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THE DESIGN OF PHOTOGRAMMETRIC PLOTTERS, HELAVA 121 
analysis in any detail because after forty years of use its performance is well-known and 
appreciated. 
Before an analytical plotter can be built, a suitable mathematical formulation must 
be found to represent the central projection. However, this is a starting point only. 
Actually, the central projection is merely a theoretical concept but has never been en- 
countered in practice. In the realm of physical reality we should say that a certain rela- 
tionship exists between the coordinates of a given point on the ground, on a map, and on 
photographs. Of this relationship we know only what is permitted by our knowledge of 
the physical phenomena that lie behind the relationship. The mathematical formulation 
we use to describe it is only an approximation and must be recognized as such. However, 
an ideal formulation describes the relationship with a fidelity that accords with the 
present knowledge of the fundamental phenomena and with the practical requirements. 
It appears that for a long time to come the mathematical formulation for plotters 
will be composed of two parts. The first part is a rigid and strictly theoretical formula- 
tion which describes the process with reasonable accuracy. This part is complemented by 
the second one, which includes mainly experimental corrections to the results provided 
by the theoretical formulation. These corrections are needed to compensate for the dif- 
ference between the theoretical and real worlds. 
It is not difficult to find mathematical formulas that represent a central projection 
and could, therefore, be used in a photogrammetric plotter. These formulas may, e.g., 
deal with a central projection between two planes, and may be used to convert the 
problem in a plotter to that of the “normal case”, from where an extension to three- 
dimensional space is very simple. Formulations of this sort have been published by the 
author in 1958 [2, 3]. 
Linear coordinate transformation in space may also be used as the basic formula- 
tion. Formulas of this kind are used in most methods devised for analytical aerial 
triangulation, but it is advantageous for the plotter to “reverse” these formulas, i.e. to 
express the photographie coordinates as a function of the coordinates of the model. 
Examples of formulations are published in [4] and [5]. 
Obviously, approximations may also be used and one such possibility is based on 
the use of the well-known parallax formulas. However, these formulations offer in- 
sufficient savings in arithmetic operations to make their application economically 
attractive. 
Many kinds of formulations not mentioned above will probably be presented even- 
tually, because the principle of the analytical plotter is very flexible regarding formula- 
tions. In particular the possibility of using other than conventional parameters appears 
attractive. 
The choice of an acceptable formulation for a plotter depends mainly upon the 
computation facilities incorporated in the instrument. If, for example, an analog com- 
puter is used, considerable advantages are offered by a formulation that derives diffe- 
rences rather than total values. The same applies, although to a lesser degree, to the 
digital incremental computation. A slight gain in speed may be expected, even in the 
case of ordinary digital computations, because fewer digits need to be handled when a 
formulation for differences is used. However, this slight advantage for ordinary digital 
computation is offset in most cases by complication of formulas. 
These considerations were formerly of crucial importance; the entire feasibility of 
plotters of this kind hinged on the successful solution of the formulation problem. At 
present, however, the computer technique is so far advanced that the problem is to find 
the most suitable means from an economical point of view. In this task the details of 
the computation techniques are of greater importance than the formulation.