超焦距及其應用(續)

The Physical Theory of Hyperfocal Distance and its Application to Photography (cont.)

May, 2010

Ringo Cheung

Abstract
Last time the hyperfocal distance equation in terms of f, N and c was derived from the thin lens formulas. In this article, the relationship between H with f, N and c will be analyzed and some physical inference will be discussed.

Relationship of H with f
Recall that the equation for hyperfocal distance H is given by :

Since H varies with the square of f rather than linearly, a small increase in f will result in a large increase in H. For example, for a 25mm lens of 135 format, H = 2.72m at an aperture of F8; while that for a 50mm lens, H = 10.83m at the same F-stop.

Graph of H vs. f
The 2nd time derivative of H w.r.t. f is :

and the minimum of H exists at f = -Nc/2, the graph concaves upwards. By considering only positive values of f for practical purposes, the graph of H vs. f for 135mm format camera (coc = 0.029) and an F-stop of N = 8 is :

For focal length of 35mm and longer, H is over 5.32m. While this is useful for scene photography, it is not very convenient for street photography and snapshot.

Graph of H vs. N
For a lens of fixed f and c, H is strictly decreasing with N since

as c > 0. On the other hand, a larger f-number means a smaller aperture. Therefore a larger f-number like f8 or f11 will result in a smaller H than with f4. The graph of a 25mm lens (135 format, i.e. coc = 0.029) with f-number from 1 to 16 is depicted below :

which is a hyperbola.

Relationship of H vs. c
Roughly speaking, H is strictly decreasing with c from the hyperfocal distance equation. However, since cameras of different formats have their own definition of wide angle, standard and telephoto lenses, (e.g. a standard 50mm lens in 135 format is considered a wide angle lens in 6x6, 120 format, with c = 0.053) a direct comparison of H vs. c (i.e. different camera formats / film sizes) is not of much practical usefulness.


Conclusion
From the above discussion, a wide angle lens set at small aperture (i.e. a large f-number) is a good candidate for short hyperfocal distance (H) and is more convenient for street and snapshot photos.