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    Spherical vs Non-spherical models for navigation
    From: Lu Abel
    Date: 1999 Mar 11, 12:46 PM

    In a previous message, Tom Feuerhelm states:
    
    - However, in astronavigation, we all use a spherical model !
    
    and then asks a bunch of questions about why and whether it produces any
    inaccuracies, give that we have all sorts of ellipsoidal models of the earth.
    
    Latitude and longitude, which are what we are determining when we do
    celestial navigation, are defined on a sphere.  To find one's position at a
    certain latitude and longitude, one would draw a line out from the center
    of the earth at the proper angles and wherever it penetrated the earth's
    surface that would be the location on the earth given by that L/Lo.  It
    doesn't matter whether the surface of the earth is a perfect sphere or
    slightly squished as it is in reality.
    
    With celestial observations, one goes directly to an angular position with
    respect to the center of the earth (we call it latitude and longitude) --
    the shape of the earth doesn't enter into even the most precise calculation.
    
    It's when one tries to go the other way -- from a point on the surface of
    the earth back to it's latitude and longitude -- that the shape of the
    surface becomes important.  We're all taught in our earliest navigation
    classes that a nautical mile is equivalent to a minute of latitude on the
    earth's surface.  Not true -- it's an *average* minute of latitude.  The
    length of a minute of latitude varys depending on where you are on the
    surface of the earth; Bowditch has tables which cover this.  So if I go
    north by one nautical mile from the equator, I will not end up at precisely
    0 deg 1 min north latitude.  Similarly, if I come south one nautical mile
    from the North Pole, I will not be at exactly Lat 89d 89m N.
    
    These differences are on the order of 60 feet (0.01 percent) or less.
    Measuring distances of hundreds of miles to an accuracy of a few feet or
    less has been something land surveyors have been able to do for 200 years
    or more, so mapmakers have been worrying about the precise shape of the
    earth (and developing all those map datums and ellipsoids) for an equally
    long time.
    
    On the other hand, a celestial navigator is lucky to get a sextant reading
    with one minute accuracy.  Even if latitude and longitude somehow depended
    on the shape of the earth, the differences between a precise model and a
    sperical model of the earth would be way below the magnitude of other
    errors in our observation and sight reduction processes.
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