This is the continuous question of how to compute Riemannian logarithm and exponential on SO(3)
According to Terse Notes on Riemannian Geometry (Example 4.7.), the matrix exponential w.r.t. $SO(3)$ is $$ \exp (X)= \begin{cases}I, & \theta=0, \\ I+\frac{\sin \theta}{\theta} X+\frac{1-\cos \theta}{\theta^2} X^2, & \theta \in(0, \pi)\end{cases} $$ where $\theta=\sqrt{\frac{1}{2} \operatorname{tr}\left(X^T X\right)}$, and $X$ is skrew symmetric.
The matrix logarithm on $SO(3)$ is $$ \log (R)=\left\{\begin{array}{cc} 0, & \theta=0, \\ \frac{\theta}{2 \sin \theta}\left(R-R^{\top}\right) & |\theta| \in(0, \pi) \end{array}\right. $$ wehre $\operatorname{tr}(R)=2 \cos \theta+1$ and $R \in SO(3)$.
So, what about $\theta=\pi$ for matrix logarithm and exponential?
I think the above expression of exponential is also valid for $\theta=\pi$. but I don't know how to deal with the case of $\theta=\pi$ for the logarithm