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>
> 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" <geo...@huxtable.u-net.com>
> 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-
> >
> > AAA, AAT, ATA, ATT, TAA, TAT, TTA, TTT
> >
> > 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 geo...@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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