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    Re: Poor St. Hilaire
    From: George Huxtable
    Date: 2007 Oct 18, 23:57 +0100

    John Karl wrote-
    
    | It looks like it's boiling down to our definition of the St. Hilaire
    | intercept method.
    |
    | To quote from the 1962 edition of the glossary in Bowditch:  "St.
    | Hilaire -- The establishing of a line of position from the observation
    | of the altitude of a celestial body by use of an altitude difference
    | and azimuth".
    
    Well, if we're playing with words, and the Bowditch definition becomes
    important, John's quote differs significantly from the definition in vol. 2
    of my 1981 edition of Bowditch, which runs as follows-
    
    "St Hilaire method - The establishing of a line of position from the
    observation of  the altitude of a celestial body by the use of an assumed
    position, the difference between the observed and computed altitudes, and
    the azimuth. This is the method most commonly used by modern navigators. The
    assumed position may be determined arbitrarily, by dead reckoning, or by
    estimation. The method was devised by Marcq St. Hilaire, a French naval
    officer, in 1874. ..."
    |
    | This is also the definition in other glossaries, and is used in the
    | books I have seen.  In St. Hilaire's original papers he doesn't bother
    | with a one-line definitions, of course,  but discusses all sorts of
    | sight reductions under a general section headings.
    |
    | I had hoped that it was clear that I'm using the traditional Bowditch
    | definition of the method - establishing a single LOP.  As I've stated,
    | the error in the straight-line approximation is only due to the
    | curvature of the true LOP compared to the distance along the straight-
    | line LOP.
    |
    | This error is well known and can be corrected without iterations.  HO
    | 229 has a table of offsets that does exactly this (or you can
    | calculated them yourself).
    
    Yes, I agree with John about that. A one-page table of offsets can also be
    found as table 4 in Bowditch vol 2, and probably other editions. It allows a
    straight-line LOP to be bent into an arc of the right curvature, so that on
    the chart it corresponds to an arc of the circular LOP. If this has been
    done, for both bodies, then the intersection of those arcs should determine
    the true position precisely, independent of how good or bad the initial
    estimated positions were (within limits). Therefore, no reiteration would be
    called for. Is that part of a "St. Hilaire procedure", though? I doubt it.
    
    John goes on to say-
    
      My book presents a direct calculation (no
    | iterations) at the bottom of page 77.  It uses the same two equations
    | of the St. Hilaire method, five times.  Whereas a two-body St. Hilaire
    | fix uses four of these equations.  So with just one more equation, of
    | the same type, we eliminate the straight-line approximation, using no
    | iterations.
    
    That is indeed of great interest. It's at the bottom of page 78, by the way,
    not 77. It takes the simultaneous altitudes of two bodies, together with the
    decs and LHAs of those bodies, and deduces the position (actually, two
    alternative positions) of the observer to satisfy those requirements. It
    requires no estimations, and no chartwork. I haven't come across it before,
    and wonder whether it's an entirely new piece of work, from John. Certainly
    very different from any St. Hilaire procedure.
    
    But I wonder whether there are shortcomings that might show up on closer
    investigation. A few questions come to mind-
    
    How does that method handle a situation in which there's a distance-run, or
    just an interval, between two sights?
    As equations 7.5b, 7.5c, and 7.5e use an arc-cos, how well do they manage
    when that angle nears zero degrees, and its cos is changing hardly at all?
    Are there problems of putting angles into the right quadrant?
    
    | Yes, one CAN use St. Hilaire iterations to improve the straight-line
    | approximation, using other information.  But the St. Hilaire method
    | itself as defined by Bowditch (and others) doesn't use iterations.  So
    | this part of the discussion has boiled down to semantics.  I was using
    | the traditional Bowditch definition when I wrote
    |
    | >  Now if members think that this is unnecessary and unproductive
    nit-picking
    | >  of terminology,
    
    Yes, I wasn't sure, before, that that was the case, but I'm becoming more
    convinced that it is.
    
    consider that:
    |
    | >  1.  All CN books (well, all that I have seen) either don't attempt to
    explain the reason for the > assumed position, or they explain it
    incorrectly.  For example, some say that an
    | >  assumption is necessary because the distance between
    | >  the sun's GP and the ship is too great to plot, some because there's
    insufficient
    | >  information to plot the LOP,
    | >  and others because we don't know how to plot the exact LOP.
    |
    | >  2.  A List member has stated that the accuracy of the St. Hilaire
    result depends on how
    | >  good the initial estimated position is.
    |
    | Speaking of a single LOP now, I thinks it's more accurate, and
    | informative, to say the accuracy depends on the distance along the
    | straight-line LOP compared to the curvature of the true LOP.  I think
    | it's misleading to say it depends on an estimated position. After
    | all, no matter what AP (or estimated position) is used, my accuracy
    | statement still applies.
    |
    | >  3. And therefore the St. Hilaire method is really an iterative method.
    |
    | If we want to expand the definition of St. Hilaire method to include
    | iterations, that's a new usage to me - and considering that this whole
    | discussion is about the wisdom, and implications, of certain
    | terminology, I'm not sure it's a good idea.
    
    St. Hilaire himself though that iterations were called for in certain
    circumstanses, as has been pointed out.
    
    | >  We've just seen that all of this is wrong.  This misunderstanding may
    not stem from the
    | >  unfortunate
    | >  terminology of "estimated" position and "assumed" position.  But if it
    doesn't,
    | > where does it come from??
    |
    | So let's forget iterations.  My major point is item (1) above.
    
    Should we gather, then, that John withdraws points 2 and 3? Those were the
    only hard conclusions that resulted from his argument, and Andres Ruiz and I
    questioned them. His item 1 is sufficiently imprecise and woolly that it's
    hard to agree or disagree.
    
    George.
    
    contact George Huxtable at george---.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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