The Rotation Group

The set of three-dimensional rotation matrices is denoted by

$$SO(3) = \left\{ \mathbf{R}\in\mathbb{R}^{3\times3} \;\middle... ...}^{T}\mathbf{R}=\mathbf{I}, \quad \det(\mathbf{R})=1 \right\}.$$ (A.29)

The composition operation is matrix multiplication.

The symbol $SO(3)$ derives from Special Orthogonal Group:

• “Orthogonal” because $\mathbf{R}^{T}\mathbf{R}=\mathbf{I}$;
• “Special” because $\det(\mathbf{R})=1$;
• “3” because it acts in three-dimensional space.

The set $SO(3)$ does not constitute a vector space. Indeed, the sum of two rotation matrices is not generally a rotation.

Rotations instead belong to a differentiable manifold and can be studied locally through their tangent space.