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Right. What I have in mind is to prove a result that says "if you are
within some specific distance of the correct position th iteration
converges", and I think it is pretty clear it can fail if you are far
enough away. As I said it is pretty much Newton's method on a sphere,
and there are plenty of general results about convergence of that.

Bill

On Thu, 23 Jan 2020 at 03:03, Frank Reed  wrote:
>
> Gary LaPook, you wrote:
> "I can't believe that this discussion has been going on so long. Just pick 
any spot on the earth as your first AP (just throw a dart at a map!), develop 
a fix and then use that as the second AP. Three iterations and, voila!, you 
have a normally accurate fix."
>
> I wouldn't go quite that far. A dart-at-the-map approach often fails on the 
first pass and sometimes fails spectacularly -- you don't get a second 
iteration at all. For a specific set of choices that are guaranteed to fail, 
if your first choice for an AP happens to lie on the great circle that passes 
through the subStar points of both bodies (a place where both bodies are on 
the same or opposite azimuths), then your lines of position from that AP are 
necessarily parallel, and they never cross. Note that this has nothing to do 
with the azimuths of the bodies as observed. Your azimuths are determined 
from your AP. There are other arbitrary AP locations that will fail because 
the lines of position cross somewhere "off the map", for example, at latitude 
110° N. And that's a symptom of the main the problem with this iteration 
procedure. It depends on a flat chart.
>
> The alternative that I have already described leapfrogs this issue by going 
straight to a "spheroid". Get an orange or a beach ball or an inflatable 
globe and draw the circles of position with radii equal to zenith distances 
measured off from the subStar positions of each body (latitude of the subStar 
point is Declination, longitude is GHA). Where the circles cross provides a 
good AP for a more detailed analysis. The intercepts from this AP will 
typically be less than a couple of degrees. They're safe for iteration.
>
> And just to repeat, there is an analytic approach that corresponds to this 
process of drawing on a sphere. We can solve directly for the fix from a pair 
of altitudes. If you can remember the equations, you don't need an orange. :)
>
> And to repeat some more, in any real-world scenario, even historically, you 
would never have more than a couple of degrees uncertainty in your position. 
With moderate uncertainty like that, you could iterate from your best guess 
without any problem. I'm just trying to make clear that I'm not objecting to 
the idea of iterating per se, but rather the notion of iterating from any 
arbitrary starting point picked at random on the globe. This, in general, 
won't work.
>
> Frank Reed
>
> 

   

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