# NavList:

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

**Re: Ellipse of Confidence in position finding**

**From:**Bill Lionheart

**Date:**2019 Mar 15, 08:58 +0000

Thanks for that detailed list to the history. Its not so easy to catch up with previous NavList discussions! One of the article says "In celestial navigation, the bisector of the azimuths of two Marcq Saint Hilaire LoPs ia a true line of position that cancels out systematic errors." I see that is interesting - if the only error was for example a fixed index error, or a habit of misestimating the horizon one way, and direction towards the GP was opposite for the two lines this makes sense. But then the incentre (Kimberling centre X(1) ) gives the intersection of the angle bisectors as the true position assuming the direction of the GPs is either towards the inside of the hat or away from it. The Index error can be measured though, and perhaps once can assume that eventually one would estimate other systematic errors either from either taking sights on more bodies or from taking sights when the position is known be an independent fix of some kind. One is then left with zero man random errors, and the (possibly weighted) symmedian. If we want to model the errors with a systematic and random error, eg Gaussian with the the same non zero mean and variance in each LOP, variance known but not the mean, we need more than three LOPS. I have not finished scouring the archive, has someone explained how to get the ML estimate in this case yet? Bill L On Thu, 14 Mar 2019 at 21:32, Andrés Ruizwrote: > > Bill, see only: > > An outside fix example > http://fer3.com/arc/m2.aspx/3-sights-SR-example-AndrésRuiz-jul-2017-g39403 see "CelNav.bisectors.pdf" > http://fer3.com/arc/m2.aspx/3-sights-SR-example-AndrésRuiz-jul-2017-g39410 > http://fer3.com/arc/m2.aspx/3-sights-SR-example-AndrésRuiz-jul-2017-g39411 > Fix NOT inside EoC > Fix = intersection of bisectors > > An example of an inside fix > http://fer3.com/arc/m2.aspx/3-sights-SR-example-AndrésRuiz-jul-2017-g39404 > Fix IS inside EoC > Fix = intersection of bisectors = CoG( cocked hat) = symmedian =... > > > El mié., 13 mar. 2019 a las 15:33, Bill Lionheart ( ) escribió: >> >> >>> >>> Another important thing about ellipses. Remember my post with an example where the EoC, the symmedian point, and others give us the wrong position. >>> Fix by bisectors >>> http://fer3.com/arc/sort2.aspx?subj=3+sights+SR+example&author=&y=201401&y2=201912 >>> http://fer3.com/arc/sort2.aspx?subj=That+darned+old+cocked+hat&author=&y=201001&y2=201912 >>> >> >> I had a bit of a look at that thread and saw a lot of confusion. You get a probability density function and that is correct given the assumptions. As navigators we have sometimes to choose a point estimate from the PDF, for example as departure point for our next dead reckoning. One should always understand a "fix" as just a point estimate of a probability distribution and so consider the risks to navigation of being at the "most dangerous" location within a probability contour. >> >> I think practically the common reason the symmedian is not a good point estimate is that the variances of the LoPs are different, eg they were obtained from linear regression fit for a different number of altitude observation of the same body. Then we need the weighted symmedian (weighted least squares solution) as our maximum likelihood estimate. It is still inside the cocked hat for three LoPs but it could be anywhere in the interior depending on the weights. >> >> Bill Lionheart >> >> View and reply to this message > > > fair winds! > -- > Andrés Ruiz > Navigational Algorithms > http://sites.google.com/site/navigationalalgorithms/ > > View and reply to this message