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    Re: Still on LOP's
    From: Chuck Taylor
    Date: 2002 Apr 27, 23:32 -0700

    Here is some more food for thought:
    
    We take 3 bearings on fixed objects.
    
    Assume that the error in taking a bearing follows a Gaussian
    (Normal) distribution with mean 0 and standard deviation 1
    degree.  These numbers seem to me to be not unreasonable.
    
    The probability that any one bearing is within +/- 1 degree of
    the true bearing is 0.6826.
    
    The probability that all 3 bearings are within +/- 1 degree of
    the true bearing is (0.6826)^3 = 0.32.
    
    The probability that any one bearing is within +/- 2 degrees of
    the true bearing is 0.9544.
    
    The probability that all 3 bearings are within +/- 2 degrees of
    the true bearing is (0.9544)^3 = 0.87.
    
    The probability that any one bearing is within +/- 3 degrees of
    the true bearing is 0.9974.
    
    The probability that all 3 bearings are within +/- 3 degrees of
    the true bearing is (0.9974)^3 = 0.99.
    
    
    Using this information, we can define confidence regions of a sort.
    
    Converting 1 degree of angle to distance depends on distance from
    the landmark.  Here is a rough table, computed by multiplying the
    tangent of 1 degree by the distance to the landmark:
    
       Distance         Width of
       to landmark      1 degree
    
       1 nm             0.017 nm
       2                0.035
       3                0.052
       4                0.070
       5                0.087
       6                0.105
       7                0.122
       8                0.140
       9                0.157
      10                0.175
    
    Interpreting these numbers for a particular instance, if all 3
    landmarks are at a distance of just under 6 miles, then we can
    say that each line of position is within +/- 0.1 nm (about 200
    yards) of the true line of position with probability 0.32.
    Similarly, we can say that each line of position is within +/-
    0.2 nm (about 400 yards) of true position with probability 0.87,
    and within +/- 0.3 nm (about 600 yards) with probability 0.99.
    
    Note that I have said nothing about whether a particular cocked
    hat covers the true position.  Still, a feel for how far off a
    particular bearing line might be can be useful information to a
    navigator.
    
    Chuck Taylor
    Everett, WA, USA
    
    
    

       
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