381 
the origin of 
function of r. 
ry. About this 
(2) 
(3) 
gin. This with 
ent angle a. 
lutocollimation, 
perpendicular 
of images on 
und is known 
:h the photo- 
and camera 
is will need to 
icient number 
Equation (1) 
as a research 
systems. It is 
nera is isolated 
is found from 
This is usually 
ority of calibra 
te practice has 
leasurements on 
action, aircraft 
on, are looked 
is an I.S.P. 
papers on the 
isher, 121 Half 31 
the calibration 
of ± 2 fx can 
r, and further 
is to be sure 
independent 
agree within 
choice of glass 
this to perfec- 
design distor 
tions down to 
20 [x or less, and such lenses can for many purposes be considered as distortion 
free. Using an aspherical surface on one component, a still closer approximation to 
zero distortion can be obtained. 
Departures from design distortion occur because of 
(a) minor departures from the design values of the radii of curvature and 
thicknesses of the optical elements, of the glass indices and component 
separations, 
(b) asymmetry in optical surfaces or the mounting of the components, wedged 
filter glasses or register plates, 
(c) local random variations in the optical components. 
Minor departures from design values are usually kept small by the lens manu 
facturer. The main effect is on the principal distance, although the distortion will 
vary from lens to lens of the same nominal type. Results have been published for 
measurements on numbers of lenses of the same type. Half 21 quotes results for 
fifteen Aviogon cameras, which had principal distances in the range 1534 mm. to 
151-9 mm.; the maximum departure of the distortion of any lens from the average 
distortion was less than 10 ¡x. Hothmer 161 quotes results for other lenses, for example 
Ross, Aviogon, Planigon, Metrogon, with rather similar variations of the distortion. 
Asymmetry in a lens acts as though a small prism were added to a perfectly 
centred lens. No single point can then be found about which the radial distortion is 
completely symmetrical for all values of r and 6. A centre of symmetry, about which 
the residual asymmetries are a minimum, can be found, and it can be shown 161 that 
the distance between the centre of symmetry and the principal point of auto- 
collimation is an adequate measure of the equivalent prism. 
5. Tangential Distortion 
Tangential distortion cannot arise in a perfectly symmetrical lens. For well- 
made lenses, the lack of symmetry should be small; and the tangential distortion 
can be deduced from the value of the equivalent prism. For a 90° lens, the maximum 
tangential distortion is 04 times the separation of the centre of symmetry and the 
principal point. Tangential distortion is not often measured during the calibration 
process, and indeed it needs rather more elaborate equipment to do so. For this 
reason alone, it is important to determine both the centre of symmetry and the 
principal point during calibration and to set some limit to their separation for 
acceptance of the camera. 
Although the theory of the equivalent prism has often been stated, and indeed for 
simple lenses it can be deduced from optical theory, there appears to have been no 
attempt to justify this for the more complex lenses used in air cameras, with widely 
spaced and very numerous components. Now that we have computer programmes 
set up for ray tracing, it would surely be easy to calculate the effect of the de-centring 
of one or more lens components. If it could be shown that each de-centring gave rise 
to an equivalent prism, and that these added vectorially as do real prisms, then one 
would have greater confidence that tangential distortion need not be measured during 
calibration. We could also make some real effort to neutralise any residual asymmetry 
in the lens, by fitting a prismatic filter of the correct angle and orientation. 
6. Residual Distortions 
Local variations from the design distortion can arise from non-spherical 
polishing of the lens surfaces, variations of refractive index within the individual lens 
components, local departures of flatness in the focal plane, and so on. When 
calibrating a camera, the measurements also show random deviations arising from