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    Poor St. Hilaire
    From: John Karl
    Date: 2007 Oct 15, 08:29 -0700

    To all criminals against GPS,
    
    Have you noticed Poor St Hilaire?  It seems that he is misunderstood
    by most CN navigators, even the authors of CN books.  Of the more than
    20 authors I've read, I think none has it correct.  Now recent
    postings on the List have reminded me of the St. Hilaire discussion in
    my recent book, "Celestial Navigation in the GPS Age."
    
    It goes back to Capt. Marcq himself.  In my English translation of his
    1873 publication, he refers to the "estimated position" when
    explaining his intercept method.  This unfortunate terminology has
    guided many CN students to a Sea of Mystery engulfing the DR position
    and the confusing concept of "assumed position".
    
    The confusion even dates back to Sumner.  I'm sure most List members
    are familiar with his 1837 calculation of a points on the ship's LOP
    by calculating its longitudes from its stipulated latitudes. Sumner
    called these "assumed" latitudes.  At first he thought he was using
    unreliable latitudes.  But after plotting three such "trial"
    positions, he had an epiphanic realization that his calculations
    actually gave exact locations of points on his celestial LOP.  Two
    such calculations give two exact points on the LOP.  The straight line
    connecting these two points has an error between the points that
    depends on several factors, such as the LOP's curvature and the
    chart's projection.
    
    Hilaire's intercept method invites further confusion.  First, we
    observe that the altitude observation and the knowledge of the body's
    GP completely determines the celestial LOP.  Therefore no sight
    reduction method uses estimations or assumptions.  Again, it's the
    terminology that screws up the concepts.
    
    There are several ways to determine the exact LOP by directly
    computing and plotting latitudes and longitudes of as many points as
    we wish (Sumner's being one such method).  But St. Hilaire is
    different -- it doesn't compute Lats and Lons at all.  Rather, it
    specifies a point's latitude and longitude which I'll call the
    reference point, RP, even though conventionally it's called the
    "assumed" position -- a misnomer.  St. Hilaire calculates the great-
    circle distance and azimuth from this RP to the nearest point on the
    LOP.  This point is exactly on the LOP; let's call it the St. Hilaire
    point (SHP).  Both the great-circle distance and the azimuth to the
    SHP are exact.  No errors.  Even if we use inspection tables (such as
    HO 214, 249, or 229) with their whole-degree limitation, the SHP is
    located exactly on the LOP within the table's accuracy.  (HO 229
    states a max altitude error of 0.3'; and for altitudes less than 86
    degrees, the max azimuth error is 0.2'.)  If we use a $12 calculator
    we get 10-digit accuracy.
    
    However, unlike the Sumner method that calculates Lats and Lons
    directly, the St Hilaire method produces errors when the LOP is
    plotted.  On a gnomonic chart the intercept distance is correct, but
    the azimuth is in error.  On a Mercator chart it's the reverse.  This
    statement is true no matter what RP is used, using whole degrees of
    latitude and LHA with inspection tables, or not.
    
    In common practice at sea, we use a Mercator projection where the
    intercept's azimuth is correct and its intercept distance has
    negligible error when approximated by a rhumb line (error less than
    0.3 nm for a 300 nm intercept, unless in polar latitudes).  Therefore,
    in practical terms, several SHPs can be calculated with inspection
    tables (or other means), tracing out the LOP exactly.  This method
    uses no iterations, no estimations, and no assumptions.
    
    However as List members know, these multiple SHPs are seldom
    calculated because drawing a straight line through the SHP
    perpendicular to the azimuth is usually an acceptable approximation to
    the curved LOP at altitudes below about 75 degrees.  If the LOP's
    curvature deviates unacceptably from the straight-line approximation,
    another (or even a different) calculation can be made.  But in any
    case, with an 75 degree observed altitude, a point along the straight-
    line LOP 40 nm from the SHP is offset from the true curved LOP by 0.9
    nm.  So at lower altitudes, the whole-degree limitation of inspection
    tables would never contribute significant error.
    
    My book also discusses why the St. Hilaire method is superior to other
    methods even though (1) it requires exactly the same amount of
    calculation as other methods, (2) it doesn't give latitudes &
    longitudes directly, (3) it has plotting inconveniences, and (4) it's
    only approximate.  I'm new to the NavList; so I apologize if members
    find this all verbose,  irrelevant, or old hat.
    
    John Karl
    
    
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