Geodesics on the Earth's Surface

Imagine that you have a plane ticket from New York to London and you'd like to figure out the route that the pilot will most likely take. The pilot naturally does not want to "turn" his plane; he would like to follow a geodesic, which is the most natural and direct route he can take.

To predict the pilot's path, would you be better off looking at a map or a globe? Since globes are the most accurate representations of the Earth, you'd be better off consulting a globe. If you did, you'd find that the geodesic route would take you northward from New York over New England, Nova Scotia, and Newfoundland rather than straight east over the Atlantic. This geodesic, or "great circle" line represents the shortest path between two points on the surface of the Earth. If you wanted to draw this path on a Mercator projection map, you'd have to draw a curved line:

The Earth's surface is curved in 3-D!! On a 2-D map, this curvature is impossible to conceptualize, but it results in curved geodesics. Likewise, the universe is curved. In the 4-D space/time, this curvature is impossible to conceptualize, but it results in curved (accelerated) geodesics, such as parabolic paths or elliptical orbits.