This paper provides two approaches to establish a proximity network among a collection of unmanned aerial vehicles (UAVs) that are initially scattered in space. The goal is to find the shortest trajectories that bring the UAVs to a connected formation where they are in the range of detection of one another and headed in the same direction to maintain the connectivity. Constant-speed unicycles are chosen to represent UAV kinematics flying steady around cruising speed. Pontryagin Minimum Principle (PMP) is used to determine the control law and path synthesis for the UAVs under the turn-rate constraints. We introduce an algorithm to search for the optimal solution when the final network topology is specified; followed by a nonlinear programming method in which the final configuration is emerged from the optimization routine under the constraints that the final topology is connected. Simulation results along with the discussion on the performance of both methods are provided.