Video-equipped unmanned aerial vehicles are highly useful in missions of surveillance, monitoring, and sensing. In these flights, the unmanned aerial vehicles must maintain the ability to see the targets as well as to save energy for prolonged endurance. This paper examines basic patterns of unmanned aerial vehicle flights that can maximize the ability to see targets, or “seeability,” and/or minimize power consumptions to assist unmanned aerial vehicle mission planning. In this paper, a point-mass model is used to describe unmanned aerial vehicle motions. A seeability model is established that peaks when the unmanned aerial vehicle is flying at a certain angle from the normal vector perpendicular to the surface of the target. Unmanned aerial vehicle flights are formulated as nonlinear periodic optimal control problems. The performance indices are selected to maximize the average seeability, to minimize the average power consumption, or to achieve a balance of the two. Motion constraints due to unmanned aerial vehicle performance capabilities and safety are imposed. The effects of different levels of constant wind velocities are considered. These nonlinear optimal control problems are converted into parameter optimization for numerical solutions. Extensive numerical solutions are obtained for unmanned aerial vehicle level flights with constant airspeeds and variable airspeeds, as well as three-dimensional flights. Clear tradeoffs between maximum seeability and minimum power are established.