Editing Great-circle distance (section)

== Formulae ==
[[File:Central angle.svg|thumb|An illustration of the central angle, Δσ, between two points, P and Q. λ and φ are the longitudinal and latitudinal angles of P respectively]]

Let <math>\lambda_1, \phi_1</math> and <math>\lambda_2, \phi_2</math> be the geographical [[longitude]] and [[latitude]] of two points 1 and 2, and <math>\Delta\lambda, \Delta\phi</math> be their absolute differences; then <math>\Delta\sigma</math>, 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:<ref>{{cite book |last1=Kells |first1=Lyman M. |last2=Kern |first2=Willis F. |last3=Bland |first3=James R. |title=Plane And Spherical Trigonometry |url=https://archive.org/details/planeandspherica031803mbp |access-date=July 13, 2018 |year=1940 |publisher=McGraw Hill Book Company, Inc. |pages=[https://archive.org/details/planeandspherica031803mbp/page/n344 323]-326 }}</ref>

:<math>\Delta\sigma = \arccos\bigl(\sin\phi_1\sin\phi_2 + \cos\phi_1\cos\phi_2\cos\Delta\lambda\bigr).</math>

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