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My responses are interspersed within what Peter Fogg said -

>Some time ago what turned out to be the biggest controversy of my experience
>of the Nav. List erupted, with the claim by George Huxtable that the fix
>position was 3 times more likely to lie outside 3 intersecting position
>lines (Lines of Position, or LOPs in Americanese) than it was to lie within
>them. I was mightily impressed by the efforts different people put into this
>discussion; diagrams drawn, computer programs written, websites set up,
>people 'reluctantly converted'. All good stuff.

Well, the Great Cocked-hat Controversy was fun in its time, though it had
started to pall toward the end. If Peter Fogg insists on blowing on the
embers, he does it at his own risk.

>To recap briefly (and hopefully accurately enough): The claim was that the
>fix position cannot lie along any LOP since one is a line, by definition of
>infinitely thin width, and the other a point without area. Since an
>infinitely fine point cannot occupy the same (non-existent) space as an
>infinitely thin line the fix position must lie on one side or the other,

the argument is about how the TRUE position, not the fix position, relates
to the LOP, but otherwise agreed so far.

>thus a 50% chance of either.

Only 50% if the error in determining the line is entirely random and has no
systematic component. If there's a big-enough systematic error, enough to
overwhelm random error, then the LOP will always pass to one side of the
true position, never the other side.
>
>When 2 LOPs intersect this becomes a 25% chance of it lying in any of the 4
>quadrants.

True for four QUADRANTS, but is it this relevant? The two lines crossing at
a point divide the plane they lie in into four triangles which snug
together at their intersection. Only if the lines are at right-angles are
the triangles quadrants. Otherwise they are two wide triangles and two
narrow ones, and the chances of the true position being in any one is the
angle at its apex divided by 360 degrees, if only random errors exist.

With 3 LOPs there are 3 of these intersections, thus a 25% chance
>of the fix lying inside the triangle and a 75% chance of it lying outside.

That wasn't the basis of the argument, though that was its conclusion.

>I was intrigued by this 3 to 1 claim and followed the discussion with great
>interest and not a little scepticism. After a while I stated that it is an
>irrelevance.

The proposition we were argung about is a very simple one and says nothing
about a "fix". It's this- "With errors in LOPs that are entirely random,
only one time in four will a triangle of three LOPs embrace the true
position." Is Peter arguing here that this is irrelevant (which is a matter
for him); or is he arguing that it's WRONG, in which case he will have to
convince us.

>My reasoning was that if you take any triangle of LOPs or
>'cocket hat' there is only one fix position that can be found by the
>doctrine of least squares.

Yes, that's the MOST LIKELY position. Being an infinitely small point, it's
actually impossible that the observer will be precisely there, but never
mind. If you draw a small circle around that point, it's somewhat more
likely that he will be in that circle than in any similar small circle
drawn in the vicinity.

If you move these LOPs outwards to encompass any
>possibily of the fix position lying outside them they must be moved in
>proportion. You end up with a larger triangle and an identical fix position
>at its centre. Once the triangle is bigger than the earth the possibility of
>it encompassing the fix is 100% but the fix position hasn't changed.

Well, whatever position you started from, that would become true, but I
don't see its relevance.
>
>What would be useful is a method of calculating where, if not in the
>triangle, the fix might lie. And even better, a method of quantifying the
>error; putting a number to it; establishing a better fix position than the
>centre of the triangle.

Again, exactly what is Peter referring to when he says "the fix"? What he's
asking for might be useful, but is impossible with random errors. Because
they are random, you don't know what they are. If you did, you could allow
for them, and then they wouldn't be random errors any more. They would
become corrections.
>
>In what seems so far a little noticed posting by George Bennett on the 24th
>of May called 'Position Line Plots' a practical method for identifying and
>quantifying systematic error has been proposed:

Yes, I agree. I was intrigued by references to references to his book
"Field Astronomy for Surveyors" and have just tracked-down and obtained a
secondhand copy. It seems to be rare in this hemisphere.

>'There is a technique ..... that allows the navigator to make a simple
>analysis and assessment of his work'
>
>I am still working my way through this posting but the implications for the
>LOP controversy already seem profound.
>
>This discussion will be easier with diagrams so please draw your own as we
>go along.
>
>Lets start with those 2 intersecting LOPs. If there is a systematic error
>then it will be either towards the direction of the 2 azimuths or away.

Sorry, but I'm lost already. Presumably Peter is discussing here LOPs from
compass bearings of landmarks. Systematic error (allowing for variation the
wrong way, let's say), causes those compass bearings to be too high (let's
say). This would twist each erroneous bearing clockwise, so not "toward the
direction" of the relevant azimuth, but at right angles to that direction.
Then each LOP, drawn on the chart from that landmark, will be twisted in
the same way.

Perhaps it would help if Peter sent a simple diagram, if not through the
list then backstage to those requesting it (me, please).

>So instead of a 25% chance of the fix lying in 4 quadrants there is a 0% chance
>of it lying in 2 opposing directions and 50% each for the other 2. In this
>case the fix position lies along the line bisecting the two 50% quadrants,
>passing through the intersection. It is still just a position line. For a
>fix we need more.

I need that diagram to understand this.
>
>With 3 LOPs it gets a little more complicated and a little more interesting.
>This is where I came in, prompted by a half-remembered claim in a book by
>George Bennett that with an azimuth spread of less than 180 degrees the fix
>position will lie outside the triangle. So, draw 3 LOPs with an azimuth
>spread of less than 180 degrees and add a direction arrow (or more as they
>get extended) to each indicating the azimuth. Use the same technique as
>above for each of the 3 intersections of 2 lines. You now have 3 bisectors
>with a common meeting point in the area away from the direction of azimuths,
>and the distance from this point to the 3 extended LOPs is the radius of a
>circle that touches each LOP. Where the bisectors meet is the indicated fix
>position.
>With any 'cocked hat' triangle, if all the azimuth arrows point outwards or
>all inwards then the fix position lies within the triangle. But with 1 in 2
>out, or 2 in 1 out (as with an azimuth spread of less than 180 degrees) the
>fix position lies outside the triangle.

Bennett was discussing how to estimate the effects of systematic error,
which is always the same amount in the same direction, though that amount
and direction may not be known. The cocked-hat proposition relates to
random errors only, and says so.
>
>The technique George Bennett describes, of drawing auxiliary LOPs some
>(large enough) distance from the originals, consistently either all towards
>the azimuth directions or all away, induces a large artificial systematic
>error. A circle is then drawn that touches each of the 3 auxiliary LOPs. The
>difference between the radius of this circle and the distance used for
>inducing the large systematic error is the indicated extent of the real
>systematic error.

If there were only systematic error, common to all observations, and no
random error, then the method Bennett describes would indeed establish that
error, and the true position, from a single cosked-hat.

When the situation is muddied by the reality of random error, then
(depending on the relative magnitudes of random and systematic error) it
may be possible to disentangle the two types of error by analysing a large
number of cocked-hat triangles statistically.

What we were arguing about, in the Great Cocked-hat Controversy, related
ONLY to random error, so I think that the Bennett method, though very
relevant to understanding what's going on, throws little light on that
particular argument.

What concusion does Peter draw from all this? Does he now dispute, or
accept, the proposition that "With errors in LOPs that are entirely random,
only one time in four will a triangle of three LOPs embrace the true
position."?

It happens to be a bad moment to raise this matter, as George Bennett's own
thoughts on the matter are rather relevant. It's good that he has joined
our group, but at present he is away and out of touch.

George Huxtable



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