  An articulated probe is modeled in the plane as two line segments, $ab$ and
$bc$, joined at $b$, with $ab$ being very long, and $bc$ of some small length
$r$. We investigate a trajectory planning problem involving the articulated
two-segment probe where the length $r$ of $bc$ can be customized. Consider a
set $P$ of simple polygonal obstacles with a total of $n$ vertices, a target
point $t$ located in the free space such that $t$ cannot see to infinity, and a
circle $S$ centered at $t$ enclosing $P$. The probe initially resides outside
$S$, with $ab$ and $bc$ being collinear, and is restricted to the following
sequence of moves: a straight line insertion of $abc$ into $S$ followed by a
rotation of $bc$ around $b$. The goal is to compute a feasible
obstacle-avoiding trajectory for the probe so that, after the sequence of
moves, $c$ coincides with $t$.
  We prove that, for $n$ line segment obstacles, the smallest length $r$ for
which there exists a feasible probe trajectory can be found in
$O(n^{2+\epsilon})$ time using $O(n^{2+\epsilon})$ space, for any constant
$\epsilon > 0$. Furthermore, we prove that all values $r$ for which a feasible
probe trajectory exists form $O(n^2)$ intervals, and can be computed in
$O(n^{5/2})$ time using $O(n^{2+\epsilon})$ space. We also show that, for a
given $r$, the feasible trajectory space of the articulated probe can be
characterized by a simple arrangement of complexity $O(n^2)$, which can be
constructed in $O(n^2)$ time. To obtain our solutions, we design efficient data
structures for a number of interesting variants of geometric intersection and
emptiness query problems.
