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

Mikael Rittri Mikael.Rittri at carmenta.com
Mon Oct 20 03:36:56 EDT 2008

Here is a variation of the first/principal/forward geodetic problem:

   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

Actually, I should say that I am not completely lost:  
   For ellipsoid accuracy, I would use Vincenty's formulas for the 
forward problem starting from C, and then use some numerical root-finding 
algorithm to find the azimuth at C that gives the desired azimuth at A.  
   For spherical accuracy, one can add the north pole as a third point B, 
and then ABC is a spherical triangle where two sides (a and b) and one 
angle (A) is known. For this case, textbooks usually suggest that one uses 
the spherical law of sines to find sin(B) (where B is the difference in 
longitude between point A and point C), and then one of Napier's analogies to 
find the angle C and the side c.  The most detailed formulas that I have found
are at http://encarta.msn.com/encyclopedia_761572350_2/Trigonometry.html#s7 , 
but the formula font makes it impossible to distinguish the upper case C 
from the lower case c.  I think it must be a lower case c in the 2nd formula
and a an upper case C in the 3rd formula, though.  I also think there is a 
typo in the 3rd formula: (a+b) and (a-b) should both be multiplied by 1/2, 
before the sine is taken.

But I am not quite sure about the fine details. If I know sin(B), 
I still don't known whether the angle B is acute or obtuse, or if both 
are possible solutions.  I think that if the distance between A and C is 
larger than the distance from C to the nearest pole, there would be either 
two solutions or none. (Well, of course, if C is the north pole and the 
azimuth at A should be zero, there would be infinitely many solutions.)
So, if anyone recognizes this problem, I'd be grateful for any advice 
or literature references. 

Best regards,

Mikael Rittri
Carmenta AB
Box 11354
SE-404 28 Göteborg
Visitors: Sankt Eriksgatan 5
Tel: +46-31-775 57 37
Mob: +46-703-60 34 07 
mikael.rittri at carmenta.com

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