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    Re: An analytical solution of the two star sight problem of celestial navigation
    From: Frank Reed
    Date: 2019 Mar 7, 13:00 -0800

    Andres, you wrote:
    "In a nautical sense the ellipse depends on the standard deviation of the estimated position, and it is proportional to the inverse of the square root of number of sights minus two: sigma = K sqrt(1/(n-2)), so n must be greater than two."

    Ok, but says who?? You describe this as being "in a nautical sense", but that's hand-waving... almost an appeal to tradition. The rule that you are repeating here is founded on one very specific --and woefully misguided-- approach to the problem which has confused and confounded those who have approached this topic for at least thirty years. And after decades, far too many navigators and navigation enthusiasts have forced themselves to believe an absurdity: that there is no error ellipse defined for a two-body fix! 

    You added:
    "Real world, not the imaginary one. Mathematics and the practicality of things. The difference between a theoretical science an engineering...
    Yes, parallel lines intersect at a point which lies at infinity.

    No You have it quite backwards. The approach that you are accustomed to is the more mathematical, the less practical, and the less useful.

    The problem here is that the traditional navigation dogma on error ellipses is founded on the principle that we must determine the statistics only from the current round of sights. By this logic, since two lines of position cross in a single perfect point there can be no estimation of error. Thus we must prohibit ourselves from thinking about an error ellipse.

    That traditional statistical logic is easily over-turned by allowing our statistics to have prior knowledge of the expected accuracy of sights. We change the mathematical assumptions to make them more realistic. And of course every navigator in the real, not imaginary world, every navigator interested in a practical, not theoretical solution, does, in fact, have some reasonable estimation of the error of sextant sights.The methodology and interpretation of the statistical statements in navigatiion make a hell of a lot more sense when we do it this way. The standard deviation of sights is treated as an input, not an output of the sight-clearing process. Does a practical navigator who shoots a sight of the Sun really have no idea what the error bars are on one sight? When you cross two lines of position, does a practical person really say that there is no practical error ellipse associated with the fix derived from the crossing?? 

    There is absolutely no question that there is a valid navigational meaning to an error ellipse for two sights. And yes, that's in a nautical sense. You just need to change your methodology ever so slightly. In addition, three and four and five sight fixes make more sense when a prior estimate of the accuracy of individual sights is considered an input to the process rather than an output. 

    Frank Reed
    Clockwork Mapping / ReedNavigation.com
    Conanicut Island USA

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