Axis-Angle and Exponential Map

Definition

Any rotation about the origin can be written with a unit axis $\omega \in {\mathbb{R}}^3$ and an angle $\theta$. The exponential map turns this (an element of the Lie algebra) into a rotation matrix (an element of the Lie group $SO(3)$):

$$R(\theta ,\omega )=\exp (\theta \hat{\omega })$$

Skew-symmetric matrix and Rodrigues formula

$$\hat{\omega }=\left[\begin{array}{ccc}0& -{\omega }_3& {\omega }_2\\ {\omega }_3& 0& -{\omega }_1\\ -{\omega }_2& {\omega }_1& 0\end{array}\right],\hat{\omega }\mathbf{p}=\omega \times \mathbf{p}$$ $$\exp (\theta \hat{\omega })=I+\hat{\omega }\sin \theta +{\hat{\omega }}^2(1-\cos \theta )$$

Where the exponential comes from

A point rotating about $\omega$ moves with $\dot{\mathbf{p}}(t)=\omega \times \mathbf{p}(t)=\hat{\omega }\mathbf{p}(t)$. The solution of this linear differential equation is $\mathbf{p}(t)=\exp (\hat{\omega }t)\mathbf{p}(0)$.

Only 3 numbers and good for optimization, which answers the question left open by quaternions (Slide 31).

Appears in

• Lecture 2.2, Axis-Angle and the Exponential Map: Lie groups, hat operator, proof, Rodrigues
• Lecture 2.2, Twists: extension to translation
• Lecture 3, Epipolar constraint: the cross product $t\times R{x}_1$ written as $[t{]}_{\times }R{x}_1$

Related concepts

• Twist
• Rotation Matrix
• Quaternion