The Haversine Formula

To compute great-circle distances on a sphere cleanly, we use the haversine function, defined for any angle $\theta$ as

$$\operatorname{hav}(\theta) = \sin^2\!\left(\dfrac{\theta}{2}\right) = \dfrac{1-\cos\theta}{2}.$$

The two forms are equivalent via the half-angle identity. The haversine is non-negative, equals $0$ at $\theta = 0,$ and grows to $1$ at $\theta = 180°.$ It is also an even function: $\operatorname{hav}(-\theta) = \operatorname{hav}(\theta).$

For example,

$$\operatorname{hav}(90°) = \sin^2(45°) = \left(\dfrac{\sqrt{2}}{2}\right)^2 = \dfrac{1}{2}.$$

The haversine is the building block of the Haversine formula for great-circle distance. It is preferred over the law-of-cosines formula because it remains numerically stable for points that are close together on the sphere.