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Gary LaPook wrote, about the probability of the true position of an
observer lying within a "cocked hat".

| Well George, I always was partial to "T's" and "A's."
|
| I follow you argument and can close my eyes and visualize what you
are
| saying.
|
| How's this for another description of your point.
|
| Draw three LOPs surrounding the position of the observer. If there
had
| been no random errors, the three LOPs would have plotted at a point
at
| the location of the observer. Now indicate on the LOPs with an arrow
| the azimuth of the body, lets say all pointing outward so this
| represents the "TTT" case. Now if you flip one of the LOPs and make
it
| an "A" it will now plot closer to the interesection of the two
| remaining "T" LOPs forming a smaller triangle over in that corner of
| the original "TTT" triangle and the position of the observer wil not
| be within the new small triangle. Then you can conceptionalize this
to
| the other possible combinations and come up with the 1 in 4 ratio.
|
| Do I understand you point?

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

From George.

Yes, we agree. Can we take it, then, that Gary now accepts that 1 in 4
figure?. In that case, the matter's settled.

Except that a minor point remains. If the three azimuths can be
enclosed within a 180-degree arc (and we make the middle-radius of
that fan of arcs corresponds to the middle letter of our three-letter
code) then the AAA and TTT conditions will not enclose the true
position. Instead, the ATA and TAT conditions will do so. Still 1 out
of 4, just the same.

Separating those two scenarios, of azimuths that embrace more than 180
degrees and those that embrace less, is the dividing case of azimuths
that are just 180 degrees apart. As Fred points out, in that case the
triangle degenerates to a straight line. It isn't a triangle at all,
and there's little point in considering that special case in any
detail, though I'm a bit puzzled how Fred manages to make 7
possibilities out of it.

Robert Eno, in Navlist 2343, points to the relevance of other aspects
of Gary's posting, and I agree with all he says. Of course, when you
have to plot a point to represent your position, the centre of the
triangle is as good as any. What's important, though, is that the
navigator should have some notion of the uncertainties involved when
he plots that point.

In Navlist 2330, I picked on Gary's disagreement with the 1 in 4
statistics, because that was all I found to question. I think that the
1 in 4 argument is worth getting right, because it's conclusions are
so simple, and it conflicts so much with what many navigators assume,
or have been taught, and because the consequences of misundertanding
the matter could be so serious.

Fred contributed- "A small triangle suggests that sigma is small for
each line, and a large
triangle that sigma is large". I agree that a large triangle implies
that there's a lot of scatter, but the contrary isn't true of a small
triangle; it depends a bit on what Fred means by "suggests".

Indeed, a single cocked-hat gives little clue to the possible scatter
in the true position. Just by chance, it might well happen to be a
tiny triangle, but that doesn't allow a navigator to deduce that the
resulting position is a precise one. It's only after assessing the
sizes of a large number of such cocked hats that a navigator could
come to such a conclusion. Not a practical proposition really; in its
place he has to fall back on common sense and experience, in
estimating what his observation errors are likely to amount to.

Robert also raises the question of how this matter is presently taught
in navigation classes, and I would like to take that a bit further,
and ask what Navlist members have been taught, and what they believed
to be the truth, before this matter ever came up for discussion on
this list (or its predecessor).

When I was taught navigation in an evening class, over 40 years ago,
the matter was simple. The triangle, whether between compass bearings
of landmarks or between astronomical position lines, gave you your
position. You must be somewhere within it, if you didn't know exactly
where. To be absolutely safe (we were told), if there was a nearby
danger, you had to assume you were in that bit of the triangle that
lay closest to the danger. At the time, I accepted that nonsense. Not
that I was less argumentative, or more docile, in those days, but I
knew no better.

What enlightened me was an Open University programme on TV, not on
navigation, but on statistics, in which the triangle between magnetic
bearings was used as an example of statistical methods. The
one-in-four conclusion was a real eye-opener.

Robert asks, of navigation teachers- "Do you teach them to take the
fix at the intersection of the LOPs and/or centre of the cocked hat or
do you hose them down with cold statistical sea water just as they are
beginning to grasp the fundamentals?". That's a somewhat tendentious
way of putting it. It's not asking a lot, of teachers or students, to
put across the simple message as follows. "Plot a point, at the centre
of the triangle, but be aware that the true position could lie well
outside that triangle."

George.

contact George Huxtable at george@huxtable.u-net.com
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.




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