Bas, Hüseyin Gazi 
The diferantial equations for the influence of small errors of inner and outer orientation on the 
image coordinates x, y are 
dx = dxo - (x/f)df ^ (f/z)bx + (x/z)bz * (xy/f)do - f (1+x2 /f 2)d¢ + ydx 
(1) 
dy = dyo - (y/df * (f/z)by * (y/z)bz * £ ( 1* y?/P) do - (xy/f) do - xdx 
(2) 
where dx and dy represent the differences, at photo scale, between the corrected and measured 
image coordinates of any point; x and y are measured comparator coordinates of point; f is 
principal distance; dxo and dyo are the error of the determination of position of the principal point 
from the fiducial marks in the photograph; df is the error in the principal distance; do, dó, dx are 
small rotations about the X, Y and Z axes respectively; bx, by, bz are translations of the camera in 
the X, Y and Z directions; and Z is object istance. In a similiar way differantial formulas can be 
derived for arbitrary cases of photogrammetry. The differantial formulas of convergent case which 
is fundamental importance for the terrastrial photogrammetry have been derived following: 
dx = dxo - (x/f)df * (f/z)bx + (x/z)bz + y/f (x coso - f sin )do - f (1*x? /f 2)d6 * y/f( f cosb * x 
sino )dx (3) 
dy = dyo - (y/8df * (f/z)by + (y/2)bz + {(x/f sine ) - (1+ y2/f cost )}f do - (xy/f) dé - {x cose - f 
sino (1* y?/P )}dx (4) 
In equations (1) and (2) the unknown orientation errors dixo , dyo, df, ...... etc. are solved by means 
of the same equations to correct the measured image coordinates. Control points are used in 
solving these elements. Then corrected photo coordinates of all points on photo are obtained by 
using these elements in matrix equations (5). The detail of application is explained in the Practical 
Application paragraphs. 
  
  
  
  
b X X dxo 
- f/f* P y cf by bz/ y |-dflf y + dyo 
(3) 
f 0 0 0 
Where f* = dD; (xy, IT 
dD; = (do -do 1) 
x*,y*- corrected image coordinates 
X,Y = image coordinates measured in comparator and corrected for systematic 
errors. 
4 PRACTICAL APPLICATION 
The present mathematical model has been applied to the normal case of close-range 
photogrammetry. The used data in this application have been obtained from the test field the 
detail of which was explained by Müftüoglu (1980). The application has been performed following 
steps: 
a. 25 test points were chosen from the test field. 
b. Theoretical correct image coordinates of these points were computed by using the 
geodetic coordinates of related points and the interior and exterior orientation data of 
camera. The equations (6) and (7) have been used to compute these coordinates. 
aii( X - Xo) * aio( Y - Yo) * ais( Z - Zo) 
  
40 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 
  
(6) 
(7) 
Table 
correc 
photo 
  
Tesi 
poin 
numb 
  
3- 
  
3- 
  
6- 
  
9- 
  
9-