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    Re: Computaion of table of offsets.
    From: George Huxtable
    Date: 2007 Oct 30, 17:11 -0000

    A Table of offsets, such as the one-page job which is table 4 in my 1981
    Bowditch, rests on an unstated, but perfectly acceptable, approximation.
    
    It does NOT show the deviation of a position circle from a rhumbline drawn
    at a tangent to that circle. It shows the deviation of a position circle
    from a GREAT CIRCLE drawn at a tangent to the circle. [If that were not the
    case, the offsets would depend on the observer's nearness to a pole, and
    would be asymmetrical, different when offsetting in a direction that
    approached the pole, compared with the opposite direction.] It's a pretty
    good approximation, except in very high latitudes, but it limits the
    validity of the offset table to small distances, over which there's really
    no distinction between the rhumbline and the great-circle, passing through
    the same point,  and pointing there toward the same azimuth.
    
    Because the offset table presumes a great-circle, we can apply it just the
    same to any point on Earth, and it may help Gary LaPook's conceptual
    difficulty if we take a simple case. On a globe, put yourself at my
    favourite hypothetical spot, in the Gulf of Guinea, at lat = 0, long = 0.
    Now consider celestial objects at varying declinations, all on the Greenwich
    meridian. The relevant great-circle is now the equator. Draw a small-circle
    around each of these objects, so that each circle just meets the equator at
    a tangent, at (0,0),  and see how much each circle deviates from the equator
    as you travel a few miles East or West from our origin point. That's exactly
    what the offset table does. Think of it in terms of a Mercator map of a
    small area centred on our origin point.
    
    Take one of these objects to be a true pole-star, with dec. = 90 deg. It
    would be just on the horizon, at altitude 0 deg, viewed from our origin.
    It's obvious, isn't it, that in that case our small-circle of equal altitude
    is also a great-circle, the circle of the Equator. There are no offsets at
    all. Our Mercator map of the relevant bit, near the Equator, of the
    zero-altitude circle round the true pole-star would be a dead straight-line,
    the Equator itself. No curvature. Infinitely-great radius of curvature.
    
    Now take another object, near the pole but not quite there at dec = 89 deg,
    say, still on the Greenwich meridian. Viewed from our origin point that will
    have an altitude of 1 deg. The1-deg altitude circle will touch the Equator
    at (0,0), but will be indistinguishable from it until you go a long way East
    or West. It will have an enormous radius of curvature on the Mercator map,
    many times greater than the Earth's radius. There's nothing incongruous in
    that fact.
    
    George.
    
    contact George Huxtable at george---.u-net.com
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    
    
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