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2. Best fit among neighbouring sheets is guaranteed. 
3. Unique co-ordinates of points are guaranteed even if 
they appear in different sheets. 
This technique can be employed with help of different kinds of 
transformation depending on the type of the co-ordinate system 
(two dimensional, three dimensional), nature of deformations, 
precision of observations, geometry of the area, existence of 
other data, etc. 
  
  
= 
  
  
  
  
v 
<< 
Local systems 
State system R: 
R 
9 
5 
  
  
Figure 2. Local and state co-ordinate systems 
2.3 Principle of simultaneous adjustment 
Let R; be a co-ordinate system of the sheet i (where every point 
on the sheet has a known co-ordinates (x, y) and R, a state co- 
ordinate system. We pass from R; to R, by applying a co- 
ordinate transformation defined by parameters p, The 
transformation can be linear or non linear: 
The simultaneous adjustment minimises the relative 
discrepancies at tie points and the absolute discrepancies at 
control points. The result provides simultaneously the 
parameters p; of the transformations T;. 
For a control point j of the sheet i, the observation equation is: 
X TG (1) 
where X; is a known co-ordinate vector (X Y 2)", of the point j 
in the state system and X; is a known co-ordinate vector (x y z)! 3 
of the point j in the local system i. 
For a tie point j appearing in sheet i, the observation equation 
is: 
X- T (x3) 0 (2) 
where X; is a unknown  co-ordinate vector of the point j in the 
state system . 
The observation system can be set as: 
Ap * Bc-e (3) 
Generally the system is overdetermined and a least squares 
solution is applied. The normal equation system is given by: 
A" Ap+A"Be=A"e 
B' Ap+B'Bc=B'e 
or (4) 
N,1p+Ny2e=f, 
N 'op*N»c-f; 
where p is the transformation parameter vector and c is the 
unknown co-ordinate vector of all tie point in the state system 
R,. p is obtained by: 
p-(Nu-NoN5 Nl)! (f-NuN'5£) (5) 
and c is obtained by: 
c-7N5S f; NS Nf. (6) 
After transformation parameters are calculated the new co- 
ordinates of all points n of each sheet m may be obtained in the 
common system R, by: 
Xn 55 T (Xn)m (7) 
3. APPLICATION 
3.1 Practical considerations 
Control point j appearing on sheet i has measured co-ordinates 
(X;,y;) in the system i and known co-ordinates (X, Y).in the 
reference system. Tie point j appearing on sheets k and | has 
measured co-ordinates (x;,y;), in the system k and measured co- 
ordinates (x;,y;) in the system |. For each tie point the number of 
measured pairs of co-ordinates corresponds to the number of 
sheets on which it appears. Its unique position in the reference 
system (X, Y;) is unknown. 
The procedure allows application of various types of 
transformations (Helmert, affine etc.). After a series of tests we 
decided to apply affine plane transformation because existence 
of affine paper deformation became evident. 
The observation equation for point j on sheet i is given by: 
X 9 bx CX 
+ 
= (8) 
Y, c d)\y " CY) 
or in another form: 
a 
b 
oss 0e 01 1 0 C 
x 90 x y 6 1// à ©) 
CX 
CY