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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

Bill

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 
difference.
>
> Alternatively, build up the vitamin D by getting out and taking a few real fixes outside.  DaveP
>
> 

   

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