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    Re: mid-longitude sailing
    From: George Huxtable
    Date: 2007 Jan 7, 18:58 -0000

    Oh, no, I've done it again!
    In my recent message in this thread, on which I have already sent one
    correction, there was another error. Geoffrey Kolbe has kindly pointed
    it out, diplomatically, in a backstage message. In the denominator of
    the "simple expression", I had transcribed the longitudes, in the
    denominator, as latitudes, which made no sense at all. The next
    paragraph, describing the action in words seems to have been correct.
    So, best I can do is to repeat the original message, with all
    corrections that I am now aware of. No guarantee is offered that
    there are no more to be found.
    Here goes-
    The Journal of Navigation has little to offer, these days, about
    "old-fashioned" navigation, but in the recent issue, there's a useful
    contribution about great-circle sailing, on a spherical Earth.
    Effectively, what they give is an easy way to get the latitude at the
    "mid-point" of a great circle course. Not the mid-point in distance,
    not the mid-point in latitude, but the latitude at the mid-point in
    longitude; that is, at the longitude halfway between the start
    longitude and the end longitude.
    Consider a great-circle journey, from (lat1, long1) to (lat2, long2).
    What's the latitude, at the mid-longitude point, where the long is
    (long1 + long2) / 2 ?
    It's given by the simple expression   tan lat3 = (tan (lat1) + tan
    (lat2)) / 2 cos ((long2-long1)/2)
    That is , you average the tangents of the latitudes at both ends,
    divide by the cos of half the longitude difference, that's the tan of
    the latitude you are after.
    Having the coordinates of that middle-point, you can then easily split
    each half further, and so on, using the same method, until your
    point-to point legs are short enough to treat each one as a
    I haven't come across that method before. It seems a simple way to
    split up a long ocean passage. It's exact, not an approximation, and
    it does seem to give the right amswers. Can anyone see snags?
    Authors are Wei-Kuo Tsieng and Hsuan-Shih Lee, from Taiwan, title is
    "Building the latitude equation of the mid-longitude", in Journal of
    Navigation, vol 60, No1, Jan 2007, pages 164 to i70.
    contact George Huxtable at george@huxtable.u-net.com
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    To post to this group, send email to NavList@fer3.com
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