# 3.2 Infinity Focus

In landscape photography, it is often recommended to focus at the hyperfocal distance to maximize the scene in focus. But not everyone agrees (and we are not talking about artistic aspects here).

Harold Merklinger, a passionate photographer with a sound background in the sciences, wondered why his images taken with the hyperfocal distance did not really meet his expectations. In particular, he noticed that the more distant objects often appeared substantially fuzzy. The following is based on his studies of the subject [Merklinger 1992, Merklinger 2002]. The basic problem is illustrated in the figure below.

Consider a lens with a focal length f, which is focused at a certain distance. In this case, the object distance g and the image distance h are adjusted according to the lens equation. As indicated by the purple lines, the rays emitted from a point in the object plane towards the lens form a cone. The diameter of the cone is given by the aperture a of the lens.

If an object is somewhat nearer or further away, it intersects the cone where it has some finite diameter s. If the object is smaller than s, only a portion of the rays that reach the image plane at the original point will come from the object; the recorded intensity will thus depend on some other contents of the scene as well. In other words, the image of the object gets blurred. The diameter s thus determines the smallest detail the lens can (surely) resolve at that distance.

## Minimum object size

In the following, the diameter s is called the minimum object size. It can be calculated as follows.

1. If an object is located at a distance gn nearer to the lens, we get

 s / a = (g – gn) / g (I1)

Solving for s gives

 s = (1 – gn / g) a (I2)

According to equation (I2), an object located somewhere near the focus distance can be very small and still be resolved. An object somewhere near the camera must be similar in size to the aperture of the lens.

2. If the object is located at a distance gf farther away from the lens, we are facing a more serious problem. In this case, we get

 s / a = (gf – g) / g (I3)

Solving for s results in

 s = (gf / g – 1) a (I4)

According to equation (I4), the minimum object size is thus not limited, but will grow proportional to the distance gf. Objects may thus completely disappear, depending on their size and distance.

## Merklinger’s approach

Fortunately, this problem can be avoided if we focus at infinity. In this case, all objects in the scene are closer than the focus distance, such that equation (I2) applies in all cases, instead of equation (I4). Furthermore, the fraction in equation (I2) disappears, and we get a surprisingly simple description of the minimum object size

 s = a (I5)

Note that the aperture a of the lens is a distance which can be measured in mm, inches, or any other unit of length. With equation (A1), we can express a with the commonly used f-stop A and get

 s = f / A (I6)

For a camera focused at infinity, the minimum size of an object to be resolved thus depends on the focal length of the lens and the f-stop used, but not on the object distance. This formula is used for the minimum object size calculator.

For example, consider a lens with a focal length of 50 mm and the f-stop set to 8. The circle of confusion is assumed to be 0.033 mm. The hyperfocal distance is thus at about 9.6 m (31 ft). In this case, we get a minimum object size of just 3 mm (0.1 in) at a distance of 5 m, but a whopping 600 mm (2 ft) at 1000 m. Anything smaller than that (including most people) will more or less disappear. On the other hand, for the lens focused at infinity, the minimum object size is about 6 mm (0.2 in), regardless if the object is 5 m or 1000 m away. While the difference at 5 m is rather small, the effect at 1000 m may be quite dramatic.

The good news about Merklinger’s approach is that it does not only provide sharp images, but is also very easy to use. He actually recommends to focus at the most distant object of interest, and adjust the f-stop to the size of nearby objects that you still want to see. From equation (I6), the f-stop can easily be calculated as

 A = f / s (I7)

For a 50 mm lens and a targeted minimum object size of 10 mm, set your f-stop to 5. In his publications, Merklinger also investigates other uses of his approach, as in wildlife photography.

Note that equation (I6) describes a lower limit for the resolved object size. Other limitations such as heat haze, diffraction and the finite resolution of the image sensor may have the effect that the resolved object size is actually worse. However, this focusing strategy ensures that we can get as good as scene, lens and camera permit. Also note that neither the circle of confusion nor the size of the image sensor are used, so results are valid for all kinds of cameras alike.

## Hyperfocal distance revisited

Merklinger’s approach leaves us with the somewhat disturbing realization that the hyperfocal distance, even though based on a seemingly solid theory about the depth of field, does not lead to optimal results. So what’s wrong?

Simply put, it is optimizing the wrong parameter. The aim of the hyperfocal distance is to ensure that the image of a point is never spread beyond a certain size on the image plane. However, as Merklinger showed, this is not the major problem, as more distant objects will be completely smoothed out. In other words, the image of a point may be limited in size, but this doesn’t really matter if the point is not visible at all. Actually, it makes much more sense to ensure that we can see it.

So do I ever use the hyperfocal distance? No, never.