Rotation Matrix

Definition

A rotation is a linear transformation that preserves angles and distances and does not mirror. In 3D it is a $3\times 3$ matrix in the special orthogonal group:

$$SO(3):=\{R\in {\mathbb{R}}^{3\times 3}\mid R{R}^T=I,\det (R)=1\}$$

Columns

The columns of a rotation matrix are the principal axes of one frame expressed in another frame: $R_{sb}=[x_s^B{y}_s^B{z}_s^B]$. Since $R^{-1}=R^T$, the transpose goes the other way.

Pro	Con
linear, easy to apply and to chain	9 numbers for 3 degrees of freedom
entries are sines and cosines, so values are bounded (good for machine learning)	6 constraints + $\det =1$, suboptimal for numerical optimization

Trap

3D rotations don’t commute: $R_x{R}_y\ne R_y{R}_x$. 2D rotations do.

Appears in

• Lecture 2.2, Rotation Matrices: columns, two views, drawbacks
• Lecture 2.2, What is a rotation: invariances, commutativity, 2D rotation matrix
• Lecture 2.1, Extrinsic parameters: $R$ in $P_C=R(P_W-C)$
• Lecture 5, Procrustes: optimal rotation $\mathbf{R}=\mathbf{U}{\mathbf{V}}^T$

Related concepts

• Euler Angles
• Quaternion
• Axis-Angle and Exponential Map
• Extrinsic Parameters