One of the unique global behaviours of a group of under-actuated autonomous vehicles is known as cyclic pursue, which requires a group of vehicles to uniformly distribute on circles and at the same time, to orbit around a centre point, while circumnavigation further requires those vehicles to circumnavigate a predefined target of interest. Such behaviours can also be called periodic formations. This paper further studies the periodic formations for multi-agent systems composed of a group of under-actuated autonomous vehicles and a target of interest by proposing a distributed strategy to maintain any required distance between two vehicles and desired angle difference between vehicle's heading and the line of sight that takes directly towards its pursuing vehicle. The authors' distributed control law, dependent only on the information of its pursuing vehicle and itself, can not only enable a network of autonomous vehicles to circulate around a target of interest, but also achieve any physically feasible periodic formations, which include forming regular polygons as a special case.