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    Re: Applications of Complex Analysis to Celestial Navigation
    From: Andrés Ruiz
    Date: 2009 Nov 30, 12:08 +0100

    1.       The output for your examples using my source code in [NavList 10248] is:


    CoP1 = 101.491917 -7.858417   28.041667

    CoP2 = 147.297542 -7.810361   33.427778


    Zp1 = (-0.173620, -0.853988 i)

    ro1 = 0.600366

    Zc1 = (-0.325224, -1.599683 i)

    r1 = 1.454431


    Zp2 = (-0.733942, -0.471227 i)

    ro2 = 0.538132

    Zc2 = (-1.213898, -0.779383 i)

    r2 = 1.215208


    d = 1.209394

    mu = 0.218318

    nu = 0.964517

    Z1 = (-1.754769, -1.867588 i)

    Z2 = (-0.172382, -0.153304 i)


    I1 = 47.366215, -133.216088 = 47º21’58”, -133º12’58”

    I2 = -64.019435, -138.352317 = -64º01’10”, -138º21’08”


    w = (0.134118, -0.355732 i) = 0.380175 e -1.210256 i

    a = (0.996566, 0.082808 i)

    b = (-1.934951, 0.160782 i)

    mb = 1.941620

    z = (0.968460, -1.842040 i)

    w = (0.134118, -0.355732 i)


    62.266667   38.670278   35.500000   -9.500000


    Hc = 48.368899 = 48º22’08”

    Z = -69.342564 = 290º39.4’


    z = 0.513661

    z = 0.463230

    z = 0.504768

    z = 0.463433

    LD = 102.658496 = 102º39’31”





    2.       I am working in a new release of CelestialFix.exe, and it includes the following methods.




    3.       Mathematics vs. navigation consistency

    One remark Robin, for double altitude sight you take an example from “A Treatise on Nautical Astronomy - John Merrifield”, (see attached pdf. Page 241 in missing, I don’t have it). Is a running fix example, taking into account by a correction for Ho by V*(t-tfix)*COS(Z-R) = 14.17433195’. Although this is and approximation, is enough within certain limits, (the exact solution is: (47.373030146499410, -133.202318631342877)  ( 47º 22.4' -133º 12.1') )

    But the important is to note that only one of the two mathematical solutions is correct in a running fix, the one near the initial position for taking into account the motion of the observer. (I speak not of position or fix but mathematical solutions).

    Taking this initial position near the other intersection we obtain this solution: (-63º 33.7' -139º 27.9')

    And (-64º01’10”, -138º21’08”) should be discarded.


    Of couse the fix is the solution near the DR position.


    Andrés Ruiz

    Navigational Algorithms






    -----Mensaje original-----

    De: robinstuart@earthlink.net [mailto:robinstuart@earthlink.net]

    Enviado el: sábado, 28 de noviembre de 2009 22:52

    Para: NavList@fer3.com

    Asunto: Re: [NavList 10839] Applications of Complex Analysis to Celestial Navigation


    I have put together an Excel spreadsheet that demonstrates the use of complex numbers in calculations performed in celestial navigation. The examples are taken from the paper Applications of Complex Analysis to Celestial Navigation, available at http://www.fer3.com/arc/img/110015.articlec.pdf.


    Each sheet consists of three sections; Inputs, Calculations, Results. The values of the Inputs are initially set to those that appear in the paper, above, and confirm the results that appearing therein. The inputs may be changed to perform the same calculations for other sets of observations.


    The spreadsheet uses the Excel Engineering functions IMSUM(), IMPRODUCT() etc. to the perform the complex number calculations. To ensure that these are available to the system, on the Excel toolbar select Tools>Addins and check "Analysis ToolPak".


    The Excel implementation using functions IMSUM(), IMPRODUCT() etc. to perform arithmetic operations on complex numbers means that formulas that appear here are less compact and transparent than they are in computer languages, such as FORTRAN, C++, PERL, in which complex numbers are a built in native data type.


    Robin Stuart


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