Over the past decade a few but increasing number of researchers have begun using Unmanned Aerial Vehicles (UAVs) to expand and improve upon existing remote sensing capabilities in the Arctic. Due to the limited flight time, it is important to make sure that the UAV follows an optimal trajectory -- in which it cover all the points from a given area within the smallest possible trajectory length. Under the usual assumptions that we cover a rectangular area and that each on-board sensor covers all the points with a given radius r, we describe the optimal trajectory. A more complex optimal trajectory is also developed for the situations in which we need to get a more spatially detailed picture of some sub-regions of interest (in which we should have a smaller value r) and it is sufficient to get a less detailed picture (with larger r) in other sub-regions. We also describe the best ways to cover the trajectory in situations in which an UAV missed a spot -- due to excess wind or to an inexact control.