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    Re: That darned old cocked hat
    From: UNK
    Date: 2010 Dec 11, 00:00 +0000

    I believe that Frank has just answered your question in his reply to Tom
    Sult. Let me just add one remark.
    Why do we chose a confidence ellipse rather than staying with the
    original cocked hat or a triangle similar to it? The ellipse is optimal
    in the sense that of all possible shapes it encloses the smallest area
    that contains the actual position with a given probability. There is
    nothing wrong per se (other than being worthless) with using the cocked
    hat itself as a 25% probability confidence area. But a 25% probability
    confidence ellipse centered on the the MPP and correctly oriented is
    MUCH smaller. This is because the probability of actually being in a
    corner of the cocked hat is very small. (Besides, in practice we want a
    95% or higher confidence interval anyway.) The nice thing about the
    confidence ellipse is that it provides a standardized measure of the
    quality of the fix and also works for more than 3 LOPs. On the downside,
    it's not trivial to derive its size, shape and correct orientation from
    the given LOPs. One really needs a calculator for that.
    Herbert Prinz
    On 2010-12-10 22:03, Hewitt Schlereth wrote:
    > Question: Is there any way to determine where around the cocked hat
    > the fix is? I mean, are there circular bands around the cocked hat
    > having greater or lesser probabilities of containing the fix, like the
    > ring road around Washington DC might have McDonalds or Wendys or
    > Burger Kings clustered around at different distances off the road, one
    > of them being the fix?
    > Hewitt
    > On 12/10/10, Herbert Prinz<666@poorherbert.org>  wrote:
    >> On 2010-12-10 02:44, John Karl wrote:
    >>> Frank&  Herbert,
    >>> I can see in the Villarceau article (without any knowledge of French)
    >>> that he obtains the construction of the symmedian point that Prince
    >>> mentions. But does he show that it minimizes the sum of the squared
    >>> distances to the three sides?
    >> Karl,
    >> As I said before:
    >> "When solving the minimum condition by partial differentiation,
    >> Villarceau found the solution to be at the point for which the distances
    >> to the sides of the triangle are as the sides of the triangle
    >> themselves. (In other words x:y:z = a:b:c)"
    >> So, yes, he did.
    >> And I further said:
    >> "This is a defining property of the symmedian point. Lemoine showed
    >> independently, but around the same time, that the symmedian point has
    >> the property of minimizing the square sum of the distances."
    >> So, Lemoine did, too.
    >> Herbert Prinz

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