Searching the Plane in the Segment Visibility Model

We study deterministic search by $n \geq 2$ unit-speed robots starting at the origin of the Euclidean plane. We introduce the concept of \emph{segment visibility} whereby a stationary target $x$ at distance $d=\|x\|>0$ is discovered at the first instant $t$ when $x$ belongs to the segment joining a pair of robots at time $t$. Discovery time is compared with the omniscient optimum $d$. We distinguish two knowledge models according to whether the robots know $d$ in advance. For $n=2$ robots, in the \emph{known-$d$} model, we give a symmetric four-phase algorithm and prove that its competitive ratio is optimal: $1+\sqrt3+π/6$; the matching lower bound follows from the known-distance two-robot shoreline-search theorem through an explicit reduction to segment detection. For $n=2$ robots, in the \emph{unknown-$d$} model, a three-circle construction and a quantitative initial-path estimate yield the stronger lower bound $\sqrt3+π/6+\sqrt{17}/4 \approx3.286425990$, giving an explicit positive cost for not knowing the radius; we also analyze a feasible antipodal-diameter logarithmic-spiral algorithm with optimized ratio $8.928516954\ldots$, thus proving that the competitive ratio when $d$ is unknown is at least $ \sqrt3+π/6+\sqrt{17}/4 $ and at most $8.928516954\ldots.$ For $n \geq 3$ robots, the competitive ratio of finding the target is exactly $\sec (π/n)$ regardless of whether or not $d$ is known to the robots.