# [Proj] direct geodesic problem

Aleksander Yanovskiy yanouski at yandex.by
Thu Aug 21 07:02:01 EST 2014

```Thank you Charles,

the points are quit clear.

Sincerely,
Aleksander

21.08.2014 13:45, Charles Karney пишет:
> 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

```