Example 2. Find the length of the minor arc of the circle $\mathrm{O}: \mathrm{x}^{2}+\mathrm{y}^{2}=9$ intercepted by $\odot \mathrm{O}_{1}$ : $(x-3)^{2}+y^{2}=27$.
Solve: Convert the equations of the two circles into parametric equations (one with parameter $\theta$, the other with $\varphi$), and eliminate $x$ and $y$, then
$$
\left\{\begin{array}{l}
\sqrt{3} \cos \varphi=\cos \theta-1 \\
\sqrt{3} \sin \varphi=\sin \theta .
\end{array}\right.
$$

Eliminating $\varphi$ yields $\cos \theta=-\frac{1}{2}$,
$$
\theta_{1}=\frac{2 \pi}{3}, \quad \theta_{2}=\frac{4 \pi}{3} .
$$

Let the intersection points be $M$ and $N$, then
$$
\angle \mathrm{MON}=\theta_{2}-\theta_{1}=\frac{2 \pi}{3} \text {. }
$$
$\therefore$ The length of the minor arc $\widehat{M N}$ is $\frac{2 \pi}{3} \cdot 3=2 \pi$.