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## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**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 ---------------------------------------------------------------------------- =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=--=-=-= TO UNSUBSCRIBE, send this message to majordomo{at}ronin.com: =-=-= unsubscribe navigation =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=--=-