[Proj] direct geodesic problem

[Charles Karney]

On 08/21/2014 03:35 AM, Aleksander Yanovskiy wrote:
> 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

Your formulation of the problem is the same as the standard formulation
of the direct geodesic problem except that you've substituted a
different measure for the distance along the geodesic.  I can think of a
couple of reasons why the problem isn't posed in such terms:

(1) The problem now depends on how the surface is embedded in three
dimensions.  In the standard approach, distances are an intrinsic
property of the surface.

(2) This measure does not typically pass through 180 degrees because
most geodesics do not pass through the antipodal point on the first
circuits around the earth.

A couple of other points can be made

(3) The direct problem is easily solved in terms of the distance.

(4) The direct problem can also be solved in terms of arc distance on
the auxiliary sphere of Legendre and Bessel.  However this is not what
you refer to as the auxiliary sphere but a trick for mapping a geodesic
on an ellipsoid to a great circle on a sphere.

Finally, an API for solving these problems is included in proj 4.9.0.
For documentation, see

   http://geographiclib.sourceforge.net/html/C/
   http://geographiclib.sourceforge.net/html/C/geodesic_8h.html

   --Charles