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

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Re: Ellipse of Confidence in position finding
From: Robin Stuart
Date: 2019 Mar 15, 13:18 -0700

Andres,

Since Bill Lionheart has now provided an understanding of the mathematical basis of and assumptions that underpin your statement “The ellipse of confidence, EoC,  has math sense for three or more sights. Not for two.” made here, I think I am now in a position to comment.

If the practical sailor plots two LoPs and finds that they show rocks awash lying dead ahead does he

a)     Alter course?

b)    Declare “The ellipse of confidence, EoC,  has math sense for three or more sights. Not for two” and sail on?

Experience teaches him the typical size of altitude errors he can achieve with his sextant. He expects that his true position to be somewhere close to where the LoPs cross and hence they define a set of confidence contours. Hard aport!

Likewise for a single LoP. After taking a single noon sight our practical navigator is quite justified in believing that he is more likely to be within 1NM of the latitude he determines than 10NM. This single LoP defines a set of confidence contours. As noted previously, whether these contours should be classified under the category of “ellipse” is neither here nor there although apparently there are those who believe this is a topic worthy of deep contemplation.

I don’t really see that any of the above can be dismissed on the basis “Real world, not the imaginary one. Mathematics and the practicality of things. The difference between a theoretical science an engineering...” as you did here.

When the above considerations are formalized into an analytic framework the assumption, that the navigator has prior knowledge of the expected size of altitude errors, is accounted for by introducing a variance for each of the observations made. This has been standard practice for a long time. It is done in references I previously cited and also in

Optimal Estimation of a Multi-Star Fix, C. De Wit. NAVIGATION, Vol.21 , No. 4, Winter 1974-75, pp 320-325

which is the first reference you quote in the paper on your website. Within this standard framework Bill Lionheart’s original question is completely correct and well-founded.

My understanding now is that your statement that 3 LoPs are required to form a confidence ellipse assumes no prior knowledge of the size of the errors to be expected when taking sights. I suppose this might happen if you had dropped the sextant or the seas were more violent than you were accustomed to or you just want to calibrate your technique. In any event if you have no idea what size of error to expect then the first LoP definitely tells you nothing. The second is guaranteed to cross it somewhere so that again tells you nothing. Only the third begins to show how well the observations are clustering. This is the basis of your statement and verified by the mathematics.

Overall I believe that through this thread I have gained some interesting insight and into the variations that are possible when constructing confidence contours and am pleased by that. The differences arise from differing assumptions about what information you have a priori. This is therefore a problem in conditional probability which can be very counter-intuitive and mind-numbing.

I would say however that some of the contributions to this thread have been case studies in how not to conduct an open good-faith discussion on a technical matter. Your initial unsubstantiated use of the term “in a math sense” quickly transformed into “in a nautical sense” but neither of these vague statements addressed or gave any consideration to the concrete plots and references that had been prepared for your benefit and provided evidence that is as plain as the nose on your face.

In a later post you clarified

“the difference between of the science and the art of navigation, is practical expertise. That is what I called “nautical sense””.

although it is unclear how this adds to the topic in hand at all. It does however seem to be consistent an overall theme suggested by earlier statements I am not an armchair but a real sailor” and “The difference between a theoretical science an engineering...” that you are somehow in possession of deep and mysterious knowledge that cannot articulated and your authority should not be questioned. Maybe this is a cultural thing but I, for one, am unwilling to engage in this style of debate.

I hope that you might consider the possibility that having a rigorous grounding and understanding of scientific principles does not necessarily exclude one from also being in possession of real practical knowledge and experience. You might also entertain the possibility that there may be more sailors on Navlist than you suppose and the qualifications that you tout may not be as exceptional as you imagine.

Robin Stuart

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