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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.
    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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