# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**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...... ;-)

--

GregR

----- Original Message -----

From: <glapook@PACBELL.NET>

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" <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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