Enter the spherical plane of elliptic geometry (I know, I know, the earth isn't really spherical, but for our purposes it's close enough). On such a plane, the fastest path from point A to B is along a great circle (as opposed to a small circle, of course). A great circle is a circle around the surface of a sphere whose center coincides with the sphere's center, and divides the sphere into two equal hemisphere. For instance, the lines of longitude are all great circles, but none of the lines of latitude but the equator are. The other lines of latitude, like the tropics of Cancer and Capricorn, are aptly named small circles, and really are smaller than great circles of the same sphere-- while the tropic of Cancer is only 36,768 km long, the equator is about 40,075 km long.
So, in our example flight from Chicago to Beijing, a straight line connecting them on the average map would be at about the parallel 40 degrees north of the equator, a great circle connecting them goes way up north into the arctic instead. Alternatively, a plane could go in the opposite direction around the great circle going south, but it's a bit longer that way around.
