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    Re: More Cocked Hats
    From: Bill Lionheart
    Date: 2020 Jul 19, 10:58 +0100

    Dear David,
    It is indeed sunny to day and I should get some vit D. Actually quite
    soon even go our to sea.   The power of mathematics is that it can
    prove or disprove something for an infinite number of cases at once.
    Numerical simulations can certainly provide a conjecture though.
    I have not finished reading the paper yet (well as you say the sun is
    shining!)  but the main point seemed to be that the rays may not
    cross, as these are position "half lines" from bearings on fixed
    stations or landmarks.
    It does seem a bit of a shame no one has sorted this out properly
    before, and it is good if this paper tidies it up and is the "last
    word".  It also seems a bit sad to me that the necessary mathematics
    was not done until the problem is almost obsolete (at least for the
    moment until some noew navigational technique uses it).
    Of course this flavour of the problem is very simple... the
    probability of being in the cocked hat is 0.25 over a population of
    cocked hats.  Think of "of all the cocked hats I drew I was only in a
    quarter of them".As soon as you have a specific round of bearings in
    mind the odds change. Robin Stuart's paper did a nice thing here
    integrating the Gaussian explicitly over triangles.
    Best wishes
    On Sun, 19 Jul 2020 at 09:16, David Pike  wrote:
    > NavList members with a mathematical bent might be interested in the
    > latest thoughts on cocked hats from mathematicians.  Bárány, Steiger
    > and Toledo https://arxiv.org/abs/2007.06838
    > Bill
    > I got as far as the bottom of the first page plus a quick scan of the 
    diagrams before my head started to hurt.  Thus proving I don’t have a 
    ‘mathematical bent’.  However, it seems to me that they’re trying to prove 
    the blindingly obvious.  After all, a three PL triangle converging to a 
    single point is generally considered a good fix, but the chances of being in 
    the centre of that tiny triangle much be close to zero.
    > I would have thought in these days of the digital computer that it ought to 
    be possible to set up a trial with millions of fixes with the PLs generated 
    randomly from each source, first with the sources for the same equipment with 
    the same bands of error and the dame distance away.  Then having seen what 
    you got, you could use sources different distances away or with different 
    band of error to see if you got a different result.  After that you could try 
    all sources on one side (i.e. 60 degree not 120 degree cut), perhaps with a 
    bit of systematic error fed in.  You might also like to look at the effects 
    of quadrantal error with respect to receiver heading vs source relative 
    bearings.  It would be interesting to know which of these made much 
    > Alternatively, build up the vitamin D by getting out and taking a few real fixes outside.  DaveP
    > View and reply to this message

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