| Résumé : |
(Auteur) An important task in geodesy is determining points. The classical geodetical methods for the solution of point determination problems are predominantly based on the measurement of angles; the related mathematical theory based on trigonometry is called triangulation. Nowadays, electronic measurement of distances offers the possibility of determining distances very easily and accurately. The mathematical theory concerned with solving a point determination problem using-distances (instead of angles) is called trilateration; it is based on algebraic, geometrical, and numerical methods.
A trilateration problem {m, n, p, q} consists in determining the relative position of m pairwise distinct "observation points" F1,...,Fm and n pairwise distinct "target points" E1,...,En from the m-n distances lEiFjl, p distances among the observation points, and q distances among the target points ; i. e. the coordinates of all the points have to be determined in an appropriate cartesian coordinate system.
If all observation and target points are situated in a plane the trilateration problem is called planar, otherwise spatial. In a spatial problem the observation points F1,...,Fm usually denote terrestrial points and the target points E1,...,En denote artificial satellites.
This thesis is concerned with finding geometrical relations between any two incongruent solutions {F1,...,Fm ; E1,...,En} and {F'1,...,F'm ; E'1,...,E'n} of the general trilateration problem {m, n, p, q}. To that end, the m-n distance equations lEiFjl=lEi'Fj'l, p distance equations lEiFjl= lE'iF'jl, and q distance equations lFj1Fj2l= lF'j1F'j2l (1 i1 i2 < n) are investigated.
First of all, analytical relations between any two incongruent solutions of the general trilateration problem {m, n, p, q} are derived from the m-n distance equations lEiFjl=lEi'Fj'l. For the geometrical interpretation of these results it is necessary to distinguish between solutions in non-degenerate and degenerate position.
In both cases, most of the geometrical relations are derived by using important results of the theory of confocal quadrics. The theorems of IVORY and GREENHILL/CAYLEY are particularly important in this context. For the first time they are formulated for a general system of confocal quadrics. |
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