3D rotation

A rotation around an axis keeps one component fixed and rotates the orthogonal component inside a plane. This chapter builds that structure step by step.

• Rodrigues algorithm starts by decomposing v into the part along k̂ and the part in the orthogonal plane
• The orthogonal plane is represented by an orthonormal basis Q = [ q̂₁ | q̂₂ ], rotated in 2D and lifted back to ℝ³
• The resulting transformation is A = k̂k̂ᵀ + Q R(θ) Qᵀ
• Expanding that sequence gives A = cosθ I + (1 − cosθ) k̂k̂ᵀ + sinθ [ k̂ ]×
• The cross product matrix term supplies the quarter-turn direction inside the orthogonal plane; its derivation is developed in the separate Cross product chapter
• The rotation matrix keeps k̂ fixed and its remaining action is a 2D rotation inside the orthogonal plane
• In ℝⁿ, the same construction becomes a planar rotation with fixed subspace col(Q)⊥

The chapter treats rotation as a composition of projection, coordinate change, 2D rotation and reconstruction, first in ℝ³ and then in ℝⁿ.

Because a rotation around k̂ does not act the same way on every part of a vector. The component along k̂ stays fixed, while the component in the plane orthogonal to k̂ is the part that actually rotates.

The chapter is available as a PDF. Page links below are best-effort: most browsers support them but some viewers may ignore the page hint.

One term keeps the component along the axis, one term rotates the orthogonal component without changing its length and one skew-symmetric term gives the quarter-turn direction inside the orthogonal plane. The formula is compact, but each term comes from a separate geometric action.

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