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    Re: Cocked hats, again.
    From: Greg R_
    Date: 2007 Mar 14, 16:31 -0700
    > I always was partial to "T's" and "A's."
    Well, there goes this list's "family-friendly" rating......  ;-)
    ----- Original Message -----
    To: "NavList" <NavList@googlegroups.com>
    Sent: Wednesday, March 14, 2007 3:25 PM
    Subject: [NavList 2341] Re: Cocked hats, again.

    > Gary LaPook wrote:
    > 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?
    > On Mar 14, 12:50 pm, "George Huxtable" <
    > wrote:
    > > Gary LaPook wrote:
    > >
    > > Well at first blush is would seem then that from 3 LOPs with equal
    > > chances of being on one side or the other of each LOP that you would
    > > have 8 combinations (2^3) so one specific case would occur only 1 out
    > > of 8 times for a ratio of 7 to 1 not three to 1. That is my first
    > > thought and I haven't made any drawings yet.
    > >
    > > ==================
    > >
    > > Reply from George-
    > >
    > > Not so, Gary. Consider the case of an observer, at a known position,
    > > taking altitudes of three stars, which are not contained within a
    > > 180-degree arc. (If they were within 180 degrees, the argument would
    > > differ in detail, but not in principle, and reaches the same
    > > conclusion). Take, as a simple example, the simplest case of three
    > > stars whose bearings differ by 120 degrees, though the argument
    > > applies to other such angles just as well.
    > >
    > > Let's denote an intercept being Towards a star by the letter T, and
    > > Away by A. No doubt we will agree that in the absence of systematic
    > > error, T and A are equally likely, with 50 % probability. (We presume
    > > that the likelihood of an error of exactly zero is negligible).
    > >
    > > Then for the three stars in order, there are 8 combinations as
    > > follows-
    > >
    > >
    > > Each of those combinations is equally likely; a probability of 1/8.
    > >
    > > Only for the two combinations AAA and TTT will the resulting triangle
    > > embrace the observer's position. So the probability of that happening
    > > is exactly 25%.
    > >
    > > I should add that Gary's disbelief is characteristic of the reaction
    > > of many list members when they meet that argument for the first time,
    > > but most, if not all, seem to have come round to acceptance in the
    > > end, if somewhat reluctantly.
    > >
    > > I have tried it out as a computer simulation and it checks out.
    > >
    > > It applies just as well to the similar cocked-hat derived from 3
    > > compass bearings of distant landmarks.
    > >
    > > It has some interesting side-effects. Say that there was a systematic
    > > error; for example, say all altitudes were too great, because of an
    > > error in the index correction. Then, (in the case of all azimuths
    > > spanning more than 180) the number of TTT observations would be
    > > enhanced. If that error was large enough to overwhelm the natural
    > > scatter, then EVERY observation would be a TTT one, and then the
    > > triangle would embrace the true position in EVERY case, not just 1 in
    > > 4. Some observers may take that to show a satisfactory state of
    > > affairs, but not at all; it points to a major error!
    > >
    > > It will be interesting to discover whether Gary can be convinced. Or,
    > > alternatively, whether he can convince us that we're wrong.
    > >
    > > George.
    > >
    > > contact George Huxtable at
    > > or at +44 1865 820222 (from UK, 01865 820222)
    > > or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

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