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In the thread "Reality check",Lu asked the following perceptive and
interesting question, which then got sidetracked into how easy it was
to estimate a rough position without instruments.-

| If I've got two Ho's (and obviously know what bodies were observed,
what
| time the observations were taken, etc, etc), is there a direct
solution
| to obtaining a position that does NOT require a DR, calculating two
Hcs
| and Zns, and crossing the resulting LOPs?
|
| As someone who works in the computer field and has at least a
| rudimentary knowledge of numerical analysis, I can easily see that
this
| could be set up as an iterative solution (ie, guess at a position,
| calculate what the observed body altitudes would be, compare to
actuals,
| use differences to get a direction to move the assumed L/Lo,
repeat...).
|  This is, in fact, not much different from the way your friendly GPS
| calculates its position.
|
| But there's a part of me that says some expert in spherical trig
came up
| with a way to cross two spherical triangles centuries if not
millennia
| ago...

Yes, there is such a method, a remarkably simple one, that involves no
dead-reckoning or assumed position. To start with, it's easiest to
think about it geometrically, with a model, which is what Elizabethan
mariners would do, using a globe of the Earth, which they would carry
on board.

Let's assume that you have measured the altitudes of two stars at
pretty nearly the same moment, or the Sun at two different times from
an anchored position, so that there is negligible travel of the vessel
between the two observations.

At the moment of its observation, body1 is known from the almanac to
be at a position of dec1, GHA1, when its altitude was measured as
alt1. It was then directly above a Geographical Position (GP1) on the
Earth's surface, where lat1 = dec1, long1 = GHA1 ( measuring
longitudes Westwards, just like GHAs). You can mark that GP1 on the
surface of the globe. The vessel must be somewhere on a circle of
radius (90 - alt1) degrees, which is the "locus" of all possible
positions of a vessel making that observation. You can take a pair of
compasses, open them to a spacing of (90 - alt1), measured off the
latitude scale of the globe, stick the spike into the marked GP1, and
draw a circle. The vessel is somewhere on that circle, but where on it
is as yet unknown. It may be anywhere.

Similarly, when body2 was observed to be at alt2, its almanac position
put it at GP2, (dec2, GHA2). So mark a position for the second body,
at GP2, and draw a circle round that, at radius (90 - alt2), and the
vessel must also be somewhere unknown on that circle also.

In which case, the vessel must lie at the intersection of the two
circles. In general, any two circles which intersect must have two
such intersections, not just one, and there's no telling which is the
one the vessel must be at, from the given information. So some rough
knowledge of the vessel's position is required, in order to eliminate
the false solution. As long as the two bodies were chosen to have
well-differing azimuths, according to normal navigational practice,
the two solutions will be thousands of miles apart, and the false
solution becomes obvious by commonsense, or from a rough knowledge of
those azimuths. Other than for making that distinction, however,
there's no requirement for any sort of DR or assumed position, no
calculation of intercepts and azimuths; it couldn't be much simpler.

Having rejected the false solution, the vessel's lat and long are read
off the globe at the remaining crossing-point, and the job's done.

The snag is, of course, that working from a model globe is inherently
imprecise, when looking for a resulting position to within a mile (or
few) on a globe that's only a few inches in radius. But there's
nothing imprecise about the principle involved. So the question
arises, can the same two solutions be found mathematically, without
using the model globe as an analogue calculator? Yes they can. The
algebra for doing the job is somewhat complex, so that it would be
hard work to do the job in "longhand", but presents no challenge to a
computer or programmable calculator.

I have written a program in bastard-Basic which runs on my 1980s Casio
programmable calculator (FX 730P or FX 795P), and if anyone is
interested would be happy to send it or post it up. It would be simple
to adapt it to another machine. It takes the 6 quantities, dec, GHA,
and altitude for each of two bodies, and returns two possible
positions in terms of lat and long, for the user to choose the
appropriate one. It does not require a DR or AP, and provides an exact
result without going through an iteration process.

It's not original, in that versions of the method have been described
previously beforehand. For example, in an article by George Bennett in
the journal "Navigation"  (which is, I think, the American one) Issue
no. 4, vol 26, winter 1979/80, titled " General conventions and
solutions- their use in celestial navigation", and to the book
"Practical navigation with your calculator", by Gerry Keys, (Stanford
maritime, 1984), section 11.12. The method has also been described in
"The K-Z position solution for the double sight", in European Journal
of Navigation, vol.1 no, 3, December 2003, pages 43-49, but that
article was bedevilled by printing errors that render it more-or-less
unintelligible, which were corrected in a later issue. Not to mention
several serious errors and misunderstandings by the author, which have
never been acknowldged or corrected in that journal.

The method can be extended to simultaneous or almost-simultaneous
sights of more than two bodies, by taking all possible pairs of
bodies; for example, 3 pairs with 3 bodies. Each pair then results in
a calculated position, and for 3 bodies each such position is the
intersection at a corner of a "cocked-hat".

The method of intersecting circles becomes much more difficult
(impossible, perhaps?) when there's a need to account for the travel
of the vessel in the time interval between the two observations, the
"sun-run-sun" case. But that's another story.

George.

contact George Huxtable at george@huxtable.u-net.com
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

   

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