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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@linkline.be
Av. Daniel Boon, 27 / B-1160 Brussels / Belgium   Voice/Fax : 32-2-672 06 10
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