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    Re: Cocked hats, again.
    From: Robert Eno
    Date: 2007 Mar 14, 20:35 -0500

    I'd like to wade into this one but not to argue about stats. Seems to me
    that this is a done deal anyway.
    I think that we are all focusing on only one aspect of Gary's excellent and
    well laid out post while perhaps overlooking some rather salient points that
    he brought up. I quote (clipped):
    "So, what does this tell us. There is about a 30% chance that the
    position of the observer will be more than .48 NM but less than 1.0 NM
    and about a 61% chance that the position of the observer will be more
    than 1.0 NM from the plotted fix. So what do we do with this
    knowledge? We use the plotted fix at the intersection of the two LOPs
    for our navigational purposes such as measuring our progress and
    planning the next leg.  We also use this fix  to deal with the
    proximity of danger keeping always in mind that the vessel may
    actually be almost 4 NM from the fix in any direction. Why do we use
    the intersection as the fix, because there is no better one available
    since this spot marks the center of possible positions of the
    observer. No other spot would be as useful for planning purposes or
    avoiding danger. Also, what methodology would you use in determining
    another spot to mark the fix?"
    "Now moving onto the three line fix derived from three observations
    resulting in a triangle. The same analysis holds with the same circles
    of uncertainty since the boat doesn't know that you took three sights
    this time instead of just two. The only question left is where should
    we plot the fix to mark the center of these circles? For very
    practical reasons we take the center of the triangle as the fix. The
    size of the triangle is limited. If you wanted to plot the fix
    somewhere outside the triangle how would you decide where to place it,
    the choices are unlimited with no way to chose between. Again, no one
    is suggesting that the position of the observer is at the center of
    the triangle but this represents the center of possible positions of
    the observer.  In fact, the position of the observer will be outside
    of the triangle often but I don't agree with the three out of four
    allegation. Counter intuitively, the smaller the triangle the more
    likely that the position of the observer is outside the triangle! If
    you think about it, this should be obvious. Using   reducio ad
    absurdum, think about a triangle only one inch in size, it would be
    impossible for the observer to be within the triangle. At the other
    extreme, a very large triangle with all of the displacements of the
    LOPs from the center of the triangle equal to 3.3 NM (3.3 sigma's,
    linear sigma's are slightly different than circular sigma's, see
    Bowditch), the only place that the position of the observer could be
    is at the fix in the center of the triangle!"
    From the standpoint of a practical navigator, the above two paragraphs make
    perfect sense to me. My interpretation of what Gary is trying to say is
    the reason why navigators place the fix at the intersection of two or more
    LOPs and/or at the centre of the cocked hat formed by 3 LOPs, is because it
    is the most practical and realistic thing to do. Furthermore, even
    recognizing that these points may not be the actual fix, there is no
    practical way to determine where the actual fix is located as the
    possibilities are many and there does not appear to be a reliable way to
    ascertain which one is the real McCoy. Gary, correct me if I am wrong in my
    This gets back to the point that I was trying to make when this discussion
    first came up: for purposes of practical navigation, there is no use in
    getting wound around the axle about whether or not the fix only has a 25%
    chance of being inside the cocked hat. It is all we've got and in the middle
    of the ocean it doesn't matter.
    Having said this, I would be interested in hearing from folks out there who
    teach celestial navigation to find out what lessons they impart in their
    students. Do you teach them to take the fix at the intersection of the LOPs
    and/or centre of the cocked hat or do you hose them down with cold
    statistical sea water just as they are beginning to grasp the fundamentals?
    Further, I would be interested in knowing what is taught in military
    navigation schools (including the air force). Does this topic even come up?
    If so, how is it presented and what kind of advice is given to the students?
    By the way, welcome back George. I was getting a little worried that you
    appeared not to be biting at all of the delicious bait that has been thrown
    out over the past two weeks!
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