NavList:

Gary,

       Thank you for your assessment. If you you aren't familiar with them already I would suggest that you take a look at some of references (Huxtable 2006, Zevering 2006) to understand the back story to this. George thoroughly debunked Zevering’s paper but the correct solution to what he was trying to do was not provided until now.

You wrote: “the first LOP is a line”

No it’s not. On the sphere it’s a small circle and an advanced LoP isn’t even that (see Figure 1) which is the key reason that approaches proposed by Zevering (2006) and Metcalf (1991) are fatally flawed. On the ellipsoid who knows what you'd call it?

You also wrote: “and infinite set of points, that are advanced until one of those points intersects with the second LOP and that is exactly what the diagram shows if the entire LOP had been advanced the same intersecting point would have been found.”

Of course it has to be true that any viable approach that works on the sphere or ellipsoid absolutely has to work in the approximation where those surfaces are represented by a plane and circles of position are approximated by lines so of course it gives the same result as the graphical method. But it doesn’t work other way around and that, as I see it, is the crux of the matter. People have been fixated on the trying to generalize the operation of advancing an LoP from the plane to the case of the sphere and doing it wrong. As it turns out, that whole exercise was unnecessary and unhelpful.

You wrote: “If there is any greater accuracy to the complex mathematical process it is swamped by the normal uncertainty of the underlying celestial observations”.

Maybe. If you can live on the plane then that’s fine but if you are going to the go sphere then you’d better do it right and that’s what was done here. In the case of the sphere things just involve some meridional parts and trig functions and aren’t any more complicated than what navigators are used to. The ellipsoid…well…I do say “Practical limitations involved in following a precise rhumb line track mean that treating the running fix on the ellipsoid is likely to meet or exceed all real world requirements for accuracy.” One purpose here was to show that if the problem is approached in the right way then generalizations that would have seemed completely intractable in the past become feasible and are only modestly more complicated than on the sphere.

Robin Stuart