Caption: A map illustrating the jetstream and the great circle airway between San Francisco and Tokyo.

Features:

A great circle is a circle that cuts a sphere in half: i.e., into equal parts. A small circle is a circle that cuts a sphere into unequal parts.
1. A great-circle distance is the global minimum distance between any 2 on a sphere (i.e., an ordinary sphere or a 2-sphere) if it subtends ≤ 180° (as seen from the sphere center). If it subtends > 180°, it is a local minimum distance: i.e., it is minimum distance compared to a path with just small deviations from the great circle path.

   But if you follow a great circle in the WRONG direction, you are going the long way around.

2. Airways often follow great circles to save fuel and travel time. This is why flights to Tokyo are often over the Aleutian Islands and flights to Europe are often over Greenland even though these flights look like the long ways on a Mercator projection map. They look short on globes. The use of great circles in traveling is called great-circle navigation.

3. A great circle is an example of a geodesic: the stationary path between points in spaces with general curvature. A stationary path is one where infinitesimal variations from cause NO change in length. Minimum and maximum paths (global or local) are stationary paths. Stationary paths are analyzed by variational calculus.