Great-circle distance

The great-circle distance, orthodromic distance, or spherical distance is the distance between two points on a sphere, measured along the great-circle arc between them. This arc is the shortest path between the two points on the surface of the sphere. (By comparison, the shortest path passing through the sphere's interior is the chord between the points.)

On a curved surface, the concept of straight lines is replaced by a more general concept of geodesics, curves which are locally straight with respect to the surface. Geodesics on the sphere are great circles, circles whose center coincides with the center of the sphere.

Any two distinct points on a sphere that are not antipodal (diametrically opposite) both lie on a unique great circle, which the points separate into two arcs; the length of the shorter arc is the great-circle distance between the points. This arc length is proportional to the central angle between the points, which if measured in radians can be scaled up by the sphere's radius to obtain the arc length. Two antipodal points both lie on infinitely many great circles, each of which they divide into two arcs of length π times the radius.

The determination of the great-circle distance is part of the more general problem of great-circle navigation, which also computes the azimuths at the end points and intermediate way-points. Because the Earth is nearly spherical, great-circle distance formulas applied to longitude and geodetic latitude of points on Earth are accurate to within about 0.5%.

Let ${\displaystyle \lambda _{1},\phi _{1}}$ and ${\displaystyle \lambda _{2},\phi _{2}}$ be the geographical longitude and latitude of two points 1 and 2, and ${\displaystyle \Delta \lambda ,\Delta \phi }$ be their absolute differences; then ${\displaystyle \Delta \sigma }$, 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:

The problem is normally expressed in terms of finding the central angle ${\displaystyle \Delta \sigma }$. Given this angle in radians, the actual arc length d on a sphere of radius r can be trivially computed as

The central angle ${\displaystyle \Delta \sigma }$ is related with the chord length of unit sphere ${\displaystyle \Delta \sigma _{\text{c}}\,\!}$:

For short-distance approximation (${\displaystyle |\Delta \sigma _{\text{c}}|\ll 1}$),