of p (namely p = -6.67 and p = 5.17), one corresponds to a zero value of p and the 
remaining one (p = 1.43) corresponds to the particular value of p most closely 
approximating the critical value of 2. 
Figure 3 shows the variation in distortion at mid-field for various object 
distances relative to distortion at zero magnification (i.e., infinity focus). Al- 
though the lenses are all of the same design and of the same nominal focal length 
(127 mm), they are seen to differ considerably from one another with respect to 
variation of distortion with mid-field object distance. To cite an extreme, we see 
from Figure 3 that at a magnification of 0.5 (s = 0.38m) the distortion of the lens 
with p = 5.17 is 10696 of its distortion at infinity, whereas the distortion for 0.5 
magnification of the lens with p = -6.67 is -8496 of its distortion at infinity. Such 
wide differences in lens performance are indicative of the potential importance to 
close range photogrammetry of properly accounting for variation of distortion with 
magnification. 
Figure 4 shows the variation in distortion with magnification between near 
and far limits of focus for the same set of four lenses. Again, lens-to-lens variations 
are pronounced. With the lens corresponding to p = 1.43 distortion between near 
and far field limits is under 296 of distortion at infinity for all magnifications and 
hence is likely to be only marginally significant. On the other hand, with the lens 
corresponding to p = -6.67 such distortion is over 20% of distortion at infinity for 
magnifications smaller than 0.10 and hence can assume considerable practical 
significance. 
IMPOSSIBILITY OF ZERO DISTORTION THROU GHOUT PHOTOGRAPHIC FIELD 
We have seen that the leading coefficient of lens distortion will be zero 
when the object plane focussed on is at distance s = (2-p)f, a condition that can be 
physically realized for all p« 1. On the other hand, the condition for invariant 
distortion throughout the photographic field is p = 2, a condition inconsistent with 
the requirement that p be less than unity for zero distortion to be attainable. lt 
follows that, desirable as it may be, one cannot design a lens that will simulta- 
neously have zero distortion for some particular object distance and invariant 
distortion throughout the field. 
GENERAL PROCESS FOR CORRECTION OF DISTORTION 
Now that the fundamentals of variation of distortion with object distance 
have been reviewed, it is appropriate that we address the matter of how to apply 
the necessary corrections in practice. We begin by assuming that the photo- 
grammetric triangulation involves a camera (or cameras) for which distortion functions 
have been calibrated for two different object distances, s, ,s, . Thus the coefficients 
Ress Bes. 4 «iKys, s K;s_ /... are considered to be known. The corrective process 
for à given in image Would en be accomplished by the execution of the following 
basic steps.