[Proj] direct geodesic problem

[Aleksander Yanovskiy]

Greetings!

Maybe somebody can clarify how to solve the following direct geodesic 
problem: find the end point of a geodesic given its starting point and 
initial azimuth and the angle between the tangent planes at the 
beginning and the end points (or, equivalently, the angel between the 
normals at the points).
It seems the most easy way is to find the corresponding arc length on 
the auxiliary sphere, but I haven't found anywhere the explicit formula 
for it as the function of the point coordinates, the azimuth and the 
angle between the tangent planes at the beginning and the end points. 
And I'm not sure that such formula would be valid for the the beginning 
and the end points laying in different hemispheres.

And why is the geodesic problem in such formulation not being used in 
practice ?

Sincerely,
Aleksander