Formulas

This directory contains the formulas used to calculate the distances between points on a sphere.

Haversine

The haaversine formula is based on the Haversine formula and is used to calculate the distance between two points on a sphere. The formula is:

$$\Delta \sigma = 2\cdot arcsin\sqrt{sin^2\left (\frac{\Delta\phi}{2}\right )+\left (cos^2\left(\frac{\phi_1 + \phi_2}{2}\right ) - sin^2\left (\frac{\Delta\phi}{2}\right ) \right) \cdot sin^2\left (\frac{\Delta\lambda}{2}\right )}$$

Vincenty

The Vincenty formula is based on the Great-circle distance and is used to calculate the distance between two points on a sphere. Technically, it is not the Vincenty formula, but it is what's used for my CS314 class. The formula is:

$$\Delta \sigma = arctan \frac{\sqrt{\left (cos\phi_2 \cdot sin\left (\Delta\lambda\right ) \right)^2 + \left (cos\phi_1 \cdot sin\phi_2 - sin\phi_1 \cdot cos\phi_2 \cdot cos\left (\Delta\lambda\right ) \right)^2}}{sin\phi_1 \cdot sin\phi_2 + cos\phi_1 \cdot cos\phi_2 \cdot cos \left (\Delta\lambda\right )}$$

Spherical Law of Cosines

The Spherical Law of Cosines formula is based on the Great-circle distance and is used to calculate the distance between two points on a sphere. The formula is:

$$\Delta \sigma = arccos\left(sin\phi_1 \cdot sin\phi_2 + cos\phi_1 \cdot cos\phi_2\ \cdot cos\Delta\lambda\right)$$

Lastly...

Once $\Delta \sigma$ is calculated, the distance between the two points is simply $d = r \Delta\sigma$, where $r$ is the radius of the sphere and $\Delta \sigma$ is the calculated distance.

Note: