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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
    >                                               ---  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
    already familiar.
    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
    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 wrong answer.
    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.
    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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