haversine

The haversine formula is a mathematical method used to calculate the great-circle distance between two points on a sphere from their latitudes and longitudes. It is named for the haversine function, haversin(θ) = sin^2(θ/2), and is widely used in navigation, geospatial computing, and geographic information systems as a convenient way to estimate distances on Earth or other spherical bodies.

To compute the distance, convert the latitudes φ1, φ2 and longitudes λ1, λ2 to radians. Let Δφ = φ2 − φ1 and Δλ = λ2 − λ1. Then a = sin^2(Δφ/2) + cos(φ1) cos(φ2) sin^2(Δλ/2). The central angle between the points is c = 2 atan2( sqrt(a), sqrt(1 − a) ), and the distance is d = R · c, where R is the sphere’s radius. For Earth, a common choice is the mean radius R ≈ 6,371 km (3,959 miles). Some implementations use d = 2R arcsin( sqrt(a) ) as an alternative form.

Limitations include the spherical assumption: the Earth is an oblate spheroid, so the haversine distance is an approximation, with larger potential error over long distances. For high-precision work, ellipsoidal models such as Vincenty’s formulae are preferred. The haversine formula remains popular for its simplicity, numerical stability for small distances, and ease of implementation in software.