niknA, — WALBYKaVKD, Dok A, — uoo. 
all of them will vanish simultaneously because of con- 
taining proportional columns with njn5, and n»n. 
Therefrom follows that at least one selected point Py 
out of the subset (3...7) has to be situated outside the 
[e,.e,]-plane Gy-G4-G, in Fig. 1 and that all together 
two points (P, and G4) out of the whole set are not 
allowed to be co-planar with the rest. Otherwise, the 
projective relations refer to regular projectivities 
between bidimensional spaces. This statement agrees 
with the need for five non-coplanar homologous points 
in connection with the determination of a regular 
transformation between tridimensional projective 
spaces (Brandstátter 1991). 
If one point out of the set (3...7), for example P;, is 
considered to be a variable point, an expansion of the 
determinant (2.5) into subdeterminants D;ik may be 
performed with respect to its row k-7. Therefrom 
follows in analogy to (Rinner 1972) the quadratic 
function 
y QyD;4 +y Q42yD42 4 y QaoyDog (3.1) 
T y'Q4yDa +e y'Q,yD2 =0 
in the projective object space. It may be expressed by 
means of the short homogeneous matrix formula 
Tab. 2: The Qj, in detail 
viQu=0 
J ee 
- 
in which Q represents the sum of all QD; in (3.1). 
Therefore A becomes singular if all points of the 
subset {3...7} belong to a general quadric surface. 
The centers of projection yg, however, must be points 
of this surface as well, because, according to (1.2), all 
products of the kind m'y, equal zero and hence the 
condition (3.1) (=formula of the quadric surface) is 
satisfied identically. This statements are well-known 
from literature, especially from the famous "Vienna 
School of Geometry’ represented mainly by Josef 
Krames (Krames 1942) and Walter Wunderlich 
(Wunderlich 1941). Their results were achieved by 
admirable synthetic considerations based on a pro- 
found knowledge of constructive and projective geo- 
metry. But in regard to analytical or digital photogram- 
metry, calculational methods are essential and hence, 
in order to detect and avoid critical situations of 
projective image correlation, an algebraic analysis of 
this problem should be prefered. 
A detailed evaluation of the partial Q's in equation 
(3.1) by means of relation (2.2), and using the m; of 
(1.3) results in the contents of Tab. 2. They show that 
in general, the resultant quadric 
Q = Q;Dqq + Q42D42 + Q20D20 + Q24D241 + Q22D22 
  
  
  
  
  
  
  
  
  
  
  
  
  
Qjk | row | © 1 2 3 
Qu] 0 | 0 "ud 0 FINIT, 
1 0 py 0 usus 
2 0 dub - 1) 0 -U34(H2 - 1)H3 
3 | © | nifus(u51-0+1)] 0 usu [us (usa - 1) * 1] 
Q,| © 0 0 0 0 
1 0 0 0 0 
2 Lr0 uus 0 U31H3H3 
3 0 U32H1H3 0 U34U32H3H3 
Q„ | © | O 0 u5- pd U32H3 - U32H3 
1 0 0 Ha(uT-1)-p3(n4-1) U52H3 (H4-1)-U32H3 (14 - 1) 
2 0 0 H2-H2 u52u5 (n5 - 1) -usaus (u5 - 1) 
3 0 0 H>(n3 -1)-153(u3 -1) U32H5 (p3 - 1)-U32p3 (H3 -1) 
Qu} 0 0 0 0 0 
1 0 0 udo U32H1H3 
2 0 0 0 0 
3 0 0 U31H2H3 U34U32H3H3 
Q | © 0 0 "Bd -U$2H3 
1 0 0 -ud Qu - 1) -U32H3 (H> - 1) 
2 0 0 H3 U32H3 
3. | 0 0 ug [ps (U32 - 1) +1] uous [us (usp - 1) * 1] 
  
  
  
  
  
80 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
  
  
   
   
   
   
    
     
  
   
    
    
   
  
  
   
  
  
  
    
   
   
    
   
     
   
  
   
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