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    Re: Advancing position circles: Huxtable vs. Zevering
    From: Lars Bergman
    Date: 2006 Sep 25, 13:50 +0200

    George Huxtable wrote 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 
    59N 18E
      
    
    
    
    
    
    
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