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    Re: LOP's without DR Position
    From: Hanno Ix
    Date: 2015 Feb 23, 22:40 -0800

    May I discuss briefly the problem itself?

    It appears that  stereographic projection would have solved the so called Direct Problem in a rather "straight forward" way. Assume we have the altitudes Hc1 and Hc1 that each define a circle on Earth's surface around GP1 and GP2, and we want to know their intersections if they exist.

    1: Please consider Earth as Riemann's sphere. Than the two circles around GP1/2  can be projected upon a complex plane tangential to the the sphere by stereographic projection. These projections will be, as we know, circles in the complex plane and therefore coplanar. Now we find the intersection of these coplanar circles - a simple process - and finally re-project the intersections back to the sphere, i.e. the Earth.

    2. There is a simplification possible: if we let an arbitrary point P on circle C1 of the pair C1, C2 of circles on the sphere coincide with the point where the complex plane touches the sphere, as it were, then one of the coplanar circles in the complex plane deteriorates into a straight line. C2, however, will again be projected as a circle - again coplanar with the straight line. Now we find the intersection of a straight line with a circle - an almost trivial problem - re-project them upon the sphere, the Earth.

    I assume this approach might already have been taken. I might be forgiven if I described this not very clearly and I blush because I have not done the work in detail or might have made an error. But I believe the process I described is correct.

    May ask if you agree?


    On Mon, Feb 23, 2015 at 12:11 PM, Yves Robin-Jouan <NoReply_Robin-Jouan@fer3.com> wrote:

    Hello Paul Bedel, Antoine Couette, and All of you,

    I had a number of contacts with Georges Bodenez, when and after I have published my own method (Planar Vertices method, 1995).

    I appreciated very much every dicussion I had with him. I did again a number of his original examples.

    Often he said himself that the basis of the 3D methods have been set by Antoine Yvon-Villarceau in his famous book on 1877: "Nouvelle navigation astronomique : théorie et pratique". "New" means using chronometers but no longer Lunar Distances. This book -which is a classical one- gives a lot of justifications for Marcq Saint-Hilaire method but, in a preliminary part, Villarceau set a very clever formulation of the 3D problem. See my enclosed slide, from a conference given at Maritime Museum, in Paris. 

    I give hereafter some translation of the original text (1877!) :

    "... so that the 2  first equations formulate 2 planes and the last equation formulates the (Earth) sphere itself. Then the solution is given by the intersection of 1 straight line and 1 sphere. But for practical reasons, we do not recommend to use these formulas, because of the length of the implied computations. We present a less DIRECT solution, but quicker to implement".  Obviously, " less DIRECT" was LOP intersection method...

    You can realise that, if you add a 3rd equation to the 2 first equations, you get a linear system: this has been the essential complement brought by Georges Bodenez in the seventies (one century later, indeed!).

    I know that a lot of people were working about the same subject elsewhere, noticeably in the move of first receiver design for GPS (to begin with Ivan Getting team).

    Notice that Villarceau has a street to his name in Paris, but Marcq Saint-Hilaire has not...

    By the way, Villarceau has been a rival of Le Verrier, inside Paris Observatory.

    Yves Robin-Jouan


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