Why the route bends north

On a flat map a long flight between two northern cities looks like a curve bowing toward the pole. On a globe it is the straight one. The shortest way between two points on a sphere runs along a great circle, a circle whose centre is the centre of the Earth, and for two cities on the same northern latitude that circle swings toward the pole.

Take two cities at 40° north on opposite sides of the world, 180° of longitude apart. Following the line of latitude covers about 138° of arc (180 × cos 40°). Going straight over the North Pole covers 50° up to the pole and 50° down: 100°. At about 60 nautical miles a degree, that is roughly 6,000 miles instead of 8,300, a bit more than a quarter shorter.

The flat map draws the straight route as a curve because no flat map can keep every distance true. The Globe Table on this site shows the same pair of cities both ways.