# NavList:

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

**Re: Cocked hats, again.**

**From:**George Huxtable

**Date:**2007 Mar 14, 19:50 -0000

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 george@huxtable.u-net.com or at +44 1865 820222 (from UK, 01865 820222) or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to NavList@fer3.com To unsubscribe, send email to NavList-unsubscribe@fer3.com -~----------~----~----~----~------~----~------~--~---