Flight plans need to account for the curvature of the earth when they are created. Since the earth is three-dimensional, it would be impossible not to fly in an arc. Another reason that planes would choose to fly in a trajectory that resembles an arc is to account for the amount of traffic that will be in the air.

Airplanes do not fly in "straight lines" as seen on a flat map because the Earth is a sphere, and the shortest distance between two points on a globe is actually a Great Circle Route, which looks like a curve when flattened. For example, a flight from New York to London will often curve north over Greenland because that is physically the shortest path. Beyond geometry, flight paths are also influenced by Jet Streams (high-altitude winds); pilots will often take a "longer" curved route to catch a tailwind or avoid a headwind to save fuel. Safety also plays a major role; routes are often designed to stay within a certain distance of "diversion airports" in case of an emergency (known as ETOPS rules), and planes must avoid restricted military airspace or areas of severe weather, leading to a "zigzag" rather than a perfectly straight trajectory.