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I am glad that Peter Fogg enjoyed his holiday.

I suspect he is gradually coming round to the majority way of thinking
about the cocked-hat, but desperately trying to avoid admitting it, even to
himself. He is doing this by redefining the quantities involved. We can't
allow him to get away with it.

To start with, he redefines the Actual Position of an observer to be not a
position, but an area, which he says-

>is at least as big as our boat,
>and could be considered to be as big as our circle of view around the
>boat, a circle with a radius of a few miles.

Well, of course, if it can be defined in such a way, it's easy to arrange
that it can't fail to coincide with a Sumner line or a bearing line. But if
so, what word would Peter use to describe the ACTUAL actual position of the
observer? That is a POINT on the surface of the Earth that is occupied by
the observer, even if the observer might not know exactly where that point
is. The actual position is what the observer will do his best to determine.
Once, that would have been a difficult thing to do, to any great accuracy.
Now, any observer with GPS can measure his position to a few metres, good
enough for our present purposes. But the actual position is where he
actually is, and is quite independent of his means of knowing where he is.

I'm not saying that the concept is wrong of drawing a big smudge on the
chart and saying "we're somewhere in that smudge". That is indeed Good
Navigation. Where you are in that smudge is your Actual Position; you can't
call the smudge your Actual Position.

Next Peter redefines a Line Of Position (LOP) to be a band of position, a
smudged line, wide enough to accommodate any errors in its measurement. And
of course if he does that, then his actual position (I mean the REAL actual
position, not Peter's redefined area) has to be within that smudged line. A
navigator must, of course, be aware that any measured bearing or position
line will have an error-band ASSOCIATED with it, and should assess the
likely width of that band. But to equate the single measurement with the
error-band creates only confusion.

And so, to nobody's surprise, Peter Fogg can show that as he he has
redefined them, the "Actual Position" has to be within the triangle formed
by his three "LOPs", with 100% probability. But this is in the world of
Alice through the Looking Glass, in which a word can mean whatever he
chooses it to mean.

So I ask Peter to reconsider the problem of the cocked hat, using the
definitions that the rest of us use. If an observer measures bearings to
three known landmarks (where the probability of an error to the left is as
great as that of an error to the right), and plots those bearings as thin
lines on a chart, what does he now consider is the probability that the
triangle so formed will embrace his actual position, (a POINT on the
surface of the Earth)? Does he still maintain that it is 100%? Or accept
25%? Or what?

Can I remind Peter of his original words which started this whole thing off
and which appeared to me to show some misunderstanding? They related to
circles of equal altitude from celestial bodies, analogous to the LOP
question. He said-

>Three circles lead to a single (?) triangle, traditionally known as a 'cocked
>hat', the centre of which is our Fix position. An important exception to the
>centre (as in 'the doctrine of least squares' - more info available on request)
>being the Fix is where there is a known danger - lee shore, reef, whatever - in
>which case the Fix position becomes the closest point to the danger. This
>will be
>along one of the LOPs, so could well be the real position.-

I took that to imply that he thought (then) that it was impossible for the
"real position" to be outside the cocked hat. Does he think that now?

==================

I should add that Chuck Taylor's recent contribution seems to be a much
more useful approach. However, it seems a bit of an over-simplification
(perhaps made in the interests of brevity) to take the standard deviation
of the bearing as being one degree. This one-size-fits-all approach ignores
the effect on the accuracy of the roughness of the sea, the size of vessel,
the precision of the instrument used, and the skill, and degree of
seasickness, of the navigator. A standard deviation of one degree is fine
as a value taken for the purposes of Chuck's argument, but in practice some
value should be taken that's appropriate to the circumstances of the
moment.


George Huxtable

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george@huxtable.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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