Derivation of three physical elements of depth of field (DoF): circle of confusion (CoC) standard, blind calculation of focus point and hyperfocal distance

1. Strict optical definition of depth of field and Circle of Confusion (CoC)

In absolute geometric optics, an ideal lens can only converge object points on a specific plane in space into a volumeless point on the focal plane. All points before and after this plane are projected on the sensor as a blurred spot that gradually diverges, called a circle of diffusion.

• Physiological resolution limit of the human eye: At a normal photopic distance of 25 cm, the human eye can only distinguish details with a viewing angle of approximately 1 arcminute (approximately 0.1 mm). As long as the diameter of the blurred spot on the sensor is smaller than this physiological threshold, the brain will mistakenly think that it is "absolutely clear".
• Industry benchmark circle of confusion constant: Under the 35mm full-frame (36×24mm) film standard, the industry-wide limit allowable circle of confusion diameter is δ ≈ 0.030mm (APS-C format is 0.020mm, medium format is 0.050mm).

2. Mathematical formula of depth of field and three major physical determining variables

The exact depth of field formula is:
$$\text{Depth of field} \approx \frac{2 \cdot N \cdot c \cdot u^2}{f^2}$$
Among them: $N$ is the aperture F value, $c$ is the diameter of the circle of confusion, $u$ is the focus object distance, and $f$ is the lens focal length. Shocking and rigorous physical conclusions can be derived from the formula:

1. The impact of object distance ($u^2$) explodes at a square level! : The closer you are to the subject, the depth of field shrinks sharply at a square rate; the farther away you are from the subject, the depth of field increases sharply at a square rate;
2. Focal length ($f^2$) affects inversely square! : Telephoto lenses (such as 200mm) have extremely shallow depth of field, and wide-angle lenses (such as 24mm) have extremely deep depth of field;
3. Aperture ($N$) only has a linear linear effect! : When the aperture is doubled (such as f/2.8 to f/5.6), the depth of field is only doubled, and its impact weight is much lower than the object distance and focal length.

3. The "asymmetric proportion law" of foreground depth and back depth of field

Many people mistakenly believe that the clear range before and after the focus point is exactly the same and symmetrical. This is a huge misunderstanding:

• Close macro shooting (object distance is extremely close): The ratio of foreground depth to back depth of field is close to 1:1 (completely symmetrical);
• Normal portrait distance (2-3 meters): Foreground depth accounts for 1/3 (33%), and back depth of field accounts for 2/3 (67%);
• Landscape hyperfocal distance: The back depth of field extends infinitely to infinity (accounting for more than 90%), and the foreground depth only covers a small section in front of the focus point.
• Practical Tips: When taking a group photo of three people standing front to back, always focus on the eyes of the person standing in the front (closest to the camera)! Let the wide rear depth of field of the rear 2/3 naturally cover the companions behind you. If you focus on the last person, the person in front will be blurred because they are out of the first 1/3 of the depth of field!

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