# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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Re: No DR position. How can you get an accurate celestial fix?
From: Bill Lionheart
Date: 2020 Jan 24, 07:45 +0000

```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
>
> View and reply to this message
```
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