For points on a sphere of radius , write their position vectors relative to the centre as with unit vectors. The great-circle distance is the shorter great circle arc length, times its central angle. Hence it equals , including zero for identical points and for antipodal points. At antipodal points the shortest arc is not unique, but its length is.
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Great-circle distance is the shortest path between two points on the surface of a sphere. It is based on the concept of a "great circle," which is a circle that divides the sphere into two equal hemispheres. Great-circle distances are significant in navigation and geography because they represent the shortest distance across the earth's surface, accounting for its curvature.