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    Re: Still on LOP's
    From: Geoffrey Kolbe
    Date: 2002 Apr 21, 11:30 +0100

    "A picture is worth a thousand words" so here is a picture:
    
    http://www.pisces-press.com/graphics/hats.jpg
    
    There are eight small drawings in hats.jpg. Each drawing shows three known
    geographical positions, which could be prominent landmarks, represented as
    black spots.
    
    The correct bearings to your actual position are shown in black. The
    bearings all meet at the actual position. This is a zero error, theoretical
    construct. But if you have a map which shows your actual position and the
    three land marks, It is possible to draw such a diagram on the map.
    
    If bearings on the landmarks are taken from the actual position, there will
    be an error in the bearings so measured. 50% of the time the error will be
    to the left, and 50% of the time the error will be to the right of the
    landmark. Let us call this L and R errors.
    
    The drawings also show simulated real life bearings taken on the landmarks
    and plotted in red. The bearings have either a left or a right error from
    each landmark, indicated by the letter at each landmark. There are three
    landmarks, the bearing on each can have a left or right error, so there are
    eight possible combinations of errors - and so eight drawings showing these
    combinations.
    
    These errors result in a cocked hat where the bearings intersect. You will
    note that only in two cases do the resulting cocked hats enclose the actual
    position.
    
    A little thought will also reveal that for any given arrangement of
    landmarks, two combinations of error will ALWAYS enclose the actual
    position, and six combinations of error will NEVER (and can never) enclose
    the actual position.
    
    Since each of these scenarios is equally likely, the chances are one in
    four that the cocked hat due to the intersection of the bearings taken from
    three land marks will include your actual position.
    
    I have omitted a few considerations due to symmetry, but I am pretty sure
    that this approach constitutes the basis of a formal proof of the assertion
    that the cocked hat resulting from the observation of three objects will
    only enclose your position 25% of the time.
    
    Not 23%, or 26%, or 17.5%, but 25%.
    
    A similar argument can be put forward for LOP's, where the cocked hat is
    due to errors in measuring altitude.
    
    
    Geoffrey Kolbe.
    
    
    
    
    
    
    

       
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