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    Re: Spherical earth model vs. ellipsoid
    From: Mike Wescott
    Date: 1999 Mar 15, 6:37 PM

    My apologies if this has been seen already, but I think my previous
    attempt was lost.
    Lu Abel wrote:
    > As my mother used to say, "you're in the right church, but sitting in
    > the wrong pew."
    Actually we're both in the wrong pew (I oversimplified), But I think I'm
    closer than you :-)
    > Latitude and longitude are defined as angles with respect to the
    > earth's center.  But Mike is definitely right that our horizon will not
    > be perfectly perpendicular to the line of L/Lo eminating from the
    > earth's center.  I would correct his diagram by labelling the line
    > perpendicular to the local horizon "altitude" and not "latitude."
    No. Not exactly anyway. What follows is derived from App X, "Geodesy
    for the Navigator", Vol 1. of the 1984 Ed of Bowditch, and from the
    Glossary in Vol 2. See also the attached figure.
    The geoid (the black lumpy curve of fig-1) is the Earth's surface at
    mean sea level. I.e., it is the surface to which oceans would conform
    over the entire earth if the oceans were free to adjust to the combined
    effects of gravity and the earth's rotation. Because the distribution
    of the mass of the earth is not uniform. The surface of the geoid is an
    irregular curve.  One aspect of this surface is that gravitation is
    everywhere equal on this surface and everywhere perpendicular to the
    Because the geoid varies over the surface of the earth and is not
    easily predictable, geographers (or would that be geodicists?) have
    defined ellisoids to closely model the geoid. Different ellipsoids have
    been defined for different puposes or different areas. An ellipsoid
    that models North America well will not work as well in Austrailia, for
    example. One such ellipsoid is reresented by the red curve, and red
    axes in fig-1. Note that the axes do not necessarily coincide with the
    celestial axes (shown in black).
    Terrestrial Latitude (green) is the angle a ray from the center of the
    earth makes with the celestial equatorial plane.
    Geocentric Latitude (red) is the angle a ray from the center of the
    standard ellipsoid makes with the equatorial plane of the standard
    Astronomic Latitude (black) is the angle a normal to the geoid makes
    with the celestial equator.  This "normal" is the direction a plumb-bob
    will point. Astronomic Latitude is what is measured by a sextant or
    Geodetic Latitude (magenta) is the angle a normal to the standard
    ellipsoid makes with the equatorial plane of the ellipsoid.
    Longitudes are similarly defined.
    What is used on charts? Geodetic coordinates.
    What do we measure with a sextant (et al.)? Astronomic coordinates.
    What's the difference? Not much, because the ellipsoids are defined to
    minimize measurement errors over some specific region (or globally, as
    with WGS-84) maximum "deflections of the vertical", as such differences
    as called, are less than 0.5' if Bowditch can be believed.
    So, where do Geocentric and Terrestrial coordinates come into play?
    In short, they don't.
    > Let's clearly understand that with celestial observations, we're trying
    > to use observations made at the earth's surface and almanac data to
    > deduce our L/Lo.
    > But L/Lo have to be defined with respect to the earth's center, or else
    > how do we end up with the situation where one may move slightly more or
    > slightly less than a nautical mile when one makes a one minute change
    > in latitude near the equator or north pole?
    Because the surface of the Earth and the Geoid are closer to an
    ellispoid than to a sphere this situation will occur for any of these
    definitions of Latitude. And if you look at the mathematics behind
    the Mercator Projection and behind "Table 6: Length of a Degree of
    Latitude and Longitude", you'll find the assumption that Latitude
    is Geodetic Latitude, defined by the perpendicular to the tangent
    plane at the surface of the ellipsoid.
    > When I made my earlier comment on this thread, I did recognize that the
    > oblateness of the earth causes the local horizon to be not precisely
    > perpendicular to a line drawn from the center of the earth, but I chose
    > not to cloud a discussion of whether elipsoidal models affect celestial
    > fixes with that relatively minor point.
    > Unfortunately, my trig and calculus are sufficiently rusty that I can't
    > dash off an estimate of the error induced by the earth's oblateness.
    > Anybody??
    The difference between Geocentric and Geodetic Latitude (using WGS-84)
    is 0 at the poles and at the equator.  The difference maximizes at
    about 45d latitude with a difference of 11.5 minutes:
    Geodetic  Geocentric  Diff in
    Latitude  Latitude    minutes
    ========  =========  =========
     0.000      0.000      0.0
    15.000     14.904      5.8
    30.000     29.834     10.0
    45.000     44.808     11.5
    60.000     59.833     10.0
    75.000     74.904      5.8
    89.000     88.993      0.4
    90.000     90.000      0.0

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