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    Re: Which Cocked Hat?
    From: Bill Lionheart
    Date: 2019 Aug 2, 23:51 +0100

    I look at the cocked hat from a different direction. When I am
    teaching scientists about inferring two variables (such as a position)
    from three measurements represented by straight lines (with some
    uncertainty) I used the cocked hat in navigation as a simple example,
    and quote the Admiralty Manual of Navigation's suggestion of taking
    the corner nearest danger as a  practical  example of NOT assuming the
    least squares solution.
    Frank's point that the take home message from the symmedian point is
    that it is nearer the shortest side than the centroid is a good rule
    of thumb of course where no corner is more dangerous than another.
    Uncertainty quantification, and estimation from data with errors, are
    really important topics in experimental science and engineering.
    Marine and aeronautical navigation are  examples where mathematical
    problems have to be solved to make safety critical decisions in often
    less than ideal conditions. I think a lot can be learnt from the
    practice that has evolved in navigation.
    If you use software to plot the lines of position, no reason why it
    shouldn't indicate some probability contours as a guide to the
    reliability of the fix, and the maximum likelihood estimate as the
    fix. The person that writes the software needs to know the theory not
    the user.
    if you really want to construct an elliptical probability contour by
    hand I can show you how to do it. Maybe over a beer  at Frank's
    "Lunars" course weekend after this one! No promises I will ware an
    n-corn hat while doing it.
    On Fri, 2 Aug 2019 at 19:04, Frank Reed  wrote:
    > David C, you wrote:
    > "Several references I have found, the earliest from the Admiralty Manual of 
    Navigation 1922, use a phrase similar to [a small triangle or "cocked hat"]. 
    To me what is important is that cocked hat is in quotes which suggests that 
    in the late 19th and early 20th century it was a slang expression."
    > Yes, I would say that it clearly is a slang-y expression. And, yes, I agree 
    that they're indicating this, and maybe some formal misgivings about this 
    term, by putting it in quotes. It's a "funny" metaphor, and that's all. I 
    don't think it's relevant what sort of hat was in common use in naval 
    uniforms at the time. It seems quite clear that the intent was to provide a 
    memorable, pithy name for that little triangle, and the three corner hat, 
    which was familiar to most everyone even if it was no longer fashionable, 
    implied by "cocked hat" made a certain community of people laugh, so the name 
    > In your follow-up message, you wrote:
    > "I have looked at the index of Bowditch 1958 and also that in Dutton's 
    Navigation and Nautical Astronomy 1951 and cannot find the phrase cocked hat. 
    Because of the way the indexes are formatted I may have overlooked the 
    > This sort of thing is greatly facilitated by technology. Searching 
    mid-century editions of Bowditch on HathiTrust, I found no references to 
    cocked hats. This supports the idea that the expression originated in British 
    usage a few decades earlier and had not yet crossed the Atlantic --at least 
    not to the desks of the editors of Bowditch at that time. We can similarly 
    search the latest (2017) edition of Bowditch directly via the pdf files and 
    there are now numerous references to "cocked hats" and always in quotes. It's 
    even in the glossary: "cocked hat. Error triangle formed by lines of position 
    which do not cross at a common point."
    > You added:
    > "In the gps era navigation enthusiasts  began to  analyse cocked hats 
    [/duck] ad nauseum - reference the unintelligible (to me) discussions about 
    where in the cocked hat a ship is."
    > I agree it's been ad nauseam and frequently pointless. If you have found the 
    discussions unintelligible it's because they are almost always consumed by 
    ramblings from confused contributors who seem to think there's something 
    complicated in all of this merely because there's math that underlies it (one 
    can understand the result and ignore the derivation just as in so many other 
    cases in navigation). Really the only thing you need to remember in practice 
    is that the fix should not be marked near the center of the triangle, but 
    rather it should be placed near the short side of the triangle (if any). And 
    if you see a dozen cases properly calculated, you'll be able to do this on 
    paper without calculation.
    > The key case:
    > Suppose you have a triangle from sights taken on bearings (true azimuths) of 
    0°, 270°, and 178°. Get out a piece of paper and draw the corresponding lines 
    of position. Assuming some random errors in the sights, you'll have a long, 
    narrow triangle with a short side on the west and two very long sides on the 
    north and south. Next, if asked to indicate the fix many people without 
    thinking on it much would select the center of the triangle. But that's wrong 
    in every sense. The fix is very close to the short side. And this isn't some 
    fancy mathematical argument. It's obvious if you take this case and replace 
    the third azimuth with 180°. Draw it out, and you'll see why. This is 99% of 
    the point of all those discussions. If you can see this, the rest is not 
    really navigationally significant. All the rest is just mathematics mixed 
    with confusion (some people bring math to the table, some bring intractable 
    > Frank Reed
    > View and reply to this message

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