Great-circle distance

The great-circle distance or orthodromic distance 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.Let λ 1 , ϕ 1 {displaystyle lambda _{1},phi _{1}}   and λ 2 , ϕ 2 {displaystyle lambda _{2},phi _{2}}   be the geographical longitude and latitude in radians of two points 1 and 2, and Δ λ , Δ ϕ {displaystyle Delta lambda ,Delta phi }   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:The shape of the Earth closely resembles a flattened sphere (a spheroid) with equatorial radius a {displaystyle a}   of 6378.137 km; distance b {displaystyle b}   from the center of the spheroid to each pole is 6356.752 km. When calculating the length of a short north-south line at the equator, the circle that best approximates that line has a radius of b 2 / a {displaystyle b^{2}/a}   (which equals the meridian's semi-latus rectum), or 6335.439 km, while the spheroid at the poles is best approximated by a sphere of radius a 2 / b {displaystyle a^{2}/b}  , or 6399.594 km, a 1% difference. So long as a spherical Earth is assumed, any single formula for distance on the Earth is only guaranteed correct within 0.5% (though better accuracy is possible if the formula is only intended to apply to a limited area). Using the mean earth radius, R 1 = 1 3 ( 2 a + b ) ≈ 6371 k m {displaystyle R_{1}={frac {1}{3}}(2a+b)approx 6371,mathrm {km} }   (for the WGS84 ellipsoid) means that in the limit of small flattening, the mean square relative error in the estimates for distance is minimized.