20 ANALYTICAL AERIAL TRIANGULATION, AUTHOR’S PRESENTATION 
in one short paper, but the present paper and the 
following panel discussion may serve as a good 
start. 
My paper gives a review of possibilities in 
four important phases in the construction of a 
triangulation method: 
1. the selection of a triangulation procedure; 
2. the selection of condition equations; 
3. the solution of the condition equations; 
4. the selection of a co-ordinate system as a 
basic system for ground control. 
The possibilities in these different phases are 
discussed mainly from the point of view of 
mathematical soundness. 
In this discussion some characteristic fea- 
tures of four methods which have actually been 
coded for electronic computors are used as a 
guide and are analysed. No attempt has been 
made to be complete, especially where the selec- 
tion of condition equations and their solution is 
concerned. 
I would now like to say a few words about 
the four phases of analytical triangulation, first 
on triangulation procedures. The paper gives a 
description of the three procedures, the applica- 
tion and different methods. One of these meth- 
ods, the Herget method, uses an application that 
is rather illogical in that the result using the 
same observations depends upon the way in 
which the strip is triangulated, the result ac- 
tually depends upon the model with which the 
triangulation is started, and thus we can expect 
different results depending on whether we start 
at one end of the strip or the other. This is 
rather illogical and the paper gives one method 
to avoid this still using the same triangulation 
procedure. 
As far as condition equations are concerned, 
the paper contains the condition of intersection 
which is used by these four methods, the con- 
ditions being formulated by these methods in 
three different ways. I will explain it this way: 
two factors must be co-planar, one states that 
the Y-parallax in the model must be equal to 
zero, and the other states the minimum differ- 
ence between two factors must be equal to zero. 
This leads to condition equations which are 
slightly different. The simplest one states that 
the scalar strip products of three factors must be 
equal to zero. The others have this scalar strip 
of product also but added to it as an extra 
factor. When one examines the function of this 
factor, it becomes clear that it will serve no 
single useful purpose; having, for instance, the 
minimum number of equations necessary to 
obtain a solution, the multiplication of condition 
equation by a factor does not alter the solution 
at all. If one has more equations one must per- 
form an adjustment but in this case one must 
properly weight one's equations and find the 
correlation between them; also in this case an 
added factor has no effect. Therefore, we should 
eliminate this factor and thus arrive at the 
simplest form of the equation. This is required 
in proper theory. 
Regarding the solution of the condition equa- 
tions in these four methods, we are left with a 
condition equation which is non-linear in the 
unknowns and we have to linearise it. It has 
been shown by Professor Thompson in a paper 
some time ago, that it is sometimes unexpectedly 
possible to linearise an equation by relating it to 
another equation which is exact and linear. This 
has been done for absolute orientation but for 
other orientations we have not yet achieved this. 
Therefore, we must linearise it by relating by a 
linear equation which approximates it as nearly 
as possible. 
This has not been done in the Herget method, 
for instance, where the geometry of the model 
has served in order to find the linear equation. 
Unfortunately, this linear equation is not exact 
in that if it is satisfied the corresponding rays 
may not intersect. Therefore, it deviates too 
much from the non-linear equations and we 
cannot accept it as proper. 
Another point is the weighting of our ob- 
servations or observational equations. The 
weighting must be based properly on the weight 
and correlation of the observations and not on 
any other factor. 
The fourth point is the co-ordinate system we 
use as a basic system for ground control. Dif- 
ferent possibilities are the geocentric system, the 
local system with known relation to the geo- 
centric system, and a map co-ordinate system in 
which we use plain map co-ordinates and terrain 
heights to form a three dimensional system. 
In our National Research Council we have 
chosen the map co-ordinate system. Each 
system has its advantages and disadvantages, but 
we think we can overcome the disadvantages of 
the map co-ordinate system in a simple way. 
Its disadvantages of earth curvature must be 
corrected before triangulation and the scale of 
map projection varies. 
I would also like to say a few words on two 
of the four methods analysed in this paper. The 
ordnance survey method is one of them. It is a 
special six point method with a predetermined 
solution. It is the only method of its kind that 
is practised, and now we understand it is being 
abandoned and superseded by another method 
which does not have these features. It would 
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