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    Still on LOPs
    From: Peter Fogg
    Date: 2002 Apr 28, 12:09 +1000

    If 3 LOPs meet at a common point then according to the Doctrine of
    Infinitely Thin Lines the chances of their meeting point being the
    Actual Position (AP) is zero, if I have understood correctly, just as
    the chances of the AP falling on any LOP is zero, since an infinitely
    small point cannot occupy the same space as an infinitely thin line.
    But knowing this to be true in a theoretical mathematical sense is not
    of much use to us in a practical navigational sense. To begin with, our
    AP is not an infinitely small point, it is at least as big as our boat,
    and could be considered to be as big as our circle of view around the
    boat, a circle with a radius of a few miles.
    And as for any LOP, rather than a line of infinite thinness it may be
    more helpful to think of it as a line of infinite thickness, since the
    further we go from this line the more remote become the chances of
    encountering the AP. The converse is presumably also true, so our line
    looks extremely smudged, gradually merging from white to dark gray then
    back to white, showing the likeliness of encountering the AP on either
    side of it. We draw the line with a 2B pencil but it may be helpful to
    always think of it as this smudged line of infinite thickness, while
    recognizing that by the time you've traveled 100 miles (or much less)
    from it, it is almost as white as the paper, reflecting the very small
    chance the AP is out there.
    Now we have two of these LOPs crossing. Once again, while theoretically
    the chances of the AP being at the intersection is zero, if you use my
    smudged lines then the chance becomes 100%. And this reflects practical
    navigational practice; based on available data this is our fix position.
    And this is where this discussion started, when it was contended that 2
    LOPs were enough and I proposed that 3 were better, citing as an example
    the uncertain meeting point of 2 LOPs from similar bearings.
    When we introduce a third (smudged) LOP we have 3 intersections: each
    presumably is, 100%, the AP. Knowing we can only be at one point at any
    one time all 3 cannot be correct, but since based on available data they
    are equally correct then we find a fix position based on least squares,
    inside the hat.
    And if our AP is not contained within the cocked hat, then where is it?
    If outside, but near one intersection then those 2 LOPs are roughly
    correct but the third is quite wrong. If outside, and somewhat halfway
    along one of the LOPs between 2 intersections, then that LOP is roughly
    correct but the other 2 must be decidedly dodgy. Is this really the case
    75% of the time?
    Normally we don't have any way of knowing how accurate any or all of
    them are apart from the size of the hat. In a practical sense a small
    hat is an accurate fix, and a common meeting point is impossibly
    perfect. But even a small hat still has smudged lines, reminding us that
    the AP could be outside it, although I never heard an answer to my
    question of what good knowing this was to us. By the time we've worked
    out our fix position we're somewhere else, and promptly use the fix to
    run forward our DR.
    Thanks to the nav. list I spent some time thinking about this while
    bowling along dusty unsealed roads beneath a wide deep blue sky,
    exploring that strange phenomenon; a marsh in near desert country, a
    mostly dry swamp (although sometimes its full of water - and life)
    larger than a small European country like Denmark. It was fascinating,
    and the navigation was easy - the tracks twisted and turned, but apart
    from the shadows cast by the sun the near full moon was clearly visible
    for most of each afternoon. We saw lots of kangaroo and emu, and even,
    somewhere else, a koala in the wild, and that is very rare.
    I've avoided thinking about how multiple (more than 3) LOPs affect
    things. Personally I find them hopelessly confusing, as some intersect
    the hat and others fall outside it. My navigational calculator copes
    with them very well. If the chance of the AP being within the hat is
    25%, then how is this affected by multiple LOPs?

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