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BALLISTIC PHOTOGRAMMETRY, SCHMID 39
the basic line of thought which underlies the chosen approach will be presented. The
bundle of rays associated with a lens is the basic tool in photogrammetry. The corre-
sponding idealized geometrical analogue is given with the concept of central perspective,
namely that a multitude of straight lines have in common one point — the center of pro-
jection. Such an interpretation is free of any limitation concerning focal length or angle
of field. In other words, any conceivable photogrammetric camera can be simulated.
While the number of photogrammetric bundles present in a specific measuring pro-
ject may vary greatly, the principle of central perspective just mentioned, will enable us
to simulate any one and consequently, all of these bundles. À specific bundle, however,
can be interpreted as a population of like quantities, the individual rays. From a geomet-
rical standpoint, no one ray has any preference over any other ray within such a bundle.
Therefore, it suffices to devise a mathematical analogue for a single ray, which can be
applied to all other rays present in a specific bundle, and by the same token, this ap-
proach is then useful for handling all the bundles and consequently, all the rays present
in any photogrammetric measuring system.
The geometrical simulation of a single ray is equivalent to the condition that three
points are located on a straight line. These points are the object point, the center of pro-
jection and the corresponding image point.
The well known Von Gruber formulas display algebraically this condition of col-
linearity, by expressing a functional relation between the spatial coordinates of the ob-
ject point, the interior and exterior elements of orientation and the plate coordinates of
the corresponding image. Thus any one ray gives rise to two equations.
Speaking somewhat cursorily, it is possible to say that an optimum program for
electronie computing is obtained by designing the analytical approach to a specific prob-
lem in such a way that the numerical treatment can use, as much as possible, a repeti-
tious process based on algebraic expressions of minimum complexity.
An analytical solution, based on the conventional method of instrumental restitution
by establishing first a relative model and determining in a second step the absolute
orientation, would call, for relative control points, for an algebraic expression for the
condition of intersection and correspondingly, for given control points, for such a con-
dition at either one, two or three given control coordinates. As a direct consequence of
such an approach, different types of conditional equations become necessary for the dif-
ferent types of control points. To complicate the situation further, the number of any
particular set of such conditional equations depends on the number of camera stations
involved in any specific triangulation problem. At last four, and under certain geo-
metrical conditions even six plate coordinate measurements, together with their residuals,
would appear in certain observational equations. In case of multi-camera triangulation,
the same residuals would be present in more than one observational equation, making
a rigorous least squares treatment laborious, or in more complex cases, even impractical.
Furthermore, the unknown coordinates of the relative and partial control points become
only implicity available as functions of the adjusted plate measurements and the orien-
tation elements. The introduction of any additional geometric conditions, as they may
exist for any one or all of these coordinates, would require complex mathematical manip-
ulations, prohibitive from the standpoint of computing economy.
All these difficulties are avoided with the aforementioned approach. Because of the
simplicity of the model, not only is the bookkeeping effort in the computer minimized,
but also, a high degree of flexibility of the solution is maintained. The potential of elec-
tronic computing is used to its fullest advantage by the fact that the final normal
equation system is formed by a process of adding of partial systems, each of which con-
tains information associated with only one individual ray.
A further benefit of the simplicity of the model is obtained if we now approach the
second phase, where it will be necessary to consider these physical parameters which
