112 
The square sum [vv] of the residuals can be obtained as indicated in all textbooks 
on adjustments. 
We find: 
[dx]? + [dy]? 
fur] = [dede] t [dudul — zn 
Ng 2+ Ng? — (5N,3—4[dæ])2 + (5N,4 — 4 [dy])? 
LE BE i (36) 
Tr ies : 
y ee = 5 y/o (37) 
The same procedure can be applied to other combinations of points. If the points are 
chosen so that within each combination they have the same distances from the center 
point 33, the radial distortion will result in chan- 
ges of dz, between the different combinations. 
In this way a very convenient method for the 
determination of the distortion can be performed. 
See Diagram 1. Of course, the approximate orien- 
tation of the projector has to be close to the ideal 
orientation. In other cases the differential for- 
mulas (12) and (13) are no more strictly valid. 
The same procedure can also be used for the 
determination of the distortion of an aerial 
camera from aerial photographs of a test area. If 
the control points of the test area are located in 
the corners of a grid as shown in Figure 2 and 
the pictures are taken near vertically over point 
33, the procedure as described above can be 
directly applied. Normal equations have to be 
formed and solved for other combinations of 5 
points and also 9 points. 
If the control points have elevation differen- 
ces, the resulting displacements have to be cor- 
rected. For this purpose a preliminary determi- 
nation of the dy and do can be necessary in order 
to find the nadir point. 
A detailed derivation of the expressions dz, curve for the distortion correc- 
above and similar expressions for other point tion of an instrument. 
combinations is given in [1]. 
Since all weight- and correlation-numbers have been determined, the error propaga- 
tion in any function of the adjusted quantities can be easily studied, with the aid of the 
general law of error propagation. 
In order to study for instance the accuracy of the corrections, computed from for- 
mulas (12) and (13) the general law of error propagation has to be applied if we do not 
prefer to express the corrections as direct functions of the measured quantities. 
The corrections of the elements of the inner orientation can be found from (25)— 
(30) and (5)—(9), assuming the outer orientation as known. If the procedure is repeated 
for different values of h and the difference A h is accurately determined and the camera 
or projection plane is moved accurately parallel to itself, the inner orientation can be 
completely determined. The accuracy of the outer orientation has in such a case to be 
determined with other methods. 
  
Finally we find 
  
Diagram 1: 
   
  
   
   
     
    
   
   
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
   
   
   
   
   
   
   
   
   
   
   
    
   
    
   
      
The : 
errors in 
tions sho! 
cameras € 
taken int 
The : 
rays can | 
his funda 
  
Dist 
of d 
The c 
the residu 
the accurs 
The s 
different 
procedures: 
For tl 
tions have 
An examp 
of the st 
projector 
Formulas 
For c 
distortion 
is and how 
the system