We present an algorithm for searching a 2D environment for unpredictable moving targets using only beam-based detection. One or more pursuers move along the environment boundary, and carry a rotating beam that detects evaders. The beam could correspond in practice to a laser or a camera. The task is to compute motions for pursuers and their beams that ensure that all evaders will be detected. For a 2D polygonal environment, we solve a long-standing open problem by presenting a complete O(n/sup 3/)-time algorithm that is guaranteed to find a successful motion strategy for a single pursuer and its beam, if a solution exists. This algorithm is extended to the case of coordinating multiple pursuers, but the number of pursuers used in a solution is not necessarily optimal. An implementation is presented, and several computed examples are shown.