halfgreatcircles

A half great circle, sometimes called a semi-great circle, is a semicircular arc on the surface of a sphere that lies along a great circle. It is the portion of a great circle between two antipodal points on the sphere. Because many great circles pass through any given pair of antipodal points, there are correspondingly many possible half great circles connecting them.

The defining features are: the endpoints are antipodal, and the arc length of a half great circle is equal to πR, where R is the sphere’s radius (half the circumference of a great circle). Geometrically, the arc is a geodesic segment on the sphere, representing a distance-minimizing path between interior points along that curve. For antipodal endpoints, infinitely many distinct half great circles exist, each lying in a plane through the sphere’s center that contains the two endpoints.

In practical terms, half great circles illustrate the concept of geodesic arcs on spherical surfaces. They are exemplified by meridians on Earth: any arc from the North Pole to the South Pole along a fixed longitude is a half great circle. More generally, half great circles help describe possible shortest paths between opposite points on a sphere and clarify how great circles compose the full family of geodesic circles on a spherical surface.