The Magic Numbers of Photography - Calculating Depth of Field and the like

Over the years, in addition to an interest in photography, I have developed an interest in science. I love reading about and discovering the relationship between different variables and there is nothing more exciting than when that relationship can be boiled down to a clean equation. Luckily for me Photography is essentially applied physics and, as physics is also in love with equations, this means photography has plenty of lovely equations for those keen enough to be interested in them. As I am not a formal student of photography I have not been exposed to these equations so I am currently collecting them as part of my learning process and also with an eye to a wee project I'm working on.

Calculating Depth of Field

Depth of field is something that many amateur photographers express an interest in. Often they will tell you that an image has great depth of field or a lovely shallow depth of field but it is far rarer for them to say that the depth of field is 143mm. Depth of field refers to the amount of an image that appears to be in sharp focus. It has depth because the image is produced by light bouncing off objects in the real world. The distance from where sharp focus begins on an object to where it ends on an object gives us our measure of depth. Hence, depth of field.

While most photographers quickly learn the general relationship between focal length, aperture, subject distance, and depth of field physics allows us to specify it exactly. That being said the first step to making this determination comes from defining something else, the circle of confusion. For our purposes, we will use the circle of confusion value for an APS-C (crop) sensor.  0.019948

The next step is to determine the hyperfocal distance of the lens. The hyperfocal distance is the point at which an object at infinity appears sufficiently sharp when photographed. It varies with focal length, aperture, and the circle of confusion and is defined below:

Focal Length²/(Aperture x Circle of Confusion)

While hyperfocal distance alone tells us little about the depth of field produced by the factors mentioned ealrier it is calculated first as its value is required when calculating the nearest and furthest focal point. The distance between the nearest and furthest focal point being the depth of field.

 Nearest Focal Point = (Hyperfocal*distance to subject)/(Hyperfocal+(distance to subject - focal length))

 Furthest Focal Point = (Hyperfocal*distance to subject)/(Hyperfocal-(distance to subject - focal length))

Depth of Field = Furthest Focal Point - Nearest Focal Point