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    From: George Huxtable
    Date: 2002 May 5, 11:25 +0100

    I have some progress to report.
    
    I have implemented a simulation of the "cocked hat" problem on my old Mac,
    using True Basic. This works (to my own satisfaction, anyway) and has
    confirmed that in 500 rounds of simulated bearings, the cocked hat embraced
    the true position 133 times, and failed to do so 367 times. Close enough to
    1 in 4, statistically speaking.
    
    What does the program do? It puts the true position at the centre of the
    screen and surround it with 3 landmarks, which can be put in a fixed
    position anywhere on the screen. The furthest landmark is taken to be at a
    true bearing of zero degrees and at a unit distance. The true bearing from
    the central true position to each landmark is calculated. Then a line is
    drawn from each landmark in the reverse direction towards the centre, just
    as when plotting a 3-point fix, except that the angle is adjusted by a
    simulated Gaussian error. The amount of that random error for each bearing
    is shown numerically on the screen, but is not used in the analysis. The
    three lines form a triangle near the centre of the screen, a different
    shape and size each time. Some of them embrace the centre, others don't.
    It's very convincing when you watch it at work.
    
    I have concentrated on the simplest case of the three landmarks being on an
    equilateral triangle centred on the true position. Everything seems to work
    OK when other configurations are tried, but I haven't done any serious work
    on them. The scatter of each bearing has been chosen to be � 5 deg
    (standard deviation) in order to provide nice big triangles. There would be
    no difficulty in allowing for different scatters from the three landmarks.
    
    Whether the triangle embraces the centre is determined entirely by human
    observation. The observer looks at the screen and clicks the approriate
    button according to whether he votes that it should into the "ins" or the
    "outs". There's also a "maybes" button but it isn't needed: for resolving
    doubtful cases the central part of the screen can be magnified by X10,
    which does the trick every time. On clicking a button the total of the ins
    or outs is tallied by one, and the next case immediately comes up on
    screen. It's easy to process one case per second.
    
    Of course, it would be easy to determine whether the true position was in
    or out of the fix mathematically, but to do this we would have to assume
    quite a lot of what we are trying to prove. So I think such a pictorial
    method might be more convincing to the unbelievers.
    
    I have learned quite a few new things from this simulation. For example,
    someone mentioed on the list that he was consistently getting points into
    the triangle when the displacement of the bearings was LRL (or some such),
    at which I demurred and said that the only combinations which would do that
    were LLL and RRR. Well, he was right and I was wrong in that matter: sorry
    about that. What I said was correct in the case of three landmarks
    surrounding the true position, spaced at 120 degree intervals. And it's
    correct, I think, for all configurations where the true position is within
    the triangle with a landmark at each vertex. But the error-combinations
    will differ if you choose another configuration, such as landmarks at 0
    deg, 60 deg, and 120 deg. I hope this has cleared that matter up.
    
    Even so, this doesn't affect the 1 in 4 probability of being within the
    cocked hat. I will do some more work on that when I can, but as I am off on
    holiday from the 6th to the 17th there's not a lot of time. (To the Azores,
    since you ask, but unfortunately not by sea...)
    
    If anyone has a Mac that runs True Basic I would be pleased to send a copy
    of my program, which still needs slight tinkering but is in working order.
    
    Bill Noyce made a perceptive contribution a few days ago, about systematic
    errors in celestial observations that can increase the probability of the
    true position lying within the cocked hat. This happens because those
    errors expand the cocked hats to surround the true position.
    
    What I would like to point out is that the same effect can (and will) occur
    with land bearings. If you have a significant compass error (because
    variation and/or deviation haven't been allowed for properly) this will
    displace all your bearings by the same amount in the same direction, all
    clockwise or all anticlockwise. If your landmarks are 120 degrees apart,
    that must expand the average size of your cocked hat. If this systematic
    compass error is greater than the scatter in the bearings, then EVERY
    bearing would be in error in the same direction, and EVERY cocked hat must
    then contain the true position. That would make Peter Fogg happy, but the
    price to pay is that every cocked hat is correspondingly larger. Of course,
    that doesn't conflict with my 1 on 4 contention, because the explicit
    assumption there is that clockwise and anticlockwise errors must be equal
    in number: not true when there's a systematic compass error.
    
    George Huxtable.
    
    
    ------------------------------
    
    george---.u-net.com
    George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    Tel. 01865 820222 or (int.) +44 1865 820222.
    ------------------------------
    
    
    

       
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