where i varies from 1 to 4 for the quadrustational system, 
giving 12 equations. 
A's can then be eliminated, reducing to five equations. 
The components of vector R, are obtained from: 
LÀ 
R, = à! MT, 
x i i.i 
where: M = scale factor of location vector ri in the image 
c system 
M; = the rotation matrix 
The equations can further be reduced by taking the rotations at 
all stations relative to the lower left station S (Fig.5) 
(Babaei-Mahani,1981). These equations are nonlinéar and must be 
linearised,then the solution is obtained using the least 
Squares procedure. 
The geometry of the system can also be defined using convention- 
al collinearity conditions. 
The quadrustational system (Fig.l) is expressed by considering 
four central projections of points in a 3-dimensional object 
space on to four image planes. These conditions can be ex- 
pressed mathematically for the ith photograph as: 
i i 
x, Ma 4 111 8, 70 (12) 
lu ig du 
y, MX f MX, -o 
where | Mi d | 
= : i i i is 
M; = Mi and M; E RT "js j71,2,3 
i 
Mo 
I x -x 
p Si 
x. = Y. - Ys for each i, P.» 1,2,... N control 
; P i points. 
2. "92 
[| P 3i 
  
2. Determination of the spatial coordinates 
Once the coordinates of the camera exposure stations and their 
orientation have been established, the Spatial coordinates of 
object points may be determined. Two different procedures are 
developed, both being based on the determination of an optimum 
point to represent the intersection. This is achieved by mini- 
mising the volume of the tetrahedron formed by the mutual inter- 
section of the four rays (Fig.5), using optimisation from rays 
in the first procedure, and optimisation from planes in the 
second procedure. 
133