yz.
n=
| two
(4.1)
(4.2)
4.4)
lace
ms)
the
sing
»dt
tion
end
f a
oef-
sful
xpe-
stly
Lion
5.1)
for
—
ADJUSTMENT OF RHOMBOIDS, ACKERMANN 81
t
Its solution, K = 64 directly yields the following corrections to the observations
Uy = agio = : ud + 0 = + Vg = ++
32 2 82 32 9 16
U 4 t V4 = — : mm t = : : eo
2 16 4 82. "e à 5 Tn
Realizing, at this stage, that with a slight extension of the computations necessary
to get the linear discrepancy Ae, preliminary coordinates vor
0 ?
xp and £i of the points C, D, and E become known, it is obvious that the least squares
D )
"Ep Ups Tp» Yy
0 0 0 € €
solution (5.8) can be carried through to give corrections to the preliminary coordinates
directly. To achieve this, the formulae for the sine law and the coordinate computations
have to be linearized. Assuming again, for the coefficients in these linear relations, those
values valid for the ideal-shaped rhomboid, the resulting corrections (expressed in the
coordinate system indicated in Fig. 1) will be
C le d le
de, = — . da — . des. = da, —0
( e 4 D e À E E,
5.4a
c le : d le ( )
dy, = — m NY + . dy, — — 1 4e Y, = +4
( e 16 D e 16 Vg PA Wr * à 4e
Hence
qc Bor F dr, xp =, t de; tn = 4 (x, + x, )
( 0 ) D
= 1 (5.4b)
Voy, + dyg yp y, t dy, Ya (y, ty, )
0 0 C D
Summarized, the computations required for this simplified rhomboid adjustment are:
a. Computation of the preliminary coordinates of the points C, D, and E using the
observed directions. This gives as a by-product the linear discrepancy Ae.
b. Correction of the preliminary coordinates according to (5.4), if necessary.
In this method the main amount of work consists of computing the preliminary
coordinates. The adjustment then, which itself is hidden, and does not show up any more
as a separate step in the computation, yields in a very simple way the corrections to
the preliminary coordinates.
It should be noted that this simplified adjustment method approximates much more
closely to the rigorous solution than the method mentioned and abandoned in section 2,
because the preliminary coordinates, on which the adjustment is based, give a much
better approximation to the actual shape of the rhomboid. From experience it can be
taken that the linear discrepancies Ae are normally in the order of 10—3 to 10—4 parts
of the base b. Therefore this adjustment method should cover most practical cases. It
can be fully recommended, if the rhomboids are well shaped or alternatively where the
accuracy of the solution is of less importance than a direct and simple solution for the
coordinates.
Remark. It should be noted that the size of the linear discrepancy Ade shows immediately
if the adjustment is necessary at all. It serves as a criterion to judge whether the
corrections to the preliminary coordinates will be significant, compared with the
required or obtainable accuracy.
6. Adjustment of rhomboids which depart further from the ideal shape.
In order to study the limits up to which the solution presented in section 5 is
