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Dear Jim and George,



Thank you George for so kindly providing me with Jim's paper.

Thank you Jim and Congratulations for your Navigation paper. It is one of the 
Best and most Comprehensive I have seen dealing with a "graphs only" 
approach. I will now on keep it on file as a quite precious addition to my 
Naval Academy Navigation course and to the Book by M. Mortimer Rogoff you 
made a reference to. Longitude error quite high sensitivity to observation 
errors is also very clearly mentioned as it was in Mortimer Rogoff's Book.

I still need to dig out in further depth the (other) graphical method 
mentionned to me a few days ago with a referenced http link on this NavList 
site and which was published recently (this summer I think).

The interest of graphical methods is to avoid computations. They also show 
that LAN's can be much more extended than the restrictions I was earlier 
taught with limitations to zenit distances less than 20?.

It could be noticed that in the method you published, which is based on the 
differentiation of the well known/fundamental Height formula, such 
differentiation is first order. IN LAN's first order is generally well 
sufficient to deal with the majority of practical purposes, but it does have 
definite limitations if and when you tackle observations closer and closer to 
zenit.

"Computation only" methods using first order terms have a limitation because 
of the presence of significant UNEVEN ORDER TERMS in the higher order 
development of hights vs time when we deal with cases with either changing 
Body declination or N/S observer speed component.


For this very reason, and as a cherry on the cake, if we consider LAN methods 
based exclusively based on Computations alone (no graphs), Douglas Denny and 
George are BOTH right on what seems an apparently Contradictary and 
irreductible statement/argument between them, with Douglass stating "The 
Height curve IS skewed" and George stating with equal force "the Height curve 
is NOT skewed, only displaced left or right". If you consider cases 
sufficiently close from LAN time and to an accuracy of 0.1 arc minute for 
example, then the Height curve is NOT skewed as George points out, ... but if 
you deal with examples of heights sufficiently far away from Culmination 
time, then ODD terms will eventually come to play in some significant manner 
and the height curve when referenced to elapsed time IS definitely skewed.

Peace be with you all !   :-))

I would still guess that ALL first order methods - including the graphical 
ones such as yours Jim - might share the same practical use limitations for 
near zenit culminations with high N/S Obervers speeds and or significant 
changes in declination. It might be worth verifying this with examples. I 
should have between 10 and 15 past examples of LAN with cumlminations above 
85?. We could check results with your graphical method Jim.

Now, for most practical purposes, and assuming that you want your LAN's 
results as accurate as achievable (which we all aim at), I would offer the 
following comments from experience :

1) - Try getting an as wider "heights span" as possible in order to improve 
Longitude accuracy. "Heights span" here means the difference between 
culmination height and you lowest observed height in your set of height 
shots. A One arcminute observation error on observed heights has less 
relative effect on longitude accuracy if you have a full 8 to 10 degree 
height span than if you have only a 3 degree height span.

2) - Whenever achievable, prefer culminations closer to zenit. With low 
culminations heights, the "heights vs time" curve becomes more "flat" and it 
degrades accurate longitude determination. In addition, for low culmination 
heights, the "heights span" criteria adressed in 1) herabove requires longer 
observations time spans, therefore increasing both the constraint duration 
for a constant bottom speed, and the effect of speed knowledge uncertainties.

3 ) - Shoot as high LAN's as possible, then ? Not quite, because of at least 2 reasons :

3.1) - Sextant swinging will become more difficult and possibly more inaccurate close to zenit, and

3.2) in the cases with high N/S speeds and/or high declination speeds close to 
zenit, even if primary consequence is to move "sideways" (i.e advance or 
delay) the "heights vs time" curve, such curve may eventually become 
significantly skewed. Ignoring such skewness (right term ?) may result into 
appreciable errors mostly on Longitudes determination. Accuracies on 
Longitudes remain a "weak point" of LAN's methods since most often the 
observed span of azimuths necessarily remains quite narrow around the 
horizon.



Have a Great day, ALL of you !


Antoine


Antoine M. "Kermit" Cou?tte






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