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    Are we most likely not where we are?
    From: Herbert Prinz
    Date: 2002 Apr 15, 04:16 +0000

    While George Huxtable tells us that
    
    >  It's a surprising fact that no matter how good the navigator, only one time in four
    > will his cocked hat
    > embrace his actual position, which is three times more likely to lie outside it.
    > [...] So it's a big fallacy to imagine that the true position must be within the
    > cocked hat. Instead of a probability of 100%, it will be just 25%.'
    
    Peter Fogg insists that this be irrelevant and the fix position be within the confines
    of three intersecting LOPs, specifically
    
    > [...] the fix position is equidistant from all 3 lines.
    
    Well, George is quite right. The cocked hat is not all that meaningful. Peter, too, is
    on the right track when he suspects a semantic problem at the base of this seeming
    contradiction. The "actual" position is not the "most probable" position. The former is
    the one where God has put us, but it is the latter, that the navigator solves for. The
    most probable position (MPP) is, by definition (!), inside the cocked hat.
    
    A little more helpful than the MPP by itself, or the cocked hat,  is the confidence
    ellipse, which defines the area within which the estimated position lies with a given
    probability, say 95%. This ellipse is centered on the MPP. Its size depends on the size
    of the errors of the observations and on the chosen probability, whereas its shape
    depends on the number of observations and distribution of the azimuths. The confidence
    ellipse normally overlaps the cocked hat partly. Consequently, there are parts of the
    cocked hat where the vessel is not very likely to be, others where it is likely to be,
    and similarly, there will be areas of both kinds outside the cocked hat. A confidence
    ellipse can be drawn for any number of LOPs and therefore give a visual representation
    where the cocked hat fails to do so.
    
    Back to the most frequent case of 3 LOPs. Now that we have settled that the MPP is
    indeed inside the cocked hat, the question remains: Exactly where in the hat? Peter is
    mistaken to say that it is equidistant from all 3 lines. It is so, if, and only if the
    cocked hat is an equilateral triangle. The navigator will strive for this by choosing
    appropriate stars. To assume that it always is, is wishful thinking.
    
    I know where the misinformation comes from: Bowditch! In the 1984 ed. on p.470, and
    again in the 1995 ed. on p. 328, in different contexts.  I quote from the latter:
    "Lines bisecting the three angles of the triangle meet at a point which is equidistant
    from the three sides, which is the center of the inscribed circle." So far, so good.
    "This point is of particular interest to navigators because it is the point
    theoretically taken as the fix when the three lines of position of equal weight and
    having only random errors do not meet at a common point." This is, with due respect to
    the DMAHC, rubbish.
    
    There are two simple arguments why it cant be so.
    
    1. If the MPP were equidistant from all LOPs in the case of 3 LOPs, why would there be
    a different law for 4 or more LOPs? But in general, there exists no point that is
    equidistant from 4 lines.
    
    2. Assume 2 LOPs parallel to each other, 2 nm apart, a third one perpendicular to
    those. According to the rule, the MPP would not be on the perpendicular LOP (half way
    between the other ones), but at a mile distance from it! Why?! On which side?!
    
    Now we are down to 7998.
    
    Here is the challenge of the week:
    
    1. Where in the cocked hat is the MPP really?
    2. Can it be constructed?
    3. Is there a book on navigation in print that has it correct?
    4. Who published the correct solution first?
    
    Herbert Prinz
    
    
    

       
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