Haversine formula

Remarks

The proof starts from the spherical law of cosines, $\cos\theta = \sin\varphi_1 \sin\varphi_2 + \cos\varphi_1 \cos\varphi_2 \cos(\lambda_2 - \lambda_1)$. Substitute $\cos x = 1 - 2\operatorname{hav} x$ for $x = \theta$ and $x = \lambda_2 - \lambda_1$. Then use $\cos(\varphi_2 - \varphi_1) = \cos\varphi_1\cos\varphi_2 + \sin\varphi_1\sin\varphi_2$ and rearrange. The second formula follows because $\theta/2 \in [0, \pi/2]$, where $\sin$ is invertible. Example: two points on the equator with $\lambda_2 - \lambda_1 = \pi/2$ give $\operatorname{hav}\theta = 1/2$, so $\theta = \pi/2$ and $d = \pi r/2$.

Going deeper

For a student

The haversine form is a rewriting of the spherical law of cosines from spherical trigonometry. Its numerical weakness is the opposite case: for nearly antipodal points, the argument of $\arcsin$ is close to $1$. A formula with $\operatorname{atan2}$, as used by Vincenty for the sphere, is accurate for all pairs. On the Earth, the spherical model gives errors up to about $0.5\%$; for better accuracy one solves the geodesic problem on the ellipsoid.