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    Vector Solution for the intersection of two Circles of Equal Altitude
    From: Andrés Ruiz
    Date: 2008 Apr 8, 15:28 +0200

    Dear George,


    Yes, my paper: "Vector Solution for the intersection of two Circles of Equal Altitude", (not equal amplitude) is in The Journal of Navigation, 2008 Volume 61, Issue 02  April 2008, pp 355-365. http://journals.cambridge.org/action/displayJournal?jid=NAV


    The Abstract and the appendix not published, (There is always an intense competition for space in the Journal and, the RIN ask me to save space), is at my web site, in the [Papers] section.


    I think there is no omission, the Running Fix algorithm is based on: “Metcalf, T. R. Advancing Celestial Circles of Position, NAVIGATION, Vol. 38, No. 3, Fall 1991, pp 285-288.”

    And the appendix is clear enough after reading Metcalf´s paper. It based on iterating until the error in position between two consecutive solutions is less than 10-6, this is the key.

    First thing I did when late in 2006, you kindly sent me your paper, was to check my running fix algorithm against your test. I have attached the results, also the solution by the NA SR algorithm and by Kaplan. The program CelestialFix.exe is available at the download section, and includes the vector solution for 2 CoPs.


    Although the central issue of my paper is to present a vector solution to the two body fix problem, I have also treated the running fix problem, but this is a generalization, you can use any solution for the intersection, like Van Allen´s and the final result is the same.


    For me the best way to solve a running fix problem, is using Kaplan paper, (Kaplan, G. H. 1995: "Determining the Position and Motion of a Vessel from Celestial Observations", Navigation, Vol. 42, No. 4, pp. 631-648.), including the motion as a part of the problem. It is the natural way for many problems in engineering.



    Yours sincerely,


    Andrés Ruiz

    Navigational Algorithms






    -----Mensaje original-----
    De: NavList@fer3.com [mailto:NavList@fer3.com] En nombre de George Huxtable
    Enviado el: martes, 08 de abril de 2008 0:31
    Para: NavList@fer3.com
    Asunto: [NavList 4815] Re: Your paper in JoN.




    In Navlist 4812 I posted the following message. I didn't intend to; it was

    the result of a bit of finger-trouble, as my intention was to send it

    off-list directly to Andres Ruiz. . However, no harm done.


    But I didn't explain the topic that Andres was writing about, in his paper

    in the latest issue of Journal of Navigation, so this message is to fill in

    that omission. It was "Vector solution for the intersection of two circles

    of equal amplitude", by Andes Ruiz Gonzalez.


    And the bit I was referring to was his section 2.3, "Correction for the

    motion of the observer", which stated-

    "When the two sights are not taken at the same time, it is necessary to move

    the first CoP [circle of position] to the time of the second one, or both to

    a common instant.Many methods have been presented, but the correct way to

    move a CoP is to advance or retire the GP [Geographical Position of the

    first observed body] due to the motion of the observer [refs 6,7]


    Using the technique described in reference [6], the correction is a function

    of the estimated position of the observer (Be,Le), and his motion: course

    and speed [C.S]. And since what we are looking for is the true position, an

    iterative process is required in order to reach the solution for the running

    fix. The algorithm is described in the Appendix."


    The appendix gives a block diagram of the decision boxes rather than a



    I have little doubt that Andres understands well how to tackle this

    question, and get the right answer, as he has explained it to us on Navlist.

    What I am questioning is his use of the term CoP, or circle of position, to

    describe the non-circle of the transferred locus of position. Presumably the

    program calculates the intersection of the second circle with that

    non-circle, or else with a different circle, which has its centre displaced

    further, so that it's tangential to the locus in the region that matters.


    This is what I wrote earlier-




    Dear Andres,


    Pleased to see your paper in Jounal of Navigation.


    Coming from an earlier generation, I didn't really take in vector

    mathematics as applied to such problems. So much of your paper went over my



    What I want to ask about are your words in 2.3 about correcting for the

    motion of an observer.


    An observer who has measured the altitude of a star to be, say 30 deg, knows

    that his locus is on a circle radius 60 deg centred at the GP of the star,

    (say, at dec = 0, GHA = 0, as in the example in paragraph 4 of my paper in

    JoN, 59, No3, pages 521-524, which I think you have). And then the  observer

    moves a known distance, say 60 miles North. His new locus is a closed

    figure, somewhat egg-shaped, but NOT a circle any longer. Does your

    algorithm in A2 find the crossings of that non-circle with another circle

    centred on another star (say, at dec = 1deg N, GHA = 45 deg W, seen at 45

    deg alt, as in that example).? What result does it give?


    Yours, George.


    contact George Huxtable at george@huxtable.u-net.com

    or at +44 1865 820222 (from UK, 01865 820222)

    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.


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