  Let $\mathscr O$ be a set of $n$ disjoint obstacles in $\mathbb{R}^2$,
$\mathscr M$ be a moving object. Let $s$ and $l$ denote the starting point and
maximum path length of the moving object $\mathscr M$, respectively. Given a
point $p$ in ${R}^2$, we say the point $p$ is achievable for $\mathscr M$ such
that $\pi(s,p)\leq l$, where $\pi(\cdot)$ denotes the shortest path length in
the presence of obstacles. One is to find a region $\mathscr R$ such that, for
any point $p\in \mathbb{R}^2$, if it is achievable for $\mathscr M$, then $p\in
\mathscr R$; otherwise, $p\notin \mathscr R$. In this paper, we restrict our
attention to the case of line-segment obstacles. To tackle this problem, we
develop three algorithms. We first present a simpler-version algorithm for the
sake of intuition. Its basic idea is to reduce our problem to computing the
union of a set of circular visibility regions (CVRs). This algorithm takes
$O(n^3)$ time. By analysing its dominant steps, we break through its bottleneck
by using the short path map (SPM) technique to obtain those circles
(unavailable beforehand), yielding an $O(n^2\log n)$ algorithm. Owing to the
finding above, the third algorithm also uses the SPM technique. It however,
does not continue to construct the CVRs. Instead, it directly traverses each
region of the SPM to trace the boundaries, the final algorithm obtains $O(n\log
n)$ complexity.
