Known are the length in a triangle ABC of a tetrahedron ABCD, i.e., also all angles between A, B and C are known. The position of D is, however, unknown, but the angles between all edges to D, i.e. between DA and DB, DA and DC and DB and DC, are known as well. Is it possible to determine the position of D, or do I need an additional information. If it is not unique, how many solutions exist?
The problem, as you state it, is overdetermined, i.e., there might be no solution at all. I can prove this relatively easily, but not in this comment; it might take a brief note of a 2-3 pages. When there is a solution then, barring denenerate cases, it is unique and the position of D can be worked out. Again, that would make part of the 2-3 pages which I might write if somebody provokes me enough :-)
@ Stan Sykora: I have to admit that I am only interested in mathematical solutions which are possible :-), i.e. the ABC and the angles from D to A,B and C are meaningful values. Unfortunately, until now I do not find any analytical solution. Better said: I found one (from photogrammetry: Killian, K. Über das Rückwärtseinschneiden im Raum Österr. Z. Vermess., 1955, 43, 97-104, 171-179 ) which describes with 4 equations 3 variables (which I would call overdetermined system) but if derived (two angles) and inserted back in the 4 initial equations in order to determine the z-coordinate of D it delivers 2 slightly different solutions which can be optimized to a single one but this needs a a numerical refinement. I hope that I simply need an additional conditions or a totally different approach. Your idea and help is really appreciated!
Oh, I totally forgot that the mentioned solution in Kilians paper refers to 5 known points and not only to 4 points. However, even then it does not deliver a unique solution.
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