Great circle

[1][2] Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geometry are the natural analog of straight lines in Euclidean space.

For any pair of distinct non-antipodal points on the sphere, there is a unique great circle passing through both.

Its arc length is the great-circle distance between the points (the intrinsic distance on a sphere), and is proportional to the measure of the central angle formed by the two points and the center of the sphere.

Any diameter of any great circle coincides with a diameter of the sphere, and therefore every great circle is concentric with the sphere and shares the same radius.

The disk bounded by a great circle is called a great disk: it is the intersection of a ball and a plane passing through its center.

To prove that the minor arc of a great circle is the shortest path connecting two points on the surface of a sphere, one can apply calculus of variations to it.

Any curve on the sphere that does not intersect either pole, except possibly at the endpoints, can be parametrized by provided

-independent constant, and From the first equation of these two, it can be obtained that Integrating both sides and considering the boundary condition, the real solution of

In a Cartesian coordinate system, this is which is a plane through the origin, i.e., the center of the sphere.

Great circles are also used as rather accurate approximations of geodesics on the Earth's surface for air or sea navigation (although it is not a perfect sphere), as well as on spheroidal celestial bodies.

Another great circle is the one that divides the land and water hemispheres.

A great circle divides the earth into two hemispheres and if a great circle passes through a point it must pass through its antipodal point.

The Funk transform integrates a function along all great circles of the sphere.

The great circle g (green) lies in a plane through the sphere's center O (black). The perpendicular line a (purple) through the center is called the axis of g , and its two intersections with the sphere, P and P ' (red), are the poles of g . Any great circle s (blue) through the poles is secondary to g .
A great circle divides the sphere in two equal hemispheres.