3D Rotation Theory

SO(3) = group of 3×3 orthogonal matrices with det = +1. Every rotation has an inverse (transpose).

Representations

Representation	Parameters	Strengths	Weaknesses
Axis-angle	axis n̂ + angle θ	Intuitive, minimal	Not closed under composition
Rotation matrix	3×3, 9 numbers	Fast multi-vector transform	9 params, orthogonality drift
Euler angles	3 angles (many conventions)	Human-readable	Gimbal lock, convention ambiguity
Quaternion	4 numbers, unit	Stable interpolation, fast compose	Double-cover ambiguity

Axis-Angle (Rodrigues Formula)

Rotation by θ around unit n̂ = (nₓ, nᵧ, n_z):

v' = v cos θ + (n̂ × v) sin θ + n̂(n̂·v)(1 − cos θ)

Right-hand rule: thumb along n̂, fingers curl in positive rotation direction.

Euler Angles (ZYX Intrinsic = Yaw-Pitch-Roll)

Yaw ψ (Z), Pitch φ (Y), Roll ρ (X) — most common aerospace/robotics convention:

R = Rz(ψ) · Ry(φ) · Rx(ρ)        applied right-to-left: roll first, yaw last

Gimbal Lock

When pitch = ±90° (second rotation hits a pole), yaw and roll become degenerate — one degree of freedom is lost. Avoid for systems requiring full orientation control.

Quaternions and rotation matrices do NOT suffer gimbal lock.

Quaternion ↔ Axis-Angle

Axis-angle → quaternion (load quaternion-algebra):

q = (sin(θ/2)·nₓ, sin(θ/2)·nᵧ, sin(θ/2)·n_z, cos(θ/2))

Quaternion → axis-angle:

θ = 2 arccos(w)
n̂ = (x, y, z) / sin(θ/2)        (undefined when θ = 0; use n̂ = (0,0,1) by convention)

Composition Order (Right-to-Left)

R_total = R_second · R_first applies R_first first, then R_second. For quaternions: q_total = q_second · q_first (same convention).

Reference Files

File	Load When
references/formulas.md	Full rotation matrix for each Euler sequence, Rodrigues formula expansion, quaternion↔matrix conversion
references/numerical-stability.md	Orthogonality drift, gimbal lock detection, near-zero axis extraction