Rodrigues Rotation Formula — Geometric Construction

A linear algebra note on rotating a vector around a fixed axis in 3D

Rodrigues rotation formula rotates a vector v around a unit axis k̂ by splitting the vector into two parts: the axial component parallel to k̂ and the planar component perpendicular to k̂. The parallel part stays fixed, while the perpendicular part rotates inside the plane orthogonal to the axis.

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Visual 1 — Step-by-step construction

Step-by-step geometric construction of Rodrigues rotation formula. The vector is decomposed into its parallel and perpendicular parts, plane coordinates are built, a 2D rotation is applied and the result is mapped back to 3D.
The construction begins by separating v into v∥ and v⊥, then building plane coordinates with an orthonormal basis in the rotation plane. A 2D rotation by angle θ is applied in those plane coordinates, and the rotated planar part is mapped back to 3D before recombining with the unchanged axial component.

Visual 2 — Animation

Animated visualization of Rodrigues rotation formula showing a vector rotating around a fixed axis while its axial component remains unchanged.
The animation shows the same structure dynamically: the vector rotates around the fixed axis, the axial component remains unchanged and only the perpendicular component sweeps through the rotation plane.

Concept

Let k̂ be a unit vector along the rotation axis and let v be any vector in R³. The axial component is v∥ = k̂ k̂ᵀ v and the perpendicular component is v⊥ = v − v∥.

The vector v∥ is already aligned with the axis, so it does not move under rotation about that axis. The whole problem reduces to rotating v⊥ by angle θ inside the plane orthogonal to k̂.

Structure

Choose an orthonormal basis (q̂₁, q̂₂) for the rotation plane, where q̂₁ points along v⊥ and q̂₂ = k̂ × q̂₁. Then the planar part can be expressed in 2D coordinates, rotated by the standard 2D rotation matrix and mapped back to 3D.

This separates 3D rotation into three pieces: keep the axial part, rotate in plane coordinates and recombine.

Key equations

v∥ = k̂ k̂ᵀ v
v⊥ = v − v∥
q̂₁ = v⊥ / |v⊥|
q̂₂ = k̂ × q̂₁
Q = [ q̂₁ | q̂₂ ]
R(θ) = [ [cosθ, -sinθ], [sinθ, cosθ] ]
w = v∥ + Q R(θ) Qᵀ v⊥
w = v∥ + cosθ v⊥ + sinθ (k̂ × v⊥)
w = k̂ k̂ᵀ v + cosθ (I − k̂ k̂ᵀ) v + sinθ (k̂ × v)

These forms say the same thing in different languages: projection onto the axis, rotation in the perpendicular plane and recombination into the final rotated vector.

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Reference

External reference: Wikipedia — Rodrigues' rotation formula, derivation section