Why the route on the seatback map curves so strangely

A long flight that looks like a detour on a flat screen is usually the straightest line available.

Every flat map of a round world is a lie of some useful kind. Flattening a sphere onto a rectangle means stretching some parts and squashing others, and the popular projections stretch the regions nearest the poles enormously. On such a map a straight line drawn between two distant cities is not the shortest path between them at all. The genuinely shortest path, the one that would be a taut string stretched over a globe, appears on the flattened version as a curve arcing towards the nearer pole. The aircraft is going straight. It is the map that is bending.

This is why long flights between continents in the northern half of the world so often drift far higher up the map than intuition expects, and why a globe on a shelf settles the argument in seconds. Put a piece of string between the two cities, pull it tight, and the string lies exactly where the aircraft flew. The seatback display, which is almost always a flattened projection, then redraws that straight string as a graceful arc and quietly convinces a whole cabin that a detour is under way.

Real routes are not perfectly straight, of course, because the sky is organised as well as shaped. Aircraft follow published paths and defined corridors, they respect airspace that is closed or restricted, and they are routed to stay within reach of suitable places to land. All of these nudge the track away from the pure mathematical ideal. Even so, the gap between the ideal and the actual is usually modest, and far smaller than the gap between the actual and what a flat map appears to be showing you.