 # NavList:

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

Message:αβγ
Message:abc
 Add Images & Files Posting Code: Name: Email:
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.

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

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
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:     =-=-=
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=--=-
``` Browse Files

Drop Files ### NavList ### What is NavList? ### Join NavList

 Name: (please, no nicknames or handles) Email:
 Do you want to receive all group messages by email? Yes No
You can also join by posting. Your first on-topic post automatically makes you a member. ### Posting Code

Enter the email address associated with your NavList messages. Your posting code will be emailed to you immediately.
 Email: ### Email Settings

 Posting Code: ### Custom Index

 Subject: Author: Start date: (yyyymm dd) End date: (yyyymm dd)