NavList:

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