This page presents a variety of calculations for latitude/longitude points, with the formula and code fragments for implementing them.

All these formula are for calculations on the basis of a spherical earth (ignoring ellipsoidal effects) – which is accurate enough for most purposes [In fact, the earth is very slightly ellipsoidal; using a spherical model gives errors typically up to 0.3%].

Distance

This uses the haversine formula to calculate the great-circle distance between two points – that is, the shortest distance over the earth’s surface – giving an “as-the-crow-flies” distance between the points (ignoring any hills they fly over, of course!).

Haversine formula: \(haversin{\left(\dfrac{d}{r}\right)} = haversin{\left(\phi_2-\phi_1\right)} + \cos{\phi_1}\cos{\phi_2}\,haversin{\left(\lambda_2-\lambda_1\right)}\)

where

\[haversin(\theta) = \sin^2\left(\dfrac{\theta}{2}\right) = \dfrac{1-\cos(\theta)}{2}\]

\(d\) is the distance between the two points (along a great circle of the sphere)

\(r\) is the radius of the sphere

\(\phi_1\), \(\phi_2\): latitude of point 1 and latitude of point 2

\(\lambda_1\), \(\lambda_2\): longitude of point 1 and longitude of point 2