10 Commission III Invited paper 
Remarks on the Theory of Analytical Aerial Triangulation 
by G. H. SCHUT 
Photogrammetric Research, 
National Research Council, Ottawa. 
In recent years a number of different methods of analytical aerial triangulation have 
been published and some of them have been coded for electronic computers. New accurate 
stereocomparators have also appeared on the market and the accuracy of analytical 
triangulation through measurements on these machines is being investigated. 
While awaiting the results of these investigations it seems opportune to conduct a 
systematic review of the theory of analytical aerial triangulation. The author would like 
to make a contribution to this in the following paragraphs. For many details reference 
must be made to earlier publications. 
On the characteristics of a sound theory. 
1. The published methods of analytical aerial triangulation are generally known either 
by the name of the author or by the name of the organization where they originated. 
This seems to be a good practical approach to distinguish between them but it is not very 
helpful in establishing their place in a comprehensive theory of analytical aerial trian- 
gulation. 
These methods can all be considered satisfactory in the sense that they will give 
results. From the mathematical point of view, however, objections must be raised against 
some features of some of them. 
It is therefore not possible to construct a sound theory of analytical triangulation by 
merely listing these methods and retaining all their characteristics. Rather, each of these 
methods must be investigated, the undesirable features rejected and a sound theory built 
up from what is left over. 
All published methods make use of: 
a. a choice of one of the three triangulation procedures described in the following para- 
graph, 
b. a choice of condition equations which the measurements must satisfy, and 
c. à choice of the method used to solve the condition equations. 
In a sound theory of analytical triangulation each separate method must make spe- 
cific choises and use these consistently. 
Equations should be used in their simplest form and the use of non-essentials cer- 
tainly must not classify a method as a separate method. Instances of non-essentials in a 
condition equation are the use of factors which do not affect the result and the conver- 
sion of vectors from projection centre to image point into unit vectors or into vectors to 
a point in an imaginary vertical photograph in cases where this only adds to the amount 
of computation. 
Also each method must be based upon sound mathematical reasoning. Simplifications 
which are not based upon such reasoning must not be introduced. Weight and correlation 
factors applied to the condition equations or to correction equations derived therefrom 
must be based properly upon the weight and correlation of the measurements. 
The selection of the particular discipline — such as analytic geometry or vector anal- 
ysis — employed in deriving and formulating the equations cannot be recognized as a 
characteristic. 
The above specifications appear very reasonable or even so obvious as to be almost 
superfluous. If any one of them is not followed in setting up a triangulation method we