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## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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Intersecting position circles. (was "Sun sights by 13 year-olds")
From: George Huxtable
Date: 2004 Sep 27, 18:11 +0100

```Chas Seitz wrote-

>Obviously, the next step would be to rewrite the software to accept data
>from additional sightings and calculate the circle intersections.
>That's easier said than done.  Many hours of web searching have failed
>to reveal any equations to do this.  I can't believe that during several
>thousand years of spherical geometry study, someone has not solved this
>problem.  It might be a horrendous problem but the solution is out there
>somewhere.  GPS is based on a similar concept.
>
>I'm not a mathematician and will not attempt to derive the equations.
>I'll look into solving this problem numerically.  But, that is a brute
>force approach that is not particularly satisfying from an aesthetic
>viewpoint.
>
>
>                                               ---  CHAS

=================

Chas might usefully take a look at the stuff in the back of the Nautical
Almanac (that few get round to) "Sight reduction procedures; methods and
formulae for direct computation".

The method in the NA requires an approximate assumed position, and I hope
Chas doesn't object to that aspect. It explains how to calculate altitudes
and azimuths of the observed bodies as seen from that assumed position,
using their predicted almanac positions, and to obtain the intercept of
each body from the difference between the calculated altitude and the
measured altitude. I presume that these are steps with which Chas Seitz is

In section 11 (it's on page 282 of my 2001 almanac) a set of equations is
supplied with which to calculate a much-improved estimate of the assumed
position. If that estimate meets a test of being sufficiently close to the
assumed position (which it usually does), then that's the answer.
Otherwise, it's necessary to reiterate, recalculating the altitudes and
azimuths from the improved position. This can be done as many times as
necessary.

For two sights, giving two intersecting circles, there will always be two
solutions, because the circles always cross in two places. If a
sufficiently stupid assumed position was chosen, the method may home in to

The method can (and indeed was designed to) handle any number of
intersecting circles, not just two. In which case, there is not an exact
solution, as there was in the case of two circles. Two circles will always
intersect at a simple point (well, two points, as noted above; one of which
is unwanted). When a third circle is added it will normally pass close to,
but not exactly through, the previous intersection, because of random
errors involved. So there is a small triangle (the "cocked hat" on a
chart), in the region of which is the correct position. The program does
what it can, using statistical methods, to estimate a likely place for the
solution, and to indicate an area of uncertainty, if there are three or
more observations.

This is usually described as a "least-squares" solution to the problem. A
more detailed account can be found in the booklet "AstroNavPC and Compact
Data 2001-2005" (Willmann-Bell), which includes a CD-rom of the program
that implements it, and which has many additional features. Unfortunately,
it won't run on my old Mac. It was developed by HM Nautical Almanac Office
in the UK, and no doubt a new issue is due soon.

I've little doubt that this will provide the solution that Chas seeks; or
if, like me, he prefers to "roll his own", it will show him how to do it.
If he has difficulties, help is likely to be available on this list.

George.

================================================================
contact George Huxtable by email at george---.u-net.com, by phone at
01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
================================================================

```
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