Vincenty's Formulae - Direct Problem

Direct Problem

Given an initial point (φ₁, L₁) and initial azimuth, α₁, and a distance, s, along the geodesic the problem is to find the end point (φ₂, L₂) and azimuth, α₂.

Start by calculating the following:

Then, using an initial value, iterate the following equations until there is no significant change in σ:

Once σ is obtained to sufficient accuracy evaluate:

If the initial point is at the North or South pole then the first equation is indeterminate. If the initial azimuth is due East or West then the second equation is indeterminate. If a double valued atan2 type function is used then these values are usually handled correctly.