distance f from the camera station, C. The z'-axis coincides with the ray Cj, positive z' 
directed toward C, and the y'-axis intersects ray Cc at c'. This coordinate system, in 
effect, represents a fictitious photograph in which J is imaged at the principal point and 
the image of the Z axis coincides with the y'-axis. Hence, the special solution just developed 
is applicable to the points expressed in this new coordinate system, x'y'z'. The problem 
to be solved, therefore, is that of finding the transformation which relates x'y'z' to xyz. 
This transformation is of the form: 
  
x. x! 
1 1 
x| = IR |y! (56) 
—f| —f 
Tal T12 a3 
in which [R] = ry Toy Tyg is an orthogonal matrix. 
tl T fes 
For point j, xt = y! = 0; hence: 
r" = un *- — rm - 
X4 rl r2 ‘13 9 rh ftus 
Ys - k Ty Toy Tyg 0 = Erg (57) 
N Ir d T2 *33| SE [ iT a3 
For point c, x^ = 0; hence: 
7 = =, > 
x] ru v2 715] 0] ko mk ET, 
= ' = t ES 
^ RE RE ET F2 r3 Ye hla =F Irs, (58) 
- _ t = 
[f Fa T2 *33J f Ko ro E123 
  
  
  
  
  
  
It follows from equation (57) and the orthogonality of [R] that: 
KE = x? +y2 + f? (59) 
J J + 
kt is positive because both Kk and f are positive. From equation (57) it follows that: 
-X. 
Tr 
13 EE (60) 
r 
23 kf , (61) 
i 
E. 7 Kf (62) 
  
Multiplying the equations in (58) by F13> T93 and Tags respectively, adding and explöiting 
the orthogonality of [R], we obtain: 
kf = — (ri4X, + fy. - ra 3f) (63) 
Squaring each of the equations in (58), adding, and again utilizing the orthogonality of 
[R], we obtain: