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