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    Re: Cocked hats, again.
    From: Gary LaPook
    Date: 2007 Mar 15, 10:23 -0700
    George Huxtable wrote:

    >It will be interesting to discover whether Gary can be convinced. Or,
    >alternatively, whether he can convince us that we're wrong.

    Gary LaPook responded:

    "In the beginning God created the heavens and the earth. The earth was
    without form and void, and darkness was upon the face of the deep; and
    the Spirit of God was moving over the face of the waters.
    "And God said 'let there be a line of position which I will call "A"'
    and thus the first line of position was upon the face of the deep. And
    this line of position ran straight north and south and divided the
    waters of the earth into two halves with an observer having an equal
    chance of being either to the east or to the west of the line of position.

    "And God saw that the LOP was good and God had separated the east from
    the west.

    "And God said 'Let there be a second LOP which I will call 'B' and let
    it run straight east and west dividing the north from the south and God
    saw that this was good also and He was pleased. Then God said 'let "A"
    and "B" cross dividing the earth and the waters into four quadrants so
    that my observer will have a one in four probability of being in any one
    of the quadrants and this was done, and God was mightily pleased" (See
    diagram 2.)

    This is where probability, statistics, standard deviation and 'sigma'
    come into the story.

    Going back to diagram 1, even though an observer has a 50-50 chance of
    being on one side or the other of "A" he is limited in how far away he
    can be from "A." Looking at standard deviation and gaussian distribution
    we can be fairly certain that he is no more that 3.3 sigmas (standard
    deviations) east or west of "A" , only one observation in a thousand
    would be farther away than that. Or, to raise the certainty higher, we
    know he has less than one chance in ten thousand of being more than 4
    sigmas and less than one chance in a million of being more than five
    sigmas away. We can illustrate all the places that the observer could be
    by adding to diagram 1 two lines parallel to "A" ,one to the east and
    one to the west, spaced to show the band either 3.3, 4, or 5 sigmas
    wide, whichever you fancy. This is done in diagram 3.  Let's call these
    additional lines "A-" and "A+." We can see that the observer will still
    have a 50-50 chance of being east or west of "A" but he is constrained
    to be within the boundaries determined by sigmas. He can't be anywhere
    else in the universe. (If you need a real number to hold onto, figure
    one sigma using a sextant on the surface is one nautical mile and  2
    nautical miles if  using a bubble sextant in flight. Your milage may vary.) The observer must
    be in the bands delimited by "A-" to "A" and on to "A+" which we can
    call the bands of possible positions.

    Now, the same reasoning holds for LOP "B" and this is shown in diagram
    4. The observer has a 50-50 chance of being north or south of "B" and he
    must be within the bands of possible positions delimited by "B-" and
    "B+." Now, when you combine the two sets of probabilities for "A" and
    for "B" you get diagram 5 which shows that the observer has a one in
    four probability of being in any one of the four quadrants limited by
    "A-", "A+", B-", "B+" and centered on the intersection of the the two
    LOPs, "A" and "B." The observer cannot be anywhere else on earth.

    Now, this is important. The only places the observer can be are all
    contained within these four quadrants determined by the first two LOPs,
    "A" and "B."  No matter how many more LOPs you add to the diagram the
    observer can never be outside this original limit. Additional LOPs can
    further reduce the possible locations to just a portion of the original
    figure but can never expand it.. Read this paragraph again and make sure
    you understand why this is true..

    Diagram 6 shows in pink all of the possible positions of the observer
    based on LOPs "A" and "B" and shows that there is a one fourth chance of
    being in any one quadrant.

    Now we move on to diagram 7 which shows a third LOP, "C" crossing at a
    45 degree angle to the first two LOPs and intersecting at the same point. (
    I drew it this way for convenience but the argument would hold true for
    an equilateral triangle or any other triangle.)
    In diagram 8 we have added "C-" and "C+" spaced from "C" in the same
    manner as the first two LOPs. Diagram 9 just shows "A," "B," and "C"
    darkened. Note, we still do not have a "cocked hat" to be inside of
    outside of.,

    Diagram 10 shows in pink all the possible positions for the observer
    with the third LOP at "C" intersecting at the same point as "A" and "B." You will notice
    that "C-" and "C+" have reduced the possible locations for the observer
    by cutting off the northeast and the southwest corners of diagram 6.
    This has improved the accuracy of the fix somewhat by eliminating some
    of the possibilities. Diagram 11 shows that the northwest and southeast 
    quadrants now have somewhat more than a one in four chance and the
    southwest and northeast each have somewhat less than a one in four

    In diagram 12 we finally get a cocked hat! The only change in this
    diagram is that we have moved the third LOP away from the intersection 
    of "A" and "B" by the width of the band of possible positions so it is
    now located at "C+." This shifts the bands of possible positions based
    upon the third LOP to the area between "C" and the newly added "C++"
    which is equally spaced to the northeast from "C+." Diagram 13 more
    clearly shows the cocked hat. It is made up of lines "A," " B," and "C+."

    Diagram 14 shows in pink the possible locations of the observer based on
    this configuration of LOPs. This illustrates that the positions must
    still be within the original four quadrants. The "C" line has eliminated
    the entire original southwest quadrant and sliced in half the northwest
    and southeast quadrants. The remaining pink triangle is only half as
    large as figure 6, meaning that we have improved the accuracy of our fix
    by a factor of two by eliminating half of the area covered by the two
    LOP fix. This also shows that a large portion of the possible locations
    of the observer are outside the cocked hat and that there are two small
    corners inside the cocked hat where the observer cannot be. However, a large
    proportion of the possible locations are within the cocked hat and the
    outside positions are not three to one compared to the positions inside.

    Moving along, we repeat the process by moving the third LOP further to
    the northeast by one more band of possible positions so that it now is
    at "C++." This makes the cocked hat larger and is illustrated in diagram
    15. Diagram 16 now illustrates in pink all of the possible locations of
    the observer with this configuration. You will see that as the cocked
    hat has gotten larger the possible locations for the observer has gotten
    much smaller and all of these positions are in the northeast corner of
    the northeast quadrant and ALL  are contained within the cocked hat! 
    There is no way the observer can be located outside the cocked hat. So,
    clearly, no three to one  ratio of positions outside the cocked hat to
    those inside.  In fact it is a one hundred percent to zero percent ratio
    of inside to outside.

    Diagram 17 is slightly different. This time, instead of moving the third
    LOP one more complete band of possible positions, we move it only so
    that it is now one band away from the northeast corner of the northeast
    quadrant, the point defined by "A+" and "B+." The third LOP in now at
    "Z." The southwest limit of the band of possible positions related to
    the "Z" LOP is line "Y" which passes over the intersection of "A+" and
    "B+." Since the diagrams are now getting pretty complicated I have
    copied the pertinent lines on to diagram 18. This diagram shows that the
    only possible location for the observer is at the intersection of "A+",
    "B+" and "Y." This point is the only place that all three bands of
    possible locations touch and is the northeast corner of the northeast
    quadrant. This one point is exactly in the center of the cocked hat and
    is equidistant from lines "A" , "B" and "Z." This is the only point on
    the planet where the observer can be with the maximum sized cocked hat.
    Obviously, again, no three to one ratio of outside to inside positions.
    In fact it is still a one hundred percent to zero percent ratio of
    inside to outside.

    So, have I convinced you?

    (Attached are two files containing 18 diagrams. I broke them into two groups so that the files would be smaller. Diagrams 6,10,11,14,16,and 18 are in color and are in the second file.)

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