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    Rhumb Line Calculations on an ellipsoid
    From: Johan Machtelinckx
    Date: 1998 Mar 17, 10:05 AM

    Hi gentlemen.
    
    After 6 months contract on board of a gas tanker, I'm back home and very
    happy to find again my computer and my favorite mailing list.
    I have always appreciated the quality of the exchanges on this list and
    follow the subjects under discussion with interest.
    
    Today, I would ask your opinion about some formulas that I want to use in a
    computer program I'm currently working on.
    
    At the Nautical College of Antwerp (Belgium), we were using (if my memory
    serves me well) only 2 kinds of formulas for rhumb line calculations.
    
    The first group of formulas belongs to the mean latitude method :
    Delta(L) = D * cos C
    Delta(G) = D * sin C / cos Lm = Delta(L) * tan C / cos Lm
    Where : C = true course of the ship
            D = running distance
            Delta(L) = L2 - L1
            Delta(G) = G2 - G1
    	 Lm = mean latitude = (L1 + L2) / 2
    These formulas are very easy to deal with. If the running distance is not
    too long or if the followed course don't involve a too big change in
    latitude, this is the easiest way to compute the results.
    
    The second group uses the meridional parts :
    Delta(L) = D * cos C
    Delta(G) = tan C * Delta(M)
    Where : Delta(M) = M2 - M1
    Mi can be found in the Norie's Nautical Tables (by the way, in my 1989
    edition, the meridional parts in these tables are still based on the Clarke
    ellipsoid of 1880. I know that Tradition is important in the maritime world
    but ...)
    Mi can also be computed by means of a formula given in the "Bowditch" :
    M = a * ln(10) * log(tan(45� + L/2)) - a * e^2 * sin L
        - a * e^4 / 3 * (sin L)^3 - a * e^6 / 5 * (sin L)^5
    Where : L = latitude
            a = 21600 / (2 * PI)
            e = eccentricity of the earth = SQRT( 2 * f - f^2 )
            f = earth's flattening coefficient = 1 / 293.46500000 (Clarke 1880)
                                               = 1 / 298.25722356 (WGS 84)
    	 (ln = natural logarithm using base e = 2.71...)
            (^ = power)
    These formulas are always accurate but are also more intricated. OK, no
    problem for a computer. In fact, I'm in trouble because I don't find any
    demonstration of the formula in the Bowditch. I understand that it is a
    mathematical series which approximates the shape of the ellipsoid of the
    earth but I would have more informations !!!
    
    To avoid the use of a formula I don't understand fully, I have tried to use
    the first group of above-mentioned formulas but I have modified them in
    order to improve their accuracy. I need your opinion about these
    modifications.
    
    The formula  Delta(G) = Delta(L) * tan C / cos Lm  is only an approximation.
    In fact, what we need precisely is the mean value of the function (1 / cos
    L) between L1 and L2.
    This is a problem for the integral calculus :
    the integral of (1 / cos L). dL between L1 and L2 is equal to :
    ln( (cos A + sin S) / (cos A - sin S) )
    where A = (L2 + L1) / 2
          S = (L2 - L1) / 2
    And the mean value is :
    ln( (cos A + sin S) / (cos A - sin S) )  /  (L2-L1)
    
    So, a rigourous expression of the first group of formulas is :
    Delta(L) = D * cos C
    Delta(G) = Delta(L) * tan C * (1 / (2 * S))
               * ln( (cos A + sin S) / (cos A - sin S) )
    Where : C = course of the ship
            D = running distance
            Delta(L) = L2 - L1
            Delta(G) = G2 - G1
            A = (L2 + L1) / 2
            S = (L2 - L1) / 2
    ! L1 must be different than L2 because if L1 = L2, S = 0 and the solution
    is undetermined. If L1 = L2, Delta(G) = D * sin C / cos (L1) = D / cos (L1)
    
    Ok, I was happy. I have found something for accurate rhumb line
    calculations not making use of a mysterious formula coming from the
    "Bowditch"...
    After a few tests, I was of course very disappointed.
    Unfortunately, my beautiful formula was only correct on a perfect sphere !!!
    
    No panic. I have a formula to convert geographic latitude (L) into
    geocentric latitude (LG) : LG = arctan( (b^2/a^2) * tan L)
    Where : b = equatorial semi-diameter
            a = polar semi-diameter
    (Formula coming from : S�rane Guy, Astronomie & ordinateur, Bordas, Paris,
    1987)
    Because b = a - f.a (f = earth's flattening coefficient), we can write :
    
    LG = arctan( (fp-1)/fp * tan L)
    L  = arctan( fp/(fp-1) * tan LG)
    Where : fp = 1 / f = 298.25722356 (WGS 84)
    
    So, if I need to compute (L1,G1) & C, D => (L2,G2) :
    
    L2 = L1 + D * cos C
    LG1 = arctan( (fp-1)/fp * tan L1)
    LG2 = arctan( (fp-1)/fp * tan L2)
    G2 = G1 + Delta(LG) * tan C * (1 / (2 * S))
              * ln( (cos A + sin S) / (cos A - sin S) )
    Where : A = (LG2 + LG1) / 2
            S = (LG2 - LG1) / 2
    
    and if I need to calculate (L1,G1) & (L2,G2) => C, D :
    
    LG1 = arctan( (fp-1)/fp * tan L1)
    LG2 = arctan( (fp-1)/fp * tan L2)
    If L1 = L2, C = 90
                D = Delta(G) * cos (LG1)
    Else
    C = arctan( Delta(G) / Delta(LG) * (2 * S)
                / ln( (cos A + sin S) / (cos A - sin S) ) )
    D = Delta(L) / cos C
    Where : A = (LG2 + LG1) / 2
            S = (LG2 - LG1) / 2
    
    These formulas seem to give the same results as the "Bowditch" formulas.
    But I'm still in trouble... because I tried to use the same methodology
    (i.e. converting geographic latitude into geocentric latitude) for great
    circle calculations on an ellipsoid (and not on a perfect sphere) and this
    time it is not working !
    Where am I wrong ? Is the principle of these formulas for rhumb line
    calculations on an ellipsoid correct or not ?
    Help !!!
    
    
    
    ----------------------------------------------------------------------------
    Johan Machtelinckx          Merchant Navy Officer         mjohan{at}linkline.be
    Av. Daniel Boon, 27 / B-1160 Brussels / Belgium   Voice/Fax : 32-2-672 06 10
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