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    Re: St Hilaire not Iterative was: Finding The Symmedian
    From: George Huxtable
    Date: 2010 Dec 24, 21:24 -0000

    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.
    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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