great-circle distance

The great-circle distance, orthodromic distance, or spherical distance is the distance along a great circle.

It is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a straight line through the sphere's interior). The distance between two points in Euclidean space is the length of a straight line between them, but on the sphere there are no straight lines. In spaces with curvature, straight lines are replaced by geodesics. Geodesics on the sphere are circles on the sphere whose centers coincide with the center of the sphere, and are called 'great circles'.

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.

Through any two points on a sphere that are not antipodal points (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.

Between 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 $\pi r$, where ''r'' is the radius of the sphere.

The Earth is nearly spherical, so great-circle distance formulas give the distance between points on the surface of the Earth correct to within about 0.5%.

The vertex is the highest-latitude point on a great circle.

Formulae

Let $\lambda_1, \phi_1$ and $\lambda_2, \phi_2$ be the geographical longitude and latitude of two points 1 and 2, and $\Delta\lambda, \Delta\phi$ be their absolute differences; then $\Delta\sigma$, 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:

:$\Delta\sigma = \arccos\bigl(\sin\phi_1\sin\phi_2 + \cos\phi_1\cos\phi_2\cos(\Delta\lambda)\bigr).$

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

Computational formulas

On computer systems with low floating point precision, the spherical law of cosines formula can have large rounding errors if the distance is small (if the two points are a kilometer apart on the surface of the Earth, the cosine of the central angle is near 0.99999999). For modern 64-bit floating-point numbers, the spherical law of cosines formula, given above, does not have serious rounding errors for distances larger than a few meters on the surface of the Earth. The haversine formula is numerically better-conditioned for small distances:

:$\begin \Delta\sigma &= \operatorname\left( \operatorname\left(\Delta\phi\right) + \left(1 - \operatorname(\Delta\phi) - \operatorname(\phi_1 + \phi_2)\right)\cdot\operatorname\left(\Delta\lambda\right)\right) \\ &= 2\arcsin \sqrt. \end$

Historically, the use of this formula was simplified by the availability of tables for the haversine

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