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Subject: Still on LOP's
From: George Huxtable (george@XXX.XXX)
Date: Sun May 05 2002 - 06:25:58 EDT
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.
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george@XXX.XXX
George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
Tel. 01865 820222 or (int.) +44 1865 820222.
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