How to prove $e^{π(ω)} = cos(ω)I_3 + (1 − cos(ω))ωˆ ⊗ ωˆ + sin(ω)π(ωˆ )$?

Question

How to prove $e^{π(ω)} = cos(ω)I_3 + (1 − cos(ω))ωˆ ⊗ ωˆ + sin(ω)π(ωˆ )$,

where $ω = |ω|, ωˆ =ω/|ω|.$

My Attempt:

In my understanding, $\pi (w)$ is an element of the lie algebra, which is a transformation from $R^3$ to $skew(R^3)$. So $e^{\pi(w)}$ is a rotation matrix in $R^3$.

However, I have no idea about what the right hand side is. How to compute the right hand side and verify they are equal?

Could anyone kindly give me some help? Thanks very much!

Answer 1

Let $w=[a_1,a_2,a_3]^T$, $\pi(w)=\begin{pmatrix}0&-a_3&a_2\\a_3&0&-a_1\\-a_2&a_1&0\end{pmatrix}$ and $u=\dfrac{1}{||w||}w$. Then $e^{\pi(w)}=Rot(\Delta,\theta)$ where $\Delta$ is the line oriented by $u$ and $\theta=||w||$. Note that $u^T\otimes u=uu^T=\dfrac{1}{||w||^2}ww^T$.

Then the correct formula is $Rot(\Delta,\theta)=\cos(\theta)I_3+(1-\cos(\theta))(u^T\otimes u)+\sin(\theta)\pi(u).$

Proof. Reduce to the case when $w=[0,0,a_3]^T$.