Quaternion Algebra
SkillDev toolsAbstract math for quaternion algebra (hypercomplex numbers ℍ). USE FOR: implementing quaternion multiplication (Hamilton product, non-commutative); conjugate, norm, and inverse; unit quaternions on S³ as double cover of SO(3); rotating 3D vectors with quaternions; understanding why q and −q represent the same rotation; normalizing after operations; constructing rotation quaternions from axis-angle. DO NOT USE FOR: converting between rotation representations (use 3d-rotation-theory); slerp/lerp interpolation (use interpolation-on-manifolds); basic vector cross/dot products (use vector-algebra).
Available today. Use it from your connected AI after setup.
No other account needed.
Connect ahel once, and every AI you use reads what you have installed.
Then ask your AI: use the Quaternion Algebra skill
What this skill tells your AI
The instructions your AI receives, as published by netfabric/netfabric.numerics in .agents/skills/quaternion-algebra/SKILL.md and read by ahel’s review.
Structure
A quaternion q = (x, y, z, w) = xi + yj + zk + w where i, j, k are imaginary units.
Also written as q = (v, w) with vector part v = (x, y, z) and scalar part w.
i² = j² = k² = ijk = −1
ij = k, jk = i, ki = j
ji = −k, kj = −i, ik = −j
Core Operations
| Operation | Formula |
|---|---|
| Addition | (x₁+x₂, y₁+y₂, z₁+z₂, w₁+w₂) — component-wise |
| Conjugate | q* = (−x, −y, −z, w) |
| Norm | ‖q‖ = √(x²+y²+z²+w²) |
| Norm squared | ‖q‖² = x²+y²+z²+w² |
| Scalar multiply | kq = (kx, ky, kz, kw) |
Hamilton Product (Non-Commutative)
q₁ · q₂ = (w₁w₂ − x₁x₂ − y₁y₂ − z₁z₂,
w₁x₂ + x₁w₂ + y₁z₂ − z₁y₂,
w₁y₂ − x₁z₂ + y₁w₂ + z₁x₂,
w₁z₂ + x₁y₂ − y₁x₂ + z₁w₂)
q₁q₂ ≠ q₂q₁ in general. Order matters for composed rotations.
Inverse & Division
q⁻¹ = q* / ‖q‖²
For unit quaternions: q⁻¹ = q* (conjugate = inverse).
Unit Quaternions
‖q‖ = 1. Lives on S³ ⊂ ℝ⁴. Every unit quaternion represents a 3D rotation. Map S³ → SO(3) is 2-to-1: q and −q encode the same rotation.
Rotating a Vector
p' = q · (0, p) · q⨉ (pure quaternion sandwich; extract x,y,z from result)
Reference Files
| File | Load When |
|---|---|
| references/formulas.md | Full Hamilton product derivation, double-cover proof, rotation construction |
| references/numerical-stability.md | Normalization drift, dot product sign, re-normalization strategies |
Signals
- GitHub stars
- 36
- Forks
- 1
- Last commit
- Aug 2026
Advanced
- Catalog kind
- skill
- Gateway key
quaternion-algebra- Source
- github.com/netfabric/netfabric.numerics