in which 
(2) 
where f denotes the focal length of the lens. This formula is an extension of a 
result originally derived by Magill (1955). As written, it holds for distortion 
functions expressed in terms of field angle 9 rather than radial distance r, a 
distinction that becomes significant at higher magnifications. Thus, if org is 
expressed in one of the standard representations as 
5 
s = Kj tan®0 + Ky tan 0 + Ky tan” 6+... (3) 
it follows from (1) that 
inwhichE, KE, ,...aond Kes, Kes, 
appropriate tö object distancess, and s, . 
... denote the distortion coefficients 
VARIATION EXPRESSED AS FUNCTION OF RADIAL DISTANCE 
Because our concern here is primarily with cameras of fixed format, we shall 
find it more natural and convenient to employ radial distance r, rather than field 
angle 6, as the argument of the distortion function. Accordingly, we shall adopt 
the following representation of the distortion function 
é 7 KU T«r TES. T... (5) 
S 
To relate (3) and (5) we first note that by definition 
tan9 = - 7 (6) 
Cc 
where c denotes the principal distance of the camera. However, by the basic lens 
formula relating image and object positions we have 
" i, 0)