
32 Image Processing for Cinema
At point A, Snell’s law can be written as x = ny, which implies
x
y
= 1. The
triangle angle sum theorem for triangles ABO and AOF gives us the angle
relationship z = x − y. The length of the arc AB is Rx, and since z is small
we also have AB ' fz = f (x − y).
From all the above:
f '
AB
x−y
=
Rx
x−y
=
R
x
y
x
y
−1
=
Rn
n−1
,
that is, the focal distance f is a function of the curvature of the surface
and of the refraction index of the lens, but is independent of the location of
the point A that we used in the derivation. Therefore, all rays parallel to the
optical axis are focused at distance f, as we wanted to prove. The same result
can be