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

Karney, Charles ckarney at Sarnoff.com
Sat Apr 10 18:45:08 EST 2010

```> From: Mikael Rittri [Mikael.Rittri at carmenta.com]
> Sent: Monday, October 20, 2008 03:36
>
> Hello,
> 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?

Converting this problem into a projection you get the retro-azimuthal
equidistant projection.

A. R. Hinks,
Geographical J. 73, 245-247 (1929)
http://www.jstor.org/pss/1784715

gives a useful application: You want to know where you are, given the
distance to the VLF clock at Rugby and the azimuth of the signal.  Note
that there may be 0, 1, or 2 solutions.

The implicit assumptions here are:
You know where Rugby is
VLF signals travel along geodesics
You have an accurate local clock
You know the direction of true north

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

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