivector
  • orthogonal complement: dual subspace
  • null decomposition: split null and non-null components

conformal geometric algebra (cga)

metric

split-signature metric with two off-diagonal negatives:

[1, 1, 1, 0, 0]

basis products:

no * no = 0
ni * ni = 0
no * ni = -1
eᵢ * eᵢ = 1
eᵢ * eⱼ = 0 (i ≠ j)

point embedding

for euclidean n-space:

p = x₁ e₁ + x₂ e₂ + ... + xₙ eₙ + e₀ + 0.5(x₁² + ... + xₙ²) e∞

null condition:

dot(p, p) = 0

common rotors

rotation:

R = cos(angle / 2) + B sin(angle / 2)     # B is unit bivector

translation:

T = 1 + 0.5 * t * e∞

rotation + translation:

R * T

rotor application

p' = R * p * reverse(R)

perspective projection

see: computing perspective projections in 3-dimensions using rotors in the homogeneous and conformal models of clifford algebra

implementation notes

  • pseudoscalar scaling factors vary; some use -1/2.
  • geometric product of pseudoscalar with itself = 1 in conformal metric (without scaling).
  • sign conventions differ between libraries; may flip all results.
  • product of pseudoscalar with itself can reveal null vector handling bugs.