A great circle, also known as a Riemannian circle, of a sphere is the intersection of the sphere and a plane which passes through the center point of the sphere, as distinct from a small circle. Any diameter of any great circle coincides with a diameter of the sphere, and therefore all great circles have the same circumference as each other, and have the same center as the sphere. A great circle is the largest circle that can be drawn on any given sphere. Every circle in Euclidean space is a great circle of exactly one sphere.
For any two points on the surface of a sphere there is a great circle through the two points. The minor arc of a great circle between two points is the shortest surfacepath between them. In this sense the minor arc is analogous to “straight lines” in spherical geometry. The length of the minor arc of great circle is taken as the distance of two points on a surface of a sphere, namely greatcircle distance. The great circles are the geodesics of the sphere.
In higher dimensions, the great circles on the nsphere are the intersection of the nsphere with twoplanes that pass through the origin in the Euclidean space R^{n+1}.
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Earth geodesics
Strictly speaking the Earth is not a perfect sphere (it is an oblate spheroid or ellipsoid – i.e., slightly squashed at the poles), which means that the shortest distance between two points (a geodesic) is not quite a great circle. Nevertheless, the sphere model can be considered a first approximation.
When long distance aviation or nautical routes are drawn on a flat map (for instance, the Mercator projection), they often look curved. This is because they lie on great circles. A route that would appear as a straight line on the map would actually be longer. An exception is the gnomonic projection, in which all straight lines represent great circles.
On a spherical Earth, the meridians (or lines of longitude) are great circles, as is the equator. Lines of latitude are not great circles, because they are smaller than the equator; their centers are not at the center of the Earth  they are small circles instead. Since Earth is not a perfect sphere, the equator (which is generally considered a spherical great circle) is about 40,075 km, while a northsouth meridian line (which is an ellipse) is almost 40,008 km. The quadratic mean or root mean square of these extremes provides a decent approximation of the average greatcircle circumference, about 40041.5 km.^{[1]}
Some examples of great circles on the celestial sphere include the celestial horizon, the celestial equator, and the ecliptic.
Great circle routes are used by ships and aircraft where currents and winds are not a significant factor. Flight lengths can therefore often be approximated to the greatcircle distance between two airports. For aircraft travelling west between continents in the northern hemisphere these paths will extend northward near or into the Arctic region, however easterly flights will often fly a more southerly track to take advantage of the jet stream.
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