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    Re: Cocked hats
    From: Geoffrey Kolbe
    Date: 2006 Dec 16, 10:13 +0000
    [NavList 1907] Jim wrote

    So as has been mentioned in recent postings, cocked hats don't necessarily contain your true position and small cocked hats (or exactly crossing lines for that matter) should not imply greater accuracy but, statistically, the position that you define with the cocked hat is likely to be better than the 1 in 4 argument might suggest.

    With great respect Jim, I think you have confused yourself with your own diagram.

    But, a picture is worth a thousand words, so here is another picture which I think better represents the nature of the argument.

    http://www.pisces-press.com/graphics/Cocked_hats.jpg

    The picture shows nine diagrams of LOP plots at about 120 degrees to each other.

    The red dot is the actual position. The red lines are theoretical position lines with no error. The diagram in the centre of the picture shows the ideal plot.

    However, an actual altitude measurement will ALWAYS have some error. There are two possibilities for this error, either it will be towards or away from the celestial body. The resulting LOP can lay on one side of the actual position - towards the celestial body, or the other side of the actual position - away from the celestial body.

    So for a three body fix, there are eight possible ways to draw the LOP diagram which includes all the permutations of the LOP error, towards or away from the celestial body.

    The other eight diagrams in the picture repeat the ideal plot, but superimposed in blue are the eight possible scenarios drawn where each LOP has an error either away from or towards the celestial body.

    The cocked hat in each of these diagrams is filled in in light blue.

    It will be noted that only two of the eight cocked hats include the actual position. For the other six diagrams, there is no way to draw the LOPs to include the actual position whilst maintaining the combination of LOP errors represented in those diagrams.

    Where random error dominates, the chance of your cocked hat including the actual position is just 1 in 4.

    If you find that in your experience, the cocked hat includes the actual position rather more often than one time in four, I would suggest that some systematic error has become significant.

    If over a long period you find that your actual position is inside the cocked more often than one time in four, the most likely source of the systematic error is incorrect sextant adjustment or some error in using the sextant. This leads to a cocked hat larger than it should be given random error alone and so consequently includes the actual position.


    Geoffrey Kolbe 







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