  We study a new search problem on the plane involving a robot and an immobile
treasure, initially placed at distance $1$ from each other. The length $\beta$
of an arc (a fence) within the perimeter of the corresponding circle, as well
as the promise that the treasure is outside the fence, is given as part of the
input. The goal is to device movement trajectories so that the robot locates
the treasure in minimum time. Notably, although the presence of the fence
limits searching uncertainty, the location of the fence is unknown, and in the
worst case analysis is determined adversarially. Nevertheless, the robot has
the ability to move in the interior of the circle. In particular the robot can
attempt a number of chord-jump moves if it happens to be within the fence or if
an endpoint of the fence is discovered.
  The optimal solution to our question can be obtained as a solution to a
complicated optimization problem, which involves trigonometric functions, and
trigonometric equations that do not admit closed form solutions. For the 1-Jump
Algorithm, we fully describe the optimal trajectory, and provide an analysis of
the associated cost as a function of $\beta$. Our analysis indicates that the
optimal k-Jump Algorithm requires that the robot has enough memory and
computation power to compute the optimal chord-jumps. Motivated by this, we
give an abstract performance analysis for every k-Jump Algorithm. Subsequently,
we present a highly efficient Halving Heuristic k-Jump Algorithm that can
effectively approximate the optimal k-Jump Algorithm, with very limited memory
and computation requirements.
