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Let's make it simpler. An observer jwishes to measure the altitude of a 
single star, Polaris, to get his latitude (for argument's sake, we'll put 
Polaris at the North Pole). His vessel is at some position on the Earth's 
surface. He may or may not know where it is, it doesn't matter.

We know that because of random scatter in the observation, his deduced 
latitude may be greater or less than his true latitude, with equal 
probability. We can, if John Karl  wishes, take it to be a Gaussian, with a 
certain given standard deviation, though it's not necessary to the argument

We can then take a piece of paper, with no coordinates marked on it (but a 
scale of miles and a North arrow), and put a point somewhere on it to 
represent the observer's position, which might be anywhere.

And then draw in a number of East-West position lines, appropriately North 
or South of the observer, scattered from it symmetrically North and South, 
corresponding to the width of the Gaussian. This line is plotted with 
respect to the position of the observer, wherever he may happen to be.

It's then just a small step to repeat the argument for two additional 
stars, at different azimuths. Then the resulting triangle is drawn with 
respect to the observer's position. What that position is, whether it's 
known or unknown, does not enter into it.

George.

contact George Huxtable, at  george@hux.me.uk
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
----- Original Message ----- 
From: "John Karl" 
To: 
Sent: Saturday, December 11, 2010 4:52 PM
Subject: [NavList] Re: That darned old cocked hat


OK here’s one last go at it (maybe call it a summary).  As I said in post 
14766, George and I are discussing too very different topics -- both quite 
irrelevant for practical navigation.

George’s irrelevance
A.  He measures triangles from a known location.

B.  And then he counts the number of times (i.e., the percent of time) that 
the known location is outside of the triangle.  (So yes George, you do need 
to know the fix location in order to do this counting.)

C.  This is not navigation.  It’s not determining anything about George’s 
location – we already know that.  It’s determining information about the 
random distribution of the triangles.

D.  George measures something. He doesn’t estimate anything – his position 
is already known.  So it’s irrelevant to navigation.

Karl’s irrelevance
A.  Three LOPs are given.  It’s known that their errors are distributed 
normally perpendicular to their lines, with no information available along 
their individual lines.

B.  By multiplying these three linear distributions, we get the 
probability/area P(x,y) that a point is within that area.  We also easily 
observe that the MPP (maximum probability position) minimizes the sum of 
the squared distances to the sides of the given triangle.  We see that this 
conclusion is independent of the standard deviation.  H. Prinz pointed out 
that this was known to Villarceau in 1877.  (Darn it.)

C.  P(x,y) is everywhere rather small, even the value of the MP is small, 
typically around 2% per area.  And obviously it can never exceed one (i.e., 
100%).  Therefore a small cocked hat has a small probability that the fix 
is inside.  (No need to even plot P(x,y) for this conclusion.)   And a very 
large cocked had has a large probability that the fix is inside.  (I did 
plot P(x,y) for this conclusion, but had no need to consider numerical 
integration.)

D.  These conclusions do not conflict with George’s measurements of 
triangle distributions around a known point – that’s a completely different 
question (and not one of navigation).

E.  Since this MPP gives the best fix, and no one has claimed an optimum 
fix location according to any other criterion, I don’t understand why all 
the other discussions, such as in the “Simple 3-Body Fix Puzzle.”  Then 
there’s outright incorrect statements, such as in Dutton and Bowditch.

F.  The significance of the MPP is only relevant on a statistical basis. 
Its main use is to stimulate NavList discussions.  In real world navigation 
at sea, make darn sure you know where you are not, not where you most 
probably are.  In short, George’s topic of triangle distributions is 
irrelevant for navigation, while this MPP discussion is irrelevant for 
practical navigation (unless you’re navigating in a casino).

JK

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