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    Re: That darned old cocked hat
    From: John Karl
    Date: 2010 Dec 11, 08:52 -0800

    OK here’s one last go at it (maybe call it a summary). As I said in post 14766, George and I are discussing too very different topics -- both quite irrelevant for practical navigation.

    George’s irrelevance
    A. He measures triangles from a known location.

    B. And then he counts the number of times (i.e., the percent of time) that the known location is outside of the triangle. (So yes George, you do need to know the fix location in order to do this counting.)

    C. This is not navigation. It’s not determining anything about George’s location – we already know that. It’s determining information about the random distribution of the triangles.

    D. George measures something. He doesn’t estimate anything – his position is already known. So it’s irrelevant to navigation.

    Karl’s irrelevance
    A. Three LOPs are given. It’s known that their errors are distributed normally perpendicular to their lines, with no information available along their individual lines.

    B. By multiplying these three linear distributions, we get the probability/area P(x,y) that a point is within that area. We also easily observe that the MPP (maximum probability position) minimizes the sum of the squared distances to the sides of the given triangle. We see that this conclusion is independent of the standard deviation. H. Prinz pointed out that this was known to Villarceau in 1877. (Darn it.)

    C. P(x,y) is everywhere rather small, even the value of the MP is small, typically around 2% per area. And obviously it can never exceed one (i.e., 100%). Therefore a small cocked hat has a small probability that the fix is inside. (No need to even plot P(x,y) for this conclusion.) And a very large cocked had has a large probability that the fix is inside. (I did plot P(x,y) for this conclusion, but had no need to consider numerical integration.)

    D. These conclusions do not conflict with George’s measurements of triangle distributions around a known point – that’s a completely different question (and not one of navigation).

    E. Since this MPP gives the best fix, and no one has claimed an optimum fix location according to any other criterion, I don’t understand why all the other discussions, such as in the “Simple 3-Body Fix Puzzle.” Then there’s outright incorrect statements, such as in Dutton and Bowditch.

    F. The significance of the MPP is only relevant on a statistical basis. Its main use is to stimulate NavList discussions. In real world navigation at sea, make darn sure you know where you are not, not where you most probably are. In short, George’s topic of triangle distributions is irrelevant for navigation, while this MPP discussion is irrelevant for practical navigation (unless you’re navigating in a casino).

    JK

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