Analytical Aerial Triangulation and Comparison between it and 
Instrumental Aerial Triangulation. 
hy G. H. SCHUT, 
Photogrammetrie Research, Division of Applied Physies, 
National Research Council of Canada, Ottawa. 
ntroduction. 
In photogrammetric mapping an extensive use is made of methods to increase the 
1 mber of ground control points established by surveying. At present the most accurate 
fihese methods is triangulation of strips of photographs in first order plotting machines 
ilowed by transformation of the obtained strip co-ordinates to the map projection system. 
Since the beginning of photogrammetric mapping analytical aerial triangulation has 
wn considered a potential method. In this method the co-ordinates of the required points 
me measured in the photographs. The map co-ordinates are derived from these plate 
1sordinates by computation only. 
Until fairly recently analytical triangulation was not practical for two reasons: no 
neise stereocomparators were available to read the co-ordinates and the required com- 
Intations were too time-consuming. At present, however, there is at least one accurate 
tascomparator, and more are being designed. Computations can now be carried out 
nckly on electronic computers. This makes analytical triangulation a practical possibility. 
Analytical triangulation has a number of advantages over triangulation on first- 
lifting machines. 
order 
A greater accuracy can be achieved because the measured co-ordinates can be corrected 
ir all determinable errors in the position of the photographic image. Film distortion can 
taken into account when a grid plate is used in front of the negative or, to a lesser 
ipe, when the camera is provided with a sufficient number of fiducial marks. Lens 
iMotion can be compensated depending only upon the accuracy with which the camera 
is been calibrated. In instrumental triangulation this is not the case, or at least not to 
fis extent. For example, no corrections can be given for irregular distortion. 
Corrections can also be made for the effects of refraction and earth curvature. 
A greater accuracy will also be achieved because analytical triangulation is not 
istricted by some of the limitations of instrumental triangulation. Triangulation instru- 
ments however accurate always have their imperfections. The bundle of rays, defined 
irthe image-points in a photograph, is not reproduced with mathematical precision. The 
idel of the terrain, obtained from a strip of photographs will therefore always be more 
it less distorted. 
In analytical triangulation the bundles of rays are defined by mathematical formulas. 
(lhe accuracy of the computations is only limited by the number of decimal places used. 
lie only instrumental errors that occur are those in the reading of the co-ordinates on 
le stereocomparator. Since a precise stereocomparator is a much simpler machine than a 
istorder plotting instrument the sources of instrumental errors are fewer in number 
ul their effect can be kept much smaller. 
| Another limitation of the triangulation machines is the accuracy with which relative 
"mation ean be established. An approximate orientation is established first. It is then 
iljusted, either empirically or by a numerical procedure. With the empirical procedure 
iesu depends more or less on the preference of the operator as to how far to go in 
"wing the parallaxes and as to which elements to use. With the numerical procedure 
{le corrections to the orientation elements are computed from observed parallaxes. Ap- 
ling these corrections to the instrument readings however does not generally cause 
“expected change in the parallaxes. This is partly caused by the use of approximate 
  
  
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