CONDITION FOR ZERO DISTORTION AT MID-FIELD 
To proceed with our investigation of variation of distortion we shall now 
assume that only the leading term (the r? term) in the expansion (5) of the distortion 
function is of significance throughout the photographic format. This assumption is 
valid for the vast majority of commercial lenses, the most notable exceptions being 
lenses, such as mapping lenses, that are specifically designed for very low distortion 
at infinity. For the broad class of lenses dominated by the leading term of the 
distortion function certain interesting special results hold. First, it can be shown 
that as a consequence of (12), there exists a particular object distance so for which 
the distortion function is zero. If we define p as 
Kot … Ki er 
, (20) 
1,9 
the object distance for zero distortion is given by 
s, = 0O-ptf. (21) 
This result is reached by settings, = 2f, 52 = in (12) and solving for the value 
of s resulting when K, , is set equal to zero. The fact that distortion is zero at 
unit magnification (s = 2f) for perfectly symmetric lenses implies that p = 0 for such 
lenses. More generally, p must be less than unity if (21) is to yield a result 
capable of physical implementation (i.e., s >f). As we shall presently see, the 
value of p is also of fundamental importance in determining the performance of a 
lens with respect to variation of distortion within the photographic field. 
SPECIAL RESULTS FOR NEAR AND FAR FIELD LIMITS 
If we again sets, = 2f, s, == in (12) and substitute the result into (5) 
(truncated at r3), we shall obtain upon exercising the definition of p (equation 20) 
S 
br, = (^ ( ET ss: (22) 
q 
in which q is given by 
q = ; : (23) 
If we now evaluate ór, as given by the above expression for the near and far field 
limits (s,,s,) as defined by (17) and then exercise the result in (15) using the y's 
established in (19), we shall obtain after some algebraic manipulation