Ini 78 ADJUSTMENT OF RHOMBOIDS, ACKERMANN
Guided initially by this idea the author evolved some forms of approximate solutions
for the adjustment of rhomboids in connection with students’ exercises at the L.T.C.
Furthermore he was encouraged in developing the following concepts while assisting in
the preparation of a block of radial triangulation carried out at the Geodetic Institute
of the Technical University in Delft. The main considerations which can guide the search
for new methods of rhomboid adjustment and can pave the way for other solutions than
those presented here are summarized as follows:
The least squares solution is of no essential importance with regard to an increase
of accuracy as the ten observations contain only one superfluous observation. This is also
emphasized by the fact that in a rigorous adjustment method the weights and correla-
tions of the observations should be properly considered which is not the case in practice
up to now. Moreover, so far, few efforts have been made to correct for the systematic
influence of tilt of the photographs which should be done before entering into adjustment.
If, in spite of all this, a rigorous least squares adjustment with the simplified assump-
tion of equal weights is applied in practice, and kept as basis throughout this paper, it is
due to the methodical elimination of the discrepancies which is the second important
property of least squares solutions. Hence deviations from a rigorous adjustment can be
accepted if in return simpler and shorter computations are gained.
A possibility for simplified adjustment methods, not used so far, is offered by the
fact that the ideal rhomboid is square shaped. This ideal shape may serve as a first
approximation or, preferably, can be used in the coefficients of differential quantities.
Another consideration is that the angular adjustment of a rhomboid serves only as
an intermediate step to find the coordinates of the points C, D, and E and has no
independent importance. In this regard straight-forward solutions which yield the
required coordinates directly should be preferred.
In the following sections of this paper some of the possible solutions for the adjust-
ment of rhomboids according to the above-mentioned general points of view are present-
ed. They are marked by different degrees of deviation from the rigourous least squares
solution and may be given preference depending on the shape of the rhomboids, the
required accuracy, or the available computational means.
The additional computations in radial triangulation, the formation of chains and
blocks of rhomboids and their adjustment are beyond the scope of this paper. The treat-
ment of these problems does not depend very much on the adjustment method applied
for single rhomboids.
2. Adjustment of indirect observations.
Looking for a direct solution for the coordinates of the points forming a rhomboid,
proceeding from the observed directions, the method which comes first to mind is the
least squares adjustment of indirect observations (Standard Problem II according to
Wi Tienstra). This means introducing nine unknowns for each rhomboid: three orientation
unknowns for the sets of directions and two planimetric coordinates for each of the points
C, D, and E. The points A and B are supposed to be known or can be assumed arbitrarily.
| This adjustment problem turns out to be very simple if the ideal rhomboid is taken
| as an approximation at the start. In this case even a general solution of the normal
equations can be worked out which would give the coordinates of the corner points of
the rhomboid directly from the observations, without any intermediate or preliminary
i computations. Hence it would be an ideal solution from a methodical point of view.
i Unfortunately it cannot be applied. The actual rhomboids depart by far more than dif-
i ferential order from the square shape, therefore the assumed approximation is too much
| id of a simplification. The errors in the resulting coordinates would be in the order of 5%
a of the base b.
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