An Analytical Method of Aerial Triangulation 
by 
Dr. UGO BARTORELLI 
Assistant Professor in the Geodesy and Topography Institute of the Engineering 
AS . : & : . 
Faculty of the University of Rome. — Advisor to the '"Ottico Meccanica Italiana 
(OMI)" of Rome. 
4 PAPER PRESENTED TO THE III COMMISSION OF THE INTERNATIONAL 
SOCIETY OF PHOTOGRAMMETRY AT ITS VIII CONGRESS IN STOCKHOLM. 
Every photograph ?+1 can be chained to the 4 photograph, already absolutely oriented, 
taking into consideration as the necessary and sufficient minimum only the four directions 
merging from its viewpoint V;, ; and passing through the two already determined pas- 
sage points Pi; p,, at the edges of the zone common to the photographs i— 1, à, à + 1, 
md through the two analogous passage points P, ,,,, P(,,,,8s yet unknown, that are 
emmon to i, à--1, and ét 2. 
The tetrahedron of the said four directions, whose faces are known from the measu- 
„ments made on the photographs, assumes its absolute orientation, forcing its two corner 
dps V,,, P and Vi, P, to belong to the two already determined passage points 
p and Py, and the other two corner edges, V; ,, P,,,, and V;,, Pa,,,, to be co- 
Wmar with the rays V; P ,,,, and V, P(,,,, already determined because emerging 
tom V, The problem is thus determined and leads to the determination of the co-ordi- 
mies of the viewpoint V;,, of the new photograph i--/ and, suecessively of the co-ordi- 
mies of the new passage points P,,,,,, and P ,,,, and to the determination of the 
ms Vig Pisa) and Vi, P(,5,, thus the known elements are available that are 
pessary to. renew the whole procedure for chaining the photograph i+2, and so on 
indefinitely. 
Each chaining operation consists of the following operations: 
Ist. Measuring on each photograph the co-ordinates of the images of the six passage 
pints of the chaining, which are arranged on the photograph as follows: four near the 
omer edges and two near the mid-points of the sides in the direction of the strip; for 
{his measurement you can use either the stereocomparators (cartesian co-ordinates) or 
the radial triangulator, with which you can measure the vector ray, as well as the anomaly 
(polar co-ordinates). 
Ind. Calculation of the cosines A, B, C, D, E of the faces of the tetrahedrons under 
unsideration, through knowing the co-ordinates, measured in the 1st step, of the focal 
iistance and of the distortion of the take camera. 
34. For each photograph, in its order in the strip, the calculation of the three 
known co-ordinates Xv; , ,, Y y, p1» Zyi+ 1> Of the corner edge V,,, of the tetrahedron in 
Weston, and of the twelve unknown director cosines 
bis u 
v 
is , is , is 
bia ? Ua Vid 
tais , U(isi)s V(i+r)e 
50a , Uta , VG+1)a 
d Ils corner edges. The fifteen unknowns are determined by solving a system of fifteen 
disi each of the second degree, there being known the coordinates Xvp Yvp 7v; 9f 
* Vewpoint V; of the photograph i; there being known the co-ordinates Xpiy Ypiss 
| | ; 
and XD Np Zp;q of the two passage points P,, and P,;; and there being known 
the director cosines 
dG+1)s , Y (4..1)s , S(i+1)s 
dici) , T (1.-1)a , S(i+1)d 
  
de WE ETE 
bcm AETERNI d a ps 
    
   
   
    
    
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
    
        
      
  
     
   
   
     
  
A RE INS an ars