NavList:

David Pike, you wrote
"This bears out what I was saying.  In chart, protractor, divider, and periscopic sextant navigation using Betelgeuse, Bellatrix, and Rigel to produce a fix is not the best use of all the stars in the sky, and this would be immediately obvious when you tried to plot it.  Putting the same sights into a computer, beware GIGO.  You’ll always get a solution, but how much reliance can be put on it.? " 

In the early days of celestial navigation software and apps, authors tended to aim for products that merely replaced the human. The final result of the analysis was a simple plot of celestial LOPs, just as it was in manual work. Better more recent apps (Andres has a bunch of these!) produce an error ellipse, and that's all you really want. The individual LOPs should dissolve into the error ellipse, much the same way that we don't care about the individual timing delays from GPS satellites and only want the resulting fix and its uncertainty ellipse (ellipsoid in three dimensions). The error ellipse is the final product from a proper analysis (or a well-designed app). The individual celestial lines of position fade away...

You added:
"On the 2D 3D idea, I was wondering if the clever people at the USN laboratories could modify the mathematics within a GNSS receiver to give a solution for submarine celestial if viewing was limited to a small portion of the sky as often happens on the surface."

This has all been worked out by many clever people [some of whom are even USN :-) ]. This reminds me of something that I've been meaning to say about the slightly daft article (published several years ago) that started this discussion. It got significant social media attention when it was published, and lots of folks looked at it said, "wow, it's from the USNI! That's, like, the US Navy or something!" And yes, "like... or something". The USNI is a private publishing group with varying degrees of loose association with the US Navy. Among their many publications, maybe the most famous was "The Hunt for Red October". They publish a great many articles on a variety of topics, and some of the material is competent. Some of the material is at best "interesting". Some of it is even fiction! This particular "submarine celestial" article is founded on naivete and some ignorance of basic celestial navigation (or at least its recent history). The article does not, contrary to the impressions of many readers, reflect the "latest thinking" on celestial navigation in the US Navy. Rather, it's a young navigator --a kid-- describing his experiments and ideas. 

Now what about these low-angle azimuth cuts for fixes?

In your professional service as a navigator, David, you say that you were given a rule: no less than 30° difference in azimuth. But this is no absolute. It was strictly a rule of "navigational culture" --a good working rule for you and your comrades.

But you can get a fix with two stars separated by 10°. The result is an error ellipse with a very long axis aligned on the direction halfway between the stars and a short axis in the perpendicular direction. And we can calculate the size of the ellipse. The short axis (under some assumptions) might turn out to be 1.5 nautical miles. The long axis, for a separation of 10°, would be about 11.4 times longer, or 17 nautical miles. Useless? No. Low quality? Yes. 

Now suppose we add more data. With the same range of 10° in azimuth, we multiply the sights by sixteen. That will reduce non-systematic error and shrink the error ellipse by a factor of four (square root of N sights) in both directions. So now you have a fix with 0.4 n.m. uncertainty in one direction and 4.3 n.m. uncertainty in the other direction. That's certainly a useful position despite the narrow range in azimuths (and yes, totally impracticl for an air navigator, but never mind that for the moment!).

Suppose instead of two bodies separated by 10° azimuth, imagine shooting the Sun sixteen times in a row (maybe once every 2-4 minutes) as it changes its azimuth by 10°. By traditional evaluation just plotting LOPs, that cut is too low. Useless. But if we work up the error ellipse, its size has been reduced by that same factor of four as in the prior case, and once again, it's a useful fix.

We need to understand the math, the underlying geometry, that benefits from a wide cut angle. But we also need to understand that rules limiting cut angles are strictly cultural. Every set of sights, even with seemingly extreme cut angles, produces an error ellipse as its final product. 

Frank Reed