<P> Through any two points on a sphere that are not directly opposite each other, there is a unique great circle . The two points separate the great circle into two arcs . The length of the shorter arc is the great - circle distance between the points . A great circle endowed with such a distance is called a Riemannian circle in Riemannian geometry . </P> <P> Between two points that are directly opposite each other, called antipodal points, there are infinitely many great circles, and all great circle arcs between antipodal points have a length of half the circumference of the circle, or π r (\ displaystyle \ pi r), where r is the radius of the sphere . </P> <P> The Earth is nearly spherical (see Earth radius), so great - circle distance formulas give the distance between points on the surface of the Earth correct to within about 0.5% . (See Arc length § Arcs of great circles on the Earth .) </P> <P> Let φ 1, λ 1 (\ displaystyle \ phi _ (1), \ lambda _ (1)) and φ 2, λ 2 (\ displaystyle \ phi _ (2), \ lambda _ (2)) be the geographical latitude and longitude in radians of two points 1 and 2, and Δ φ, Δ λ (\ displaystyle \ Delta \ phi, \ Delta \ lambda) be their absolute differences; then Δ σ (\ displaystyle \ Delta \ sigma), the central angle between them, is given by the spherical law of cosines if one of the poles is used as an auxiliary third point on the sphere: </P>

The shortest distance between two points on the earth's surface is