PTE PE TO FES 
ER 
Space, O(X.,Z.) is the principal point of 
the photo, S(Xs,Ys,Zs.) is the perspective 
centre, f is the focal length. 
For two pairs of points 1,2 and 3,4 
located respectively in the two vertical 
Planes I and III, parallel to the focal 
plane for the image x-coordinate is 
obtained 
X41-A,1 
yi. Ys 
MN, LL —-X, + EL 
yirrl = Va 
The relations for the image z-coordinate 
are analogous. 
penoting by 1* and l1l'':and by L* and L':: 
the respective distances “1-2 and 3-4 in 
the image and object planes the following 
relations for the focal length f and for 
the projection centre Y,-Coordinate are 
obtained: 
Fh 
! 
  
  
ji: L:*J1 - 1*3: L! 
(3) 
l* Ii: yt - 1112 Li yiii 
3 iz TII = 17711 Li 
The deduction of these relations does not 
take into account the influence of the 
angular elements of orientation. This 
influence can be easily analytically 
evaluated in the case when the points 
form a square or a rectangle, because by 
inciining the photo in a plane, 
perpendicular to two of its sides, the 
distances between the images of the 
respective points are changed in 
different directions. 
The relations (3) are simplified in the 
case when the distances between the 
points in the object plane or in the 
image plane are equal. 
In the first case, when L'= L**1= L (ses 
Fig.1) they are: 
iz qrisfyrit = YE 
  
(4) 
  
  
       
Using the formulae (4) the errors in f 
and Y are analytically estimated as 
functions of the distances between the 
points and between the planes in which 
they are located. The following 
conclusions are drawn from this estimate: 
(i) the error in the determination of Ys 
decreases with increasing the 
distance between the points in the 
image plane and with increasing the 
ratio of the distance between the two 
planes with control points and the 
distance between the object and the 
camera station; 
(ii) the error in the focal lenght f 
decreases with increasing the 
distance between the two planes with 
control points and with increasing 
the distance between the control 
points in each plane. 
  
  
  
  
  
  
p 
3x mou TT 
\ / 
\ Y 
S B mis] 5 
E A = X > 
À X I 4 | 
iN S £5 E 
NN mtd © 
pe £7 5 
airs = 
IX 4 7) 4 À 
S 
Pie 
24 Lil 31 
I 
m 
Fig.1 
If the points in the two planes of the 
test object are located at equal 
distances, one behind the other ( for 
instance at the apices of a parallelopi- 
ped), their images will project on 
different points on the photograph. In 
this case lens distortion will introduce 
different deformations of the respective 
points lying in each of the two planes. 
If. the distortion is not eliminated at 
will influence negatively the accuracy of 
the determination of the orientation 
elements. This requires either a 
preliminary correction of the image 
coordinates, using camera certificate