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    Re: LOP's without DR Position
    From: Antoine Couëtte
    Date: 2015 Feb 24, 07:11 -0800

    Hello to all,

    You will find here-under synthetic overview about our "LOP's without DR position" problem:

    This problem (quite different from "Douwes's problem") can be summarized as follows:




    You are given a set of observations with ONLY:


    - A minimum of 2 observed celestial Bodies geocentric heights. And:

    - The UT times of these observations. And, only if relevant:

    - The course and speed of the Observer. And:

    - No more, i.e. not even any Observer’s approximate position (such as DR).


    When and by whom were FULL solutions - i.e. Latitude AND Longitude - FIRST published?




    From the numerous replies received here lately - thanks to all ! - I will attempt putting into some order our various contributions. It will also give me the opportunity to reply to queries I have left unanswered so far.



    The published solutions to our "LOP's without DR position" problem fall into 4 main categories:


    1 - 3D Geometrical solutions


    2 Observations


    2G.1 - Intersection of 2 Plans and a Sphere solved through Vector Equations (i.e. a system of 3 Equations in space).

    The first publication of this method seems to be the one by Antoine Yvon-Villarceau in 1877, i.e. 2 years only after the LOP Method credited to Marcq Saint Hilaire (1875). Many thanks to Yves Robin-Jouan for mentioning this information here:



    Our NavList Member Andrés Ruiz independently rediscovered such method (well done Andrés!) and is routinely using it. And also:


    In http://fer3.com/arc/m2.aspx/LOPs-without-DR-Position-HannoIx-feb-2015-g30453, Hanno Ix gives us one solution to our exact same problem through a NavList transcript by our late and dear George Huxtable of a Casio Calculator program dating from the 70’s. Thanks to you Hanno. From a quick look at its programming lines, this Program seems to be using method 2G.1 here-above. 


    3 Observations


    3G.1 - Intersection of 3 Plans (3 Celestial Bodies) solved through Vector Equations.

    The first currently known publication of this method is the one published by Professor Georges Bodenez in 1976.

    Re: http://fer3.com/arc/m2.aspx/LOPs-without-DR-socalled-Douwess-problem-Robin-Jouan-feb-2015-g30448


    3G.2 - Other properties of 3 Celestial Bodies solved in 3D through Vector Equations.

    One such method, i.e. the “Méthode du Plan des Sommets”was published in 1995 by our NavList Member Yves Robin-Jouan (Well done to you too, Yves!). While this methods starts from a quite different and distinct viewpoint than here-above, its formulae eventually boil down as closely related to 3G.1 Equations here-above. One of Yves’s merit is to have pointed out the “limiting case” of such 3D methods which happens when the 3 Celestial Bodies are on one same great circle.



    4 or more Observations


    4G.1 - 3D least square methods involving 4 bodies or more.

    I am not aware (yet) of such methods (maybe a refinement of G.3 here-above?)


    2 - Analytical Solutions (100% Calculus


    2 Observations


    These methods compute the intersection of 2 Circles of position.


    2A.1 - Carl Friedrich Gauss’s found one solution (before 1855). Was this method published then?

    This method is kindly mentioned by Andrés Ruiz here:

    http://fer3.com/arc/m2.aspx/LOPs-without-DR-Position-Andr%C3%A9sRuiz-feb-2015-g30459 and also here:

    http://fer3.com/arc/m2.aspx/LOPs-without-DR-Position-Andr%C3%A9sRuiz-feb-2015-g30446 where Andrés also gives us an excellent and impressive list of all Celestial Navigation Methods. Thanks again Andrés!


    2A.2 – Methods involving stereographic projection of the Circles of position from the Pole onto a plan


    2A.2.1 - In 2006 (or later) our NavList Member Robin G. Stuart (well done to you too!) published the following document where he makes a very clever use of complex numbers: Applications of complex analysis to Celestial Navigation


    2A.2.2 - And, cherry on the cake, Hanno IX just rediscovered this method yesterday! Well done again!



    2A.3 - In 1981 James A. Van Allen published the following document:


    "An Analytic Solution of the Two Star Sight Problem of Celestial Navigation."Journal of the Institute of Navigation, Vol 28, No 1 Spring 1981. Geoffrey Kolbe (Thanks to you Geoffrey) kindly mentioned this paper here:


    I do not know about his method. Would it be related to the stereographic projection method?


    3 or more Observations


    These methods compute the intersection of 3 Circles of position


    3A.1 - I earlier indicated (http://fer3.com/arc/m2.aspx/LOPs-without-DR-socalled-Douwess-problem-Cou%C3%ABtte-feb-2015-g30425 ) that to the best of my knowledge a French Air Force Colonel was the first one to publish an Analytical Solution for 3 Bodies.

    I could not find again this specific paper in my archives (too well stored!!!). Nonetheless, it looks like the following paper: http://cat.inist.fr/?aModele=afficheN&cpsidt=19474690 Solution for the general two bodies problem (Douwes' problem), through the resolution of canonical equations by G. Houneau published in the French Magazine Navigation 1991, vol. 39, no153, pp. 116-123 which I have not downloaded to-day.

    As indicated in its French abstract further down this paper claims to solve also 3 observations and even more observations. Hence there seems to be a definitive confusion in it: the problem this paper addresses and solves is not the Douwes’s problem but rather instead it is much more probably our exact “LOP without DR Position” problem.

    Whoever can shed more light on method 3A.1 is greatly welcome (Paul, Yves?).


    3A.2 – In his book “La Navigation Astronomique, Fondements, applications perspectives” (I own a copy printed in July 2000) earlier mentioned a few times on NavList lately, Philippe Bourbon published in Chapter XIV an analytical method processing 3 or more observations.


    NOTE: Moving Observers, and/or inaccurate heights and Observations on an Ellipsoid


    All the solutions here-above whether Geometric or Analytic immediately deliver - or should deliver - the same “exact” mathematical results fi the following hypotheses are made:


    Measures with no errors,

    Spherical Earth, and

    Bodies at infinite distances with zero parallax (i.e. stars)


    These methods need to be further refined to cope with moving Observers and “noisy” data.


    And finally, to accurately deliver their best results on the Ellipsoidal Earth with finite distance bodies geocentric heights (especially our so close Lady Moon), all these methods are believed to use some kind of iterations because under such circumstances, Parallax corrections depend on the Observer’s Latitude.


    3 – Local Apparent Noon Solutions


    These solutions are quite familiar to our NavList Community. Thanks to you Henry to have mentioned these methods here: http://fer3.com/arc/m2.aspx/LOPs-without-DR-Position-Halboth-feb-2015-g30452.


    Interesting to see that Calculators have revived these methods. Their classical (i.e. hand computations) limitations have now become much less constraining as regards minimum culmination heights and maximum duration around LAN. Mortimer Rogoff pioneered this field with HP-67 and HP-97 in remarkable book “Calculator Navigation” published in 1979.


    There are some variants to these methods (e.g. “Position from Observation of a Single Body” by James N. Wilson, London Institute of Navigation Vol. 32, N°1, Spring 1985 where James N. Wilson explicitly refers to Mortimer Rogoff’s book mentioned here-above).


    LAN methods seem to have been in use probably for one century now, if not more.


    4 - Other Solutions


    One such solution has just been mentioned by Henry Halboth here:



    In fact Sun Observations near the zenith are fun to do, as long as we can chase it oever the horizon and get an accurate Azimut. Only draw the Sun Center on the Navigation chart and report from it the correct Azimut line with the adequate length (complement to the geocentric height). This method has been in use certainly for many decades. And M. Robert Doniol as well as his son 4* Admiral Guirec Doniol also used it.


    There are certainly other methods and this list could be extended.


    4 – Conclusion


    Again, many thanks to all for the many contributions given. I am sorry that I have not given explicit examples, but the information given here-above should enable to retrieve or to build up such examples.


    While the list given here-above is certainly not complete, we can already see that there are quite a few manners to solve our "LOP's without DR position" problem.


    To conclude, I am still interested in knowing whether somebody published a method matching method 3.G.1 here-above before it was published in France in 1976.


    Thanks for your kind patience in reading all the way down, and


    Bets Friendly Regards to all




    Antoine M. “Kermit” Couëtte

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