whereas for objects near unit magnification (q = 2f), 
7 e». (29) 
This indicates that the percentage variation in distortion between near field and far 
field limits of the photographic field diminishes with increasing magnification, being 
only about 1/8 as much at unit magnification as at zero magnification. 
CONDITION FOR INVARIANCE OF DISTORTION BETWEEN NEAR AND FAR 
FIELD LIMITS 
From (27) it is clear that there are two conditions which will lead to 
invariant distortion throughout the photographic field, namely, when either 
q-1 = O or 2-p = 0. The first condition corresponds to the trivial situation of infinite 
magnification (and hence zero depth of field). On the other hand, the second 
condition which requires that p = 2 (or equivalently that K; = 4K, 2e ) appears to 
be physically attainable by design. Evidence for this is provided in Table 1 in which 
we have listed the values of p corresponding to 22 Hypergon lenses as computed from 
measured data presented by Magill (1955). By design, the Hypergon is a perfectly 
symmetric lens. Accordingly, p should ideally have a value of zero. As is shown 
in Table 1 only two of the samples achieve this ideal value, while the remainder 
have values ranging from a low of -6.7 to a high of 45.2. This demonstrates that 
p can be highly sensitive to manufacturing variations. 
TABLE 1. Values of p for 22 Hypergon lenses as 
computed from data given by Magill (1955). 
  
r^ 
o 
= 
wn 
Lens p p 
46 
.22 
17 
il 
86 
13 
67 
.14 
. 64 
. 00 
  
Hd o "(D wn Mou: ond ou 
Qo 4. 0 U 5» dg o 
Qo = 
o q 
>>> >>> >>> >> 
t 
+ 
PP STREET 
7 
  
  
  
EXAMPLES OF LENS-TO-LENS VARIATIONS IN CLOSE RANGE DISTORTION 
e The effect that the value of p has on the behavior of distortion is illustrated 
in Figures 3 and 4 in which results for four of the Hypergons listed in Table 1 are 
employed. Of the selected lenses two correspond to the smallest and largest values