NavList:

Any pair of LOPs with estimated errors on them yields an error ellipse. That is a complete solution to the navigation problem. The error ellipse for two perpendicular LOPs (if they are believed to have the same expected error) is a circle. So when I cross those two perpendicular Sun LOPs, I can do my calculations, get that error circle, and then erase the LOPs from the chart (figuratively, not literally). The fact that the LOPs were oriented along the cardinal directions is not relevant. Any pair of perpendicular directions would have yielded exactly the same error ellipse. Therefore when I bring a third LOP into the puzzle, its orientation cannot matter to the solution. So we can solve the problem in the graphically simple case where the third LOP is parallel to one of the original ones, and know the solution immediately for any other orientation because we know that the two LOP case yields an error "ellipse" which is exactly circular.

I'm not suggesting that we should throw out the usual methods for plotting LOPs and getting the fix from the chart. In fact, for my next trick, I'm going to go in the opposite direction. Suppose I have an error ellipse from another source, e.g. dead reckoning. Since any pair of LOPs is exactly equivalent to an error ellipse --they contain the same information-- I can turn this around and replace any error ellipse, even if it doesn't originate from lines of position, with a pair of equivalent LOPs. There's a catch unfortunately. The LOPs will have different "weights" depending on the size of the initial error ellipse. But I'll offer an example anyway. Suppose I have a DR position that I believe to be accurate within about six nautical miles in both directions. It's a circular error ellipse, a thumbprint on the chart. Then I take a Sun sight which I believe to be accurate within about two nautical miles and when I plot it, it falls three miles from the center of my DR error ellipse. What is the resulting best estimate of the fix? Better yet, what is the new and improved error ellipse?

-FER

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