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    Re: Advancing position circles: Huxtable vs. Zevering
    From: George Huxtable
    Date: 2006 Sep 25, 22:31 +0100

    In NavList 1331, Lars Bergman has got the correct solution to the
    problem I set in NavList 1314 (with a correction in 1323):
    
    ==========================
    
    "But what solutions do NavList members arrive at, I wonder, if they
    pick
    up that same challenge, using any method they think appropriate?"
    
    The problem was defined as:
    
    "1. An observer, at position P1, measures the altitude of a star S1,
    at(Dec1 = 0, GHA1 = 0), to be 30 degrees.
    2. Then he travels due North by 60 nautical miles (= 1 degree), to
    P2.
    3. From there, he observes another star S2 (then at Dec2 = N 1
    degree, GHA2 = W 45 degrees) to be at an altitude of 45 degrees.
    Where on Earth is he then?"
    
    ---
    Assume position P1 to be located at Lat, Long. Then P2 will be at
    Lat+1d, Long.
    
    We use the altitude formula sinAlt=sinLat*sinDec+cosLat*cosDec*cosLHA,
    where LHA=GHA+Long, GHA is counted westwards and longitude eastwards.
    
    Now, at P1
    sin30d=sinLat*sin0d+cosLat*cos0d*cos(0d+Long)
    which can be simplified to
    1/2=cosLat*cosLong
    
    At P2 we have
    sin45d=sin(Lat+1d)*sin1d+cos(Lat+1d)*cos1d*cos(45d+Long)
    
    Now let's make a guess: Long=-45d, i.e. 45 degrees westerly longitude.
    With this guess we find that
    sin45d=sin(Lat+1d)*sin1d+cos(Lat+1d)*cos1d=(cosLat-cos(Lat+2d)+cos(Lat+2
    d)+cosLat))/2=cosLat
    
    The last equation can be simplified to
    1/sqrt(2)=cosLat, and then Lat=+/-45d. This result satisfies the
    equation at P1 as well, and thus we are quite sure it is correct, but
    in
    order to be fully convinced we can verify the solution(s) by
    calculating
    the altitudes:
    
    sinAlt1=sin45d*sin0d+cos45d*cos0d*cos(0d-45d)=(cos45d)^2=1/2 =>
    Alt1=30d
    sinAlt1=sin(-45d)*sin0d+cos(-45d)*cos0d*cos(0d-45d)=(cos45d)^2=1/2 =>
    Alt1=30d
    sinAlt2=sin46d*sin1d+cos46d*cos1d*cos(45d-45d)=cos45d=1/sqrt(2) =>
    Alt2=45d
    sinAlt2=sin(-44d)*sin1d+cos(-44d)*cos1d*cos(45d-45d)=cos45d=1/sqrt(2)
    =>
    Alt2=45d
    
    The altitudes are correct, thus the guess was right and the two
    solutions are:
    
    P1 45N, 45W
    P2 46N, 45W
    or
    P1 45S, 45W
    P2 44S, 45W
    
    =======================
    
    Lars has it right. He solved the problem by "guessing and checking",
    and indeed, the geometry of the problem was so simple that the answer
    was nearly self-evident. I would have tried it the same way.
    
    Geoffrey Kolbe has also got the correct answer, after I had corrected
    my definition of the problem.
    
    The problem was designed to show up errors in the proposed method, to
    which I was objecting. That method envisaged taking a position circle,
    radius 60 degrees, around position S1. Then (and this is where the
    error arises) shifting that circle, by moving its centre Northward
    through 60 miles. Then looking for the intersections between that
    circle and a circle, radius 45 degrees, centred on star S2.
    
    That would give two "positions", at 45.998 N, 45.429 W, and 43.999 S,
    44.600W.
    
    Now, we can try to back-track, to see if those "positions" for P2 were
    correct-
    
    P1 must be 1 degree further South, in both cases, which puts the two
    "positions" for P1 at-
    
    44.998 N, 45.429 W, and 44.999S, 44.600W.
    
    And if we work out the altitude of a star S1, at 0N, 0W, from those
    "positions", they would be, respectively, 29.753 deg and 30.231
    degrees, both significantly different from the 30 degrees that was
    specified. So the proposed method of shifting the initial circle, by
    the "run" between observations, must be wrong.
    
    The error arises because when you take a position circle on a sphere,
    and shift every point on it by the same distance in the same direction
    (the "run"), the end result is NOT a circle with its centre displaced
    by the "run"; it's not even a circle at all.
    
    George.
    
    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.
    
    
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