notes, definitions, and references for geometric algebra (GA), including conformal geometric algebra (CGA). see also the software implementation sph-ga.
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.
[A, B] = AB - BA
{A, B} = AB + BA
A -> R A R⁻¹
v -> -R v R⁻¹
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)
for euclidean n-space:
p = x₁ e₁ + x₂ e₂ + ... + xₙ eₙ + e₀ + 0.5(x₁² + ... + xₙ²) e∞
null condition:
dot(p, p) = 0
rotation:
R = cos(angle / 2) + B sin(angle / 2) # B is unit bivector
translation:
T = 1 + 0.5 * t * e∞
rotation + translation:
R * T
p' = R * p * reverse(R)