From Tbilisi

draw the way the plane flies.

five flights out of Tbilisi on a printed route map -- draw the way you think the plane flies, then watch it fly the real way over your line and read how many km you added

Why does a flight from London to Los Angeles go up over Greenland instead of straight across? Because the Earth is a ball, and the shortest path between two places on a ball is a great circle, a geodesic -- the line a string takes when you pull it tight across a globe. A flat Mercator map stretches everything near the poles, so that great circle prints as a curve bowing toward the nearer pole, while the straight line on the map is a rhumb line, one constant compass heading, and it is longer. The gap grows the longer and more east-west a route is and the further it is from the equator, and it vanishes for a route running due north or south, because a meridian is both lines at once. Every flat map lies about distance somewhere; on the seat-back screen, the curve is the short way. So why draw the straight line at all? Because it is the one you can steer: Mercator made his map in 1569 for sailors, and on it a straight line keeps a single compass heading the whole way. Fly from Tbilisi to San Francisco on the great circle and you leave heading almost due north and arrive heading south; hold due west instead and you fly thousands of kilometres further.

Concepts: maps, geometry, sphere.

An interactive you play with one thumb, on Wundr.