[Proj] Locate a point from distance and backwards azimuth?

Karney, Charles ckarney at Sarnoff.com
Tue Jun 23 09:54:20 EST 2009

> From: Mikael.Rittri at carmenta.com
> Date: Monday, October 20, 2008 03:36
> Here is a variation of the first/principal/forward geodetic problem:
> Known:
>    Position of point C.
>    Distance from point C to an unknown point A.
>    Azimuth, at A, of the great-circle arc between A and C.
> Where is A?
> Note the difference from the first/principal/forward problem: the
> azimuth is known at the unknown point A, instead of at the known point
> C.
> I am not really sure when this is useful, but I have been asked about
> it twice by different people, so I feel I ought to solve it some day.
> Does this problem has a name, and are there detailed published
> solutions anywhere?

This is, apparently, an old chestnut.  It is "IVe cas." on p. 489 of

    L. Puissant,
    Nouvel essai de trigonométrie sphéroidique
    [New essay on spheroidal trigonometry],
    Mém. l'Acad. Roy. des Sciences de Paris 10, 457-529 (1831).

The table on p. 521 shows the other cases that Puissant treats.
Puissant discusses this in the context of a spheroidal earth.

If you're happy with a spherical solution, then I suggest

   L. Euler,
   Principes de la trigonométrie sphérique tirés de la méthode des
   plus grands et plus petits
   [Principles of spherical trigonometry taken from the method of the
   maxima and minima],
   Mém. de l'Acad. Roy. des Sciences de Berlin 9,
   223-257 (1753, publ. 1755).
   http://math.dartmouth.edu/~euler/pages/E214.html (figures missing)

His "Prob. X" on p. 254 treats the case you're interested in.

Charles Karney <ckarney at sarnoff.com>
Sarnoff Corporation, Princeton, NJ 08543-5300

Tel: +1 609 734 2312
Fax: +1 609 734 2662

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