under point 3 are of systematic character and are to be determined from the measure-
ments and computations.
2.1. Corrections due to lacking parallelism between the control plane and the plane
1-2-3
Lack of parallelism can be caused by two rotations d and d£ around axes
which are identical with the co-ordinate axes x and y (see Fig. 1). Since the two
planes can always be adjusted approximately parallel, the angles dz and d£ can
be regarded as small. A small translation dh, is also introduced, see Fig. 1.
The elevation error dh in an arbitrary point, and determined as dh — hy easured —
lyiven Can be interpreted as caused by the translation dh, and the rotations dn and
d£ according to the formula (see Fig. 1)
dh — dhq4 xd ^ yd£. oe sail)
Applying the expression (1) to the points 1-3 (Fig. 1) we find
dh, = dho + x dy + y, dé,
dh, = dhy + xod - y, d£, > 42502)
dhs = dhy + x3dn + y 3 dé.
Fhe errors dlig, dy and dé can now be determined as direct functions of the elevation
errors dh, — dh4 in solving the equation system (2).
I he condition for the solution is evidently
LX, Ji
b xe di:
| x3 Ys
This condition means that the three points must not lie on a straight line.
The corrections to the preliminary values of hy, » and £ are obtained after
changing the signs of one side of the equation system (2).
After solving the equation system we then find the corrections
= dh,(X2 V3 — X3 V2) + dha(X3 V1 — X1 Y3) + dh3(x1 V2 — X2 V1)
dh, — : s 340)
V10X2 — Xa) + yo(Xa — X1) ya(1 — x2)
dm = dh,(y2 — y3) + dho(y3 — V1) + dh3(91 — V2) (d)
yi(Xa — xa) - yo(xa — x1) - yaQ — x2) 3 S
di dh,(xa — x3) - dhs(x4 — xa) + dhy(xs—x3). (5)
yi(Xa — xa) yo(xa— X3) t ya(X1 — X3)
After substituting the expressions (3) - (5) into the expression (1) corrections
dh. can evidently be computed to arbitrary points with the co-ordinates x and y.
The final elevations A,4 are then determined from the expression
Aginar = Ameasurea + Ahcorr- 2.06)
Then the final elevations can be compared with the control elevations and the
elevation discrepancies can be directly determined.
2.2. The accuracy of the procedure ;
The accuracy of the determination of the final elevation discrepancies is evi-
dently identical with the accuracy of the Anna if the control elevations can be re-
garded as free from errors. It is consequently of basic importance for the judgement
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