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    St. Hilaire -- My Take
    From: John Karl
    Date: 2009 Nov 15, 19:08 -0800

    Here's my answers to several questions from the recent 2-body post.
    I'm starting a new post since these are unrelated to the 2-body
    question (although they've been somewhat addressed in much early
    A.   DIRECT CALCULATION:  By direct calculation I mean computing the
    Lats and Lons of points on the celestial LOP from some an equation, or
    equations.  These points trace out the LOP in Lats & Lons, just as one
    would make an x-y plot of Sin A versus the angle A, by evaluating Sin
    A and plotting the result versus A.  For a CN example, we can find LHA
    from the single equation Cos LHA = (Sin Ho - Sin Lat Sin dec)/ Cos Lat
    Cos dec, getting the longitude of a LOP point that has latitude Lat
    (from Lon = LHA - GHA).  This is an example of a direct calculation of
    the Lats & Lons of the LOP's coordinates.  Also tables could be easily
    constructed to give the LHA in terms of entries of Dec, Lat, and Ho.
    (The navigator then takes LHA - GHA to get longitude.)
    B.  PLOTTING VARIABLES:  I think (not sure) that the discussion on the
    List about parameters is what I would call plotting variables.  In the
    above, the plotting variable is the latitude.  Sumner used the
    longitude as a plotting variable to plot three points on his LOP.
    Another plotting variable is the true bearing from the GP to the LOP
    point (that is, the angle at the GP between the coH side and the coDec
    side of the nav triangle).  This is convenient in computer
    applications.  (These bearing-angle equations are discussed in
    Exercise 1.22 in my book.)  In high altitude sights using a drafting
    compass with its point on the GP, its distance equal to the observed
    coH, and striking an arc on a chart, we are mechanically using the
    bearing angle as the plotting variable.
    C.  ST. HILAIRE:  The St. Hilaire method uses two equations to plot a
    point on the LOP, the altitude and azimuths equations.  The point at
    the end of the intercept, at the computed azimuth, is exactly on the
    LOP, just as in the above examples.  (In all practical applications
    the difference between the rhumb line and great-circle distances is
    negligible. Moreover, the great circle intercept distance is indeed
    exact, just as the Lats & Lons are from the direct method.  So chart
    projections are irrelevant in this whole discussion.)  If desired,
    many other APs can be used to compute many other points exactly on the
    LOP.  Thus St. Hilaire can be used to trace out the LOP -- exactly.
    D.  ST. HILAIRE & DIRECT:  Both methods compute points on the LOP
    exactly.  There are no assumptions, no estimates, no iterations.
    These computations might be done with calculators, computers,
    logarithms, Bygrave slide rules, or tabulated results (which are the
    results of someone else doing the same math).  The means of
    calculating has no bearing on this topic.  And forget whole degrees --
    they have nothing to do with this discussion.  (Also BTW, they are
    sometimes used in log calculations and calculator applications for
    convenience, the same reason they're used in tables.)  Both methods
    provide straight line approximations to the circular LOP.  In the
    direct calculation, two points on the LOP are connected with a cord,
    giving a approximation similar to St. Hilaire's, and in some cases
    even better.
    E.  ST. HILAIRE vs. DIRECT:  At first it seems that St. Hilaire is
    inferior to direct calculation for two reasons:  It requires two trig
    equations for each LOP point, and it requires plotting because it
    doesn't give Lats & Lons directly.  But St. Hilaire has a trump card:
    IT'S ROBUST (i.e., it's not fussy, works the same in all cases, has no
    gotcha's).  But the direct calculation isn't robust.  For example, in
    calculating a Lon corresponding to a selected Lat, a nearly east-west
    LOP can yield a Lon that's too far away.  Worse yet, the LOP might not
    even intersect the parallel of latitude.  Likewise, specifying Lons
    and calculating Lats gives the same problem with nearly north-south
    The essential point is that the St. Hilaire method completely avoids
    these problems by using a completely different approach to specifying
    which section of the LOP we wish to plot.  It specifies a point by
    both Lats & Lons to identify our area of interest.  It says that we
    want the section of the LOP that is closest to this point.  This point
    is a reference point, a locator point, usually abbreviated
    "AP" (doesn't "AP" stand for A locator Point?).  There is no
    assumption, no estimation, only a decision.  We need to decide where
    we want to plot the LOP.  We must decide this before plotting any
    celestial LOP by any method (unless we wish to plot the whole
    circle).  Once we have the point on the LOP closest to our area of
    interest, a straight line drawn perpendicular to the azimuth to the GP
    gives an approximation suitable for all but very high altitude cases.
    This is similar to the direct-calculation straight-line approximation.
    F.  TERMINOLOGY:  Some list members have pointed out that our
    terminology may depend on our background: how, where, and when we
    learned celestial.  Yes, this is true of many words, but from what
    I've read by members, we have pretty good agreement on DR and EP.
    However, I was surprised to see a member write that the term "LOP"
    means a straight line.  Not in my book.  There are many LOPs that are
    not straight:  a range LOP from a lighthouse's height, the double
    horizontal-angles LOP from two objects, a LORAN LOP, and a bathometric
    contour, just to name a few.  An LOP is simply any line of position,
    no matter what shape.
    As some member's know by now, "AP" is nomenclature that really irks
    me, for two reasons:  First we don't assume anything when we pick an
    AP, we just decide where we want to plot a section of the LOP.  We
    select it -- we don't assume anything.  It's like deciding to plot Sin
    A between 45 and 63 degrees.  We're deciding, not assuming.  The 2002
    edition of Bowditch also lists CHOSEN POSITION for the term, which
    matches well with what I'm saying.  Second, if you think this is
    putting too fine a point on terminology, let me say that every CN book
    I've seen either doesn't explain why we're using an AP, or if they
    attempt to explain it, they either have it wrong, or give the wrong
    implication.  I believe the term "assumed position" contributes to
    this confusion.
    G.  Someone asked where, or how, I learned CN.  Well, when I was about
    8 or 9-yrs old in Flint MI, I purchased a new 1945 edition of Bowditch
    since we were all sailors in my family.  While it was hard going for
    me, I think I learned a lot from it, but with no sextant and no
    horizon, wasn't able to practice sights.  Then in 1960, while a
    physics undergrad at M.I.T., I stumbled upon a sextant in a Chelsea
    pawn shop.  I bought this Hezzanith for $18, went home and pulled
    Bowditch off the self, and started taking sights looking at the sky
    and ocean from near my home, then in Marblehead.  In those days I used
    logs from Bowditch, and later, tables from H.O. 211.  Many years
    later, when I was invited to teach CN aboard the S/V Denis Sullivan on
    a Milwaukee-Montreal leg, I become interested in teaching this stuff.
    So I wrote a couple of books, a course manual, a PowerPoint
    presentation, and now teach on land or sea whenever I get the chance.
    NavList message boards: www.fer3.com/arc
    Or post by email to: NavList@fer3.com
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