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A good reference I've found that gives a detailed derivation of the formula
for Rhumb line sailing on the surface of a sphere and an ellipsoid is
"Geometry of Navigation" by Roy Williams. Section 3.2 is titled Navigating
Along a Rhumb Line on the Surface of a Sphere. The resulting formula is what
is currently used for Rhumb line sailing. Section 3.4 is titled Navigating
Along a Rhumb Line on the Surface of the Ellipsoid. Here, the eccentricity
of the reference ellipsoid is accounted for. The resulting formula is very
similar to the traditional Rhumb line formula except the Dlat is replaced by
Meridian Distance.

The formula for the calculation of the Meridional Distance, (Latitude Parts
as Williams calls it) is provided in section 2.10. It involves the
evaluation of an integral so I will not attempt to write the formula here.

Sam Chan
----- Original Message -----
From: "Chuck Taylor" 
To: 
Sent: Tuesday, September 17, 2002 7:32 PM
Subject: Re: Meridional Distances


> Peter,
>
> In an earlier response I mentioned an error in the formula for Meridional
> Parts as given in Bowditch 1995.  I am happy to report that it (a missing
> exponent) has been corrected in the online version at
> http://164.214.12.145/pubs/, and presumably in later printings.
>
> At 10:09 AM 9/18/2002 +1000,  Peter wrote:
>
> >My new (again, new to me) formula for rhumb line goes
> >
> >Arctan course = DLON/DMP
> >
> >Distance = DMD/cosCourse
> >
> >where:
> >DLON means the difference in longitude expressed in minutes of arc
> >DMP means the difference in Meridional Parts
> >DMD means the difference in Meridional Distances
>
> The standard formulas for Mercator Sailing, using your notation, are
>
> Course = arctan( DLON/DMP )
>
> Distance = DLAT / cos( Course)
>
> where DLAT means the difference in latitude in minutes of arc.
>
> These formulas are given in paragraph 2417 of Bowditch 1995 and paragraph
> 1007 of Bowditch 1981 Volume 2.
>
> >there is a separate, rather more complicated, formula for when the
> >course lies close to east or west.
>
> The use of a separate formula for when the course lies close to east or
> west is characteristic of Mid-Latitude Sailing.  Like Michael Wescott, I
> suspect that what you are looking at is some sort of hybrid method.  Would
> you mind giving us a reference that describes the method?  I have never
> heard of a table of Meridional Distances, only Meridional Parts.
>
> Chuck Taylor
> Everett, WA, USA
>
> C

   

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