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Bas, Hüseyin Gazi 
  
e The error in X changes by ratio 1 / cos2x. The normal case is obtained when angle a is equal 
to zero. The error in X increases with the increasing of a. The increase in a depends two 
factors : object distance Z and the base between two theodolite stations. If it is taken into 
account that the object distance does not change from point to point in close range 
measurements and that it is a generally small distance, the amount of a can be assumed to be 
directly related to the base. 
e The error in Y is related to both horizontal and vertical angles. The error increases with the 
increase in these angles. Therefore small values of vertical angles should be preferred to 
reduce their effect on the error. 
e The error in Z increases as the object distance and horizontal angle increase. This error 
decreases as the base increases. 
The positional error Mp obtained from equations (8) is Mp= (m?x * m?y* m?z !/2 that is , 
Mp = Z / cosa((/ cos?a) * (1/(V2 cos?B) + tang tano)? + (V2- Z/B (1/ Cosa))211/2 -m 
(9) 
Furthermore, if the geometry of intersection is taken into account and if Sıl=S2l (I is the 
intersected object point); B/Z- 2tan(y/2) and Z/B = %cot((y /2) are obtained (see Ghosh, 
1979). Substituting (Z/B) into equation (9) 
M» -Z- [1/cosa {7% cos2a (1 + %cot2(y/2)) + % cos2ß (1+% (tan tana) + %(tan2ß - tan2a)}!/2] - m 
(10) 
is obtained. Equations (8), (9) and (10) can be considered as the general formulas expressing the 
object space coordinates errors as a function of the theodolite reading error (m), horizontal and 
vertical angles (a,f), errorless base (B) and the object distance (Z). 
S OPTIMUM CONFIGURATION 
As mentioned earlier, the least coordinate and the least positional errors can be obtained by 
minimizing the object distance Z, maximizing the base (B) and minimizing a and P angles. 
However, as the base increases, the horizontal angle a also increases. Therefore a base length 
which can keep the values of a between 0-45 degrees should be preferred in close range 
measurements with theodolite. Furthermore a and f are not a single parameter for measuring 
system, the value of these angles change from point to point. Therefore to get an optimum 
configuration by using the partial derivations of function (df/da=0, df/dB=0) in close range 
measurement with theodolite is a difficult operation. 
In this study equations (8), and (9) have been solved for every two meters of distances of object 
and the base, and for every five degrees of horizontal and vertical angles, and the theodolite 
reading error m have been assumed as 2 mgon. Some of the results are listed on Table 1. 
Table 1. Object Space Coordinates Errors and the Positional Error for Some 
Parameters 
  
os |pe Z B mx my mz mp 
  
0 5 2 2 
  
  
  
  
  
  
  
  
  
4.44x10 9 |4.44x10-6 |8.89x10-6 | 1.09x10-5 
  
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 49 
idee I)