2 DERIVATION OF THE PRINCIPAL DISTANCE FROM THREE PHOTOGRAPHS OF A 
PLANE OBJECT 
2.1 Direct Relative Orientation of two Metric Photographs 
If four object points forming a plane can be measured in two metric 
photographs, a direct rectification of the object plane is possible 
without iterations. This problem vas verified by Wunderlich /7/ using 
matrices, and Killian /3/ using trigonometric relations. Both methods 
solve a cubic equation and deliver two results. The correct result 
has to be pointed out by analysis of the rectified data and 
plausibility tests. 
These algorithms may be modified for calculating the relative 
orientation /2/. The derived parameters (coordinates of the 
projection center, rotation angles, local object coordinates) can be 
applied as approximate values in bundle adjustment. To get the scale 
of the model the distance between two object points has to be 
measured. 
2.2 Orientation of three non-metric Photographs 
Three photographs of a plane object are necessary to calculate the 
principal distance (c ) if the principal point (x ,y ) lies in the 
center of the image. ^ 9.4.9 
By combining two photographs at one time, three relative orientations 
can be calculated with 2.1 /2/. In the first step any value that is 
smaller than c may be taken as principal distance Cc,. Each 
combination defivers four rectified object points (Fig. 1). The 
difference between the coordinates of: points 315: 3", 3" and 4!, 4", 4" 
may be used as an indicator for the descripancy of the principal 
distance e from the real value e. 
  
  
  
aa 
— 
1'=1"=1" 21-2422" X 
Fig. 1: Three rectifications in one model coordinate system 
  
Ax; . (k) = x. (k) - x. (k) i-2,3,1 
Ay: (k) = y! (k) - y; (k) j=1,2,3 .. image (combination) 
J J k=3,4 .... point number 
9 2 Xy: eis «is rectified object coordinates 
dx (k) = 1/3 1 AX; (kK) dx,dy .... mean discrepancy in 
dy” (k) = 1/3L y, (k) the x- and y-coordinate 
= 531.7