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    Re: leg56. Chart vs formulas
    From: Lu Abel
    Date: 1999 Jul 16, 12:30 PM

    A method for figuring out a course and distance between two points without
    using a chart is called a Sailing.  There are three Sailings generally used
    by navigators, Mid-latitude, Mercator, and Great Circle.
    1.  Mercator.  This behaves as if one is plotting the course on a Mercator
    chart, but without actually drawing it.  As Dan points out, a Mercator
    sailing produces rhumb-line course cutting every meridian at the same
    angle; in other words the vessel holds a constant true heading over the
    whole trip.
    If I draw a straight-line course on a Mercator chart, I'm creating a right
    triangle, with one side part of a meridian of longitude, one side part of a
    parallel of latitude, and the hypotenuse as the course line.  But the
    vertical side of the triangle is not simply the difference in latitude, but
    the difference as it would be expanded and drawn on a Mercator chart as
    needed to keep the meridians parallel to one another.  The meridional parts
    table in Bowditch (which can be used either for drawing charts or figuring
    Mercator sailings) gives these expansions, taking into account even such
    things as the fact that the earth is not a true sphere.
    2.  Mid-latitude.  If the difference in starting and ending latitudes are
    not too great, computing a mid-latitude sailing is far easier than a
    Mercator sailing.  "Mid-latitude" doesn't mean the method works only at
    middle latitudes, but rather that one calculates the average or
    mid-latitude (Lm) between the starting and ending latitude.  One uses
    1/cos(Lm) as an approximation of the Mercator "expansion factor" for the
    latitude side of a simple right triangle and uses that to compute a course.
     (It's really a lot easier than my words make it sound -- all it takes is a
    few simple lookups in the traverse tables in Bowditch or a few keystrokes
    on a $15 scientific calculator.)
    3.  Great Circle.  This, as its name implies, calculates the great circle
    course between start and destination.  It uses mathematics very similar to
    sight reduction using either a calculator or one of the tabular methods.
    To actually follow a great circle course means periodically recalculating
    the needed course since it constantly changes over the great circle.
    Lu Abel
    At 05:36 AM 7/16/99 -0800, Dan Hogan wrote:
    >Mercator Sailing:
    >        Provides a solution to the plot as made on a mercator chart. Uses
    >meridonial differences and difference of longitude in place of
    >difference of latitude and departure, respectively.
    >Rhumb Lines:
    >        Maintains a true direction. Makes the same angle with all meridians
    >it crosses. Maintains a straight line on a mercator chart. The
    >difference between the Rhumb line and the Great Circle connecting two
    >points increases as (1) as the latitude increases, as the difference in
    >latitude between two places decreases, and (3) as the difference of
    >longitude increases.
    >>I don't know what is the difference between mercator sailing and
    >>rhumbline, for me is the same �?
    >Dan Hogan

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