Editing Great-circle distance (section)

{{Short description|Shortest distance between two points on the surface of a sphere}}{{About|shortest-distance on a sphere|the shortest distance on an ellipsoid|geodesics on an ellipsoid}}

[[File:Illustration of great-circle distance.svg|thumb|A diagram illustrating great-circle distance (drawn in red) between two points on a sphere, P and Q. Two [[antipodal points]], u and v are also shown.]]

The '''great-circle distance''', '''orthodromic distance''', or '''spherical distance''' is the [[distance]] between two [[point (geometry)|points]] on a [[sphere]], measured along the [[great circle|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 (geometry)|chord]] between the points.)

On a [[manifold|curved surface]], the concept of [[straight line]]s is replaced by a more general concept of [[geodesic]]s, 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 point|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 [[radian]]s 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 [[pi|{{math|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 [[azimuth]]s at the end points and intermediate way-points. Because the Earth [[Spherical 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%.<ref>{{citation |title=Admiralty Manual of Navigation, Volume 1 |publisher=The Stationery Office |year=1987 |isbn=9780117728806 |page=10 |url=https://books.google.com/books?id=xcy4K5BPyg4C&pg=PA10 |quotation=The errors introduced by assuming a spherical Earth based on the international nautical mile are not more than 0.5% for latitude, 0.2% for longitude.}}</ref>