The vanishing line v joins both Vi and V3. 
Similarly, the vanishing line v' in 7 is determined 
through the two vanishing points. 
U;(ao1 $031] : 0), U;(a02 =6 : asa). (6) 
3.5 The Trace t 
As shown in fig.(1), the trace t can be considered as 
belonging to both o and x. Let t' denotes it in 7 
when separated from a. t and i' must be congruent 
beside being corresponding to each other. It is well 
known that they are parallel to the vanishing lines 
in space [4]. Let K(1: k: 0) and L(1: 0 : 7) be 
the points of intersection of t with z; and z5, and 
let their corresponding points be K'(1 : k’ : 0) and 
L'(1 : 0 : l'). The metric length KL and K'L' must 
be equal. 
Since KL is parallel to v fig.(4a), then Af = a, 
hence from (5) and (4), 
AL 1 
is Pt ES ee AM 
A, K k 
  
/ / 
pe Ap ap2 
Similarly, for the corresponding points Æ'(1 : k" : 0) 
and L'(1:0 : l’), it can be shown that 
P ied (8) 
VB 7 
k 402 
Substituting the coordinates of L and K into (1), we 
get the coordinates of K' and L' as follows: 
K'(aoo -- ag1k : a11k : 0), L'(aoo 4- a02l : 0 : a23l) (9) 
Hence ak 
km LE. 10 
ago + ao1 k did 
and a: 
Lite vir imens 11 
ago + ao2 | (14) 
Since IE = UK" , then: 
I? -- k? — 2lk cos0 — l7 4- k'? —2l'k'cosóü' (12) 
where 0 and @’ are the angles subtended by the axes 
pairs, which can be measured directly or determined 
from the well-known cosine formula: 
put ATs Ar Ar = Acie 
cos = Aida EA As 421 
2A1 A2 A1 A4 13 
ATAL +ATA] -AA (13) 
WA AA, 
  
  
cos = 
  
  
1 Al A! 
1*'2*1 
Dividing (12) by k? and substituting from (7),(8) and 
(10), we get after several reductions: 
2 
(aoo + dork)” = 
x 42, a2 . 891^ — 249,292 cos 0 a5? z 52 
a, 8012 —2a01802 cos 0 +402? 
(14) 
It can be shown that the R.H.S. of (14) is a positive 
quantity and hence it is set to 6°. 
4 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
Solving equation(14), yields two values for k. The 
practical value of k is chosen, such that, for a positive 
photo, t has a position similar to that shown in fig.(1) 
and(4), in which: 
|AiK| « |A1V1|, or 
  
  
  
  
/ 
«e| [eoe (15) 
401 ao1 
Since from(14) k= mm 
hence 
m nM for aoo > 0 (16) 
= zzi for ago « 0 
Determining k, we can calculate the value of 1, k' 
and I’ from equation(7),(10) and (11) respectively. 
The homogeneous coordinates of K, L, K' and L/ are 
therefore known. 
3.6 Setting 7 to coincide with a 
Let * be put upon a such that K’ and L’ coincide 
with the corresponding points K and L respectively. 
In this position the two planes are centrally collinear. 
The collineation axis is t and the collineation center 
is So as shown in fig.(5). Mathematically, this is a 
  
à 
Figure 5: Coincidence of 7 with a 
achieved by referring both planes to a common carte- 
sian system of coordinates. 
In the planes a and v, K and K' are chosen to be 
the origins of the new systems, while the axes X and 
X' are chosen along LK and L'K' respectively. The 
Y and Y' axes are perpendicular to them as shown 
in fig.(4a,b). The two systems will coincide with each 
other, when both planes coincide as shown in fig.(5). 
The above procedure is performed through several 
coordinate transformation: 
      
  
  
  
  
   
   
   
   
  
  
  
    
   
   
   
  
   
    
   
   
  
   
  
   
   
  
   
  
  
   
  
  
   
   
  
  
  
   
      
   
   
  
    
  
  
  
  
   
    
Figure 6: R 
rectangular « 
1. For any 
then the 
coordin: 
is as foll 
and con 
2. From fi; 
space co 
and 
4. Let X =