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John Karl wrote, about an alleged "misconception"-

"We've discussed this before, but the intercept method is not iterative.
The calculation from ANY AP produces a true azimuth to the body, and the
intercept marks an exact point on the LOP (a PLOP). (And in all practical
cases, plotting the intercept as a rhumb line on a Mercator chart has
negligible error.) Drawing a straight line through the PLOP is an
approximation, but not an an iteration. If it's desired to repeat the AP
selection and associated intercept and azimuth calculations for another
PLOP, we're simple tracing out the LOP -- exactly. No approximations, no
iterations. Tracing the LOP point by point is not an iteration. Each PLOP
is exact. See the attached figure."

and Gary replied-


"Right, that PLOP is exact, but if the two straight line approximations of
the LOPs cross at great distance from the two PLOPs then the intersection
doesn't accurately mark the fix so another iteration should be done which
will result with an intersection much closer to the PLOPs."

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

We have indeed been through this before.

Yes, a true LOP is exact, if it is drawn as the arc of an appropriate
circle, centred on the GP of the observed body. The problem arises in
representing that arc by a straight line, tangent to the circle. It has
little to do with the projection that's in use, Mercator or otherwise, as
becomes obvious if we consider the small circle that represents the LOP of
an object near the zenith. Taking the fix to be the intersection of two
resulting straight lines, rather than the intersection of two arcs,  is the
approximation, just as Gary points out, and as St Hilaire recognised. And
then, to get a precise answer from an imprecise DR, the St Hilaire method
IS iterative.

The Nautical Almanac put forward their algorithm on page 282 as an
iterative procedure, and I expect that they chose a maximum offset of 20
miles to ensure that the basic resolution of their tables, of 0.1 miles,
wasn't significantly degraded. It's the basis of their software package
"AstronavPC" (and probably other commercial software).

If you can be sure that your DR is always within 20 miles or so of the
truth, or if you are prepared to accept a less-accurate result if it is
outside that, then iteration might perhaps be unnecessary. But once you
have taken the trouble to write a program on a calculator or computer, it's
a rather trivial step to include iteration. So why not?

Indeed, having done so, the algorithm seems remarkably robust, and as I
have mentioned before, seems capable of homing in from (almost) anywhere in
the world to the correct position (or two positions, in the case of a
two-body observation). However, I have made no exhaustive tests to verify
that. It might take six or seven rounds to do so, if offered a really
outlandish starting-place. But if that's really true, it implies that
there's no need to offer a starting DR position at all, and you could just
as well start off from somewhere such as the North Pole every time.

Andrew Nikitin wrote, under the thread "Finding the Symmedian"-
"The reason NA mentions that this is an iterative process only in
association with the formula and not elsewhere, is that if you are going to
use the formula, this means that you are using some kind of computer
anyway, and recalculating new azimuths and Hc for all sights is trivial. So
you should be diligent and just do it."

Indeed, that's true. It's such a trivial matter that the computer can do
it, if given the GPs and altitudes, and the user needs to do nothing. So
why not include it in Nikitin's procedure?


"On the other hand, if you are not using formula and using plotting
techniques to find fix, this means that your are probably using tables and
recalculating az and Hc would double your work. So you should be diligent
and spend your valuable time doing something more productive."

I don't follow that logic. It depends if you are prepared to accept
inferior results in some situations. A navigator using graphical techniques
is likely to recognise those situations and reiterate. A user who is simply
taking numbers from a computer may well be doing so blindly.

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

On quite another topic, Andrew Nikitin is a new name to me, and if I'm
right, these are his first postings to Navlist. They show that he has a
thoughtful turn of mind, and is knowledgeable about the very matters that
we so enjoy arguing about on this list. May I offer a warm welcome to him,
and hope we will see much more from his keyboard.

George.

contact George Huxtable, at  george@hux.me.uk
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.




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