Critical configurations of Lambert's 6-station problem
Title:
Critical configurations of Lambert's 6-station problem
Author:
Killian, Karl
Appeared in:
Marine geodesy
Paging:
Volume 2 (1979) nr. 4 pages 323-335
Year:
1979
Contents:
Let A,B,C denote three mainland-stations whose positions are unknown. Let D,E,F denote the positions of a moving ship at three different moments of time. At any of the locations D,E,F, the three azimuths to the target-stations A,B,C are measured. The problem is to determine from these nine measurements the mutual positions of all stations A,B,C,D,E,F up to a common shift and a change of scale. The problem was formulated by J. H. Lambert in 1765. He also specified a mathematical solution. In this paper so-called “critical configurations”; are investigated. It is shown that there are nontrivial configurations of the six stations A to F, such that solution of the problem is not unique (critical configuration of the first kind). There is a larger set of configurations, such that the linearized problem admits of an infinity of solutions (critical configurations of the second kind). In the latter case, the original nonlinear problem may be solvable, but the solution is highly unstable with respect to perturbations of the measurements. The main result obtained is as follows. If all stations A to F are located on a second degree curve, i.e., on a conic section, then the configuration is critical, at least of the second kind. The configuration continues to be critical if arbitrarily many observations and target-points are added which are all situated on the second-degree curve.
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