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    Truth about St. Hilaire
    From: John Karl
    Date: 2015 Nov 21, 12:21 -0800

    Frank reminds me of an old discussion, in his 18 Nov 22:13 post (g33598), by saying that the St. Hilaire method uses a simulation:

    From Frank:

    What is the intercept method? At its most basic level, the intercept method is comparison against simulation.  We observe a celestial object's altitude, and then we compare that altitude with a simulated observation from some nearby location at the same instant of time. The simulated observation calculation gives us both the altitude and the azimuth of the celestial object.

    To me, the use of the word “simulation” is misguided or misleading, just as the term “assumed position” is, as I’ve discussed at length back in 10/15/2007 to 10/28/2007 (g3417-g3680).

    St. Hilaire uses neither a simulation nor an assumption.  In a single-body sight reduction we start with its geographical position, GP (from the time & NA) and its co-altitude, coH (from the observed altitude, Ho).  We know everything there is to know:  The LOP is a circle of radius coH and centered at the body’s GP.  This is a curve, a plot, on a 2-d surface where all points are defined by the two coordinates, lat & lon.  And we have the equations for all points, any point, on this LOP.  As in plotting any 2-d curve, from its mathematical expression, you simply specify one of the 2-d variables to calculate the other.  It’s like plotting y = sin(x).  There is no assumption, no simulation.  For example, pick a series of latitudes and then calculate the corresponding longitudes of LOP points.  Repeat this as many times as you wish to calculate as many LOP points you wish.  Or visa versa.  No approximations, no assumptions, no simulations, you can plot the LOP anywhere you wish.

    Which brings us to the question of where we wish to plot it.  Again, no assumptions, no simulations.  But we do need to specify what portion of the LOP we wish to plot.  Here St. Hilaire trumps most all other sight reduction methods.  For any specified location, St. Hilaire’s approach calculates the unique Point on the LOP (call it the PLOP) that is nearest to that specified point, call that the Assigned Point, the AP.  (Calling this point an assumed position has misled students of celestial navigation for many decades.)

    Next using the body’s GP, the Ho, and the AP, St. Hilaire calculates the great-circle distance and azimuth from the AP to the PLOP.  This distance (called the intercept) is usually so small that it can be plotted on a Mercator chart with negligible error, making this PLOP an accurate point on the LOP, using no assumptions, no simulations.

    Finally, the real advantage of St. Hilaire is that we don’t need any more PLOPs.  Since this azimuth points from this PLOP to the body’s GP, a perpendicular line to the azimuth running through this PLOP (and is thus tangent to the circular LOP) is a good approximation to the LOP at that point (when Ho is less than say 75°).  Using this straight line for the LOP is the only approximation in the method, and we see no assumptions, no simulations.

    Happy St. Hilaire, John K

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