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    Re: A simple three-body fix puzzle
    From: Frank Reed
    Date: 2010 Dec 10, 14:27 -0800

    Tom, you wrote:
    "It seems to me the point of this is that the best probability of location is a "donut" that roughly follows the lop's around the hat. We have learned that we have perhaps a 25% +/- the details chance of being close to any "center" of the hat. And we can calculate the probability distribution of our location along any one of our LOP's. If the gausien distribution is centered on the LOP then we have a "fuzzy donut" probability and not a Position. "

    No, no donuts. This is one of the things that can go wrong when people have this discussion. The point with highest probability is at the fix, inside the triangle formed by the crossing LOPs. The ellipse of non-negligible probability, however, extends well outside the triangle. Picture a low, broad "hill" of probability with a "bell curve" cross-section. There is a 75% chance of being outside the triangle. But this doesn't mean that there is a "hole" in the donut. The location of maximum probability is right where we have been discussing: at the "symmedian" point. But the triangle is NOT an approximation to the error. The probability steadily decreases as you move away from the fix, and this applies to any number of lines of position (greater than one). EVERY fix yields an error ellipse (or better yet, a family of concentric error ellipses, each reprsenting a lower probability than the one before it). That's the final product. The center of that error ellipse, which is the high point on a low, flat hill, is located at that point inside the triangle, but the sides of the hill decline away much more gradually than most people expect.


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