# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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

```Hewitt,

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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