Given a polygonal domain $\cal P$ consisting $h$ pairwise disjoint convex polygonal obstacles together defined with $n$ vertices and a positive real number $ε$ in $(0, 0.6)$, this paper presents an algorithm to preprocess $\cal P$ in $O(n+\frac{h}ε(h+\frac{1}{\sqrtε})\lg(\frac{h}{\sqrtε}))$ time to compute data structures of size $O(n+\frac{h}{\sqrtε} (h+\frac{1}ε))$ so that given any two points $s$ and $t$ in the free space defined by $\cal P$, a path between $s$ and $t$ with a $(1+ε)$ multiplicative stretch and $13\ell$ additive stretch is output in $O(\frac{1}{\sqrtε}(\lg{\frac{h}{\sqrtε}})+\frac{h}{ε^{2.5}}(\lg{\lg(\frac{h}{\sqrtε})}))$ time. Here, $\ell$ is upper bounded by $(\sqrt{2ε}) (\max_{P_i \in \cal P} \max_{p, q \in P_i} |pq|)$.

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