In this article, we propose a new formation scheme for a leader-follower unmanned aerial vehicle (UAV) system inspired by a human pilot's behavior wherein the formation geometry does not necessarily remain fixed as the vehicles maneuver. In other words, the position and the orientation of the follower with respect to the leader are subject to change as they maneuver while satisfying some constraints. Our strategy ensures that the follower UAV maintains a desired fixed relative distance with respect to the leader UAV, whereas its orientation with respect to the leader UAV may change to reduce its control effort and provide it with a tactical advantage. We call this new relational maneuvering scheme flexible since the set of feasible positions for the follower UAV is not fixed, as is common in close proximity two-ship formations in air-to-air combat. By assigning the follower UAV's linear and angular velocities as its control inputs, our approach tries to emulate a human pilot's behavior in UAVs by taking anticipatory maneuvers when the leader UAV makes aggressive turns. The proposed flexible-geometry formation scheme is robust to the leader's maneuver changes since the follower UAV's control law does not need the information of the leader's angular speed control and only uses relative measurements. This makes the design lucrative even when the vehicles are heterogeneous, global measurements are unavailable, or if the leader UAV is noncooperative. Finally, we present multiple simulations to highlight the merits of the flexible formation control laws.