Die Ausgleichung und Genauigkeit Eines Polygonzuges mit Lagengemäss Ungenauen Anschlusspunkten

Summary In adjusting a bilaterally connected polygonal traverse, whether to positionally accurate or inaccurate points of departure, the mean errors are the problem of primary importance. Hitherto, the mean errors of lengths and angles were chosen; the procedure was such that the weights of both quantities were determined from the mean errors chosen, and this fixed the ratio of the mean errors. This ratio did not change as a result of the adjustment, but the absolute values of the mean errors did. Provided the adjustment was carried out on a polygonal traverse with fixed points of departure, this change did not matter. In the case of ellipses of errors in the points of departure, this change is not permissible, because it would include the change of the semi-axes of the ellipses of errors and, therefore, also of the positional rigidity of the points of departure. The contribution of this paper is in the exact method by which it is possible to compute a coefficientc, pertaining to the mean angular error selected (in the case of positionally inaccurate pointsd), which determines the mean errorm=±e √s (orm=±d √s) of the lengthss. The solution is based on the definition of the mean error of a unit weight founded on the work of deformation. In the calculus of observations, the work of deformation has so far been determined as the deformation work of internal forces from the corrections of the individual quantities after the adjustment is concluded. However, it is possible, as was demonstrated in this paper, to express it as the deformation work of external forces, which act during the adjustment in pointO (Fig. 1) on the auxiliary static system in the shape of a console, and which provide it with the necessary deformation. If the external forces are expressed by means of the tensor of the auxiliary system, the equation for the mean error of the unit weight will provide a relation between this error and the coefficientc, ord. If the mean angular error is selected and if its weight is put equal to one, an equation of the fourth degree is obtained in terms ofc (ord), from which it is possible to compute this coefficient. From the external forces, necessary to produce the deformation of the auxiliary system during its adjustment, the corrections of the individual elements of this system may be determined. If we want to determine the ellipse of errors in one of the polygonal points, it is necessary to investigated the shifts of this point, which occur if the point is acted upon by a unit force first in one and then in another direction, perpendicular to the first. Both shifts represent conjugate radii of the deformation ellipse. From this ellipse we proceed to the ellipse of errors. The general solution is supplemented by a numerical example of adjusting a polygonal traverse with positionally inaccurate points and, for sake of comparison, also by the adjustment of this traverse with positionally accurate points.

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Anschrift: Husova 5, Praha 1-Staré Město