The focus distance d is the distance between the photographic subject and the camera sensor. It can be expressed as the sum of the subject distance g between subject and lens and the image distance h between lens and sensor as
| d | = | g + h | (F1) |
When the lens is focused, it is actually moved back or forth a little such that the lens equation is fulfilled. To calculate g and h for a given focus distance d, we solve equation (F1) for h and substitute it into the lens equation (L5)
| 1 / f | = | 1 / g + 1 / (d – g) | (F2) |
With some algebra, we end up with a quadratic equation
| g² – g d + f d | = | 0 | (F3) |
This equation has (at most) two different real solutions, given by
| g | = | d / 2 ± sqrt (d² / 4 – f d) | (F4) |
where sqrt denotes the square root. With equation (F1), we also get
| h | = | d / 2 ± sqrt (d² / 4 – f d) | (F5) |
For simplicity, we define the common root term as
| r | = | sqrt (d² / 4 – f d) | (F6) |
If we add the root term r in equation (F4), we must subtract it in equation (F5) and vice versa. In practice, the vast majority of lenses are constructed such that the distance g between subject and lens can be (much) larger than the distance h between lens and image sensor. Otherwise, we could not focus at infinity (there are actually a few lenses that can’t, such as the dedicated macro lens Canon MP-E 65 mm 2.8). Thus, we set
| g | = | d / 2 + r | (F7) |
| h | = | d / 2 – r | (F8) |
3.1 Minimum focus distance
Note that equations (F7) and (F8) can only be solved if the expression under the root in equation (F6) is non-negative, i.e.
| d² / 4 – f d | ≥ | 0 | (F9) |
Division by d (positive for all meaningful cases) gives
| d | ≥ | 4 f | (F10) |
Inequation (F10) defines a lower bound for the focus distance d, i.e. we cannot get any closer than 4 times the focal length. How close we can actually get also depends on the mechanical construction of the lens. The closest possible distance dmin that still gives a sharp image is called the minimum focus distance (MFD) or closest focus distance of the lens. It is closely related to the magnification and a major characteristic of any camera lens.
This is already enough theory to calculate the magnification of a lens. For the depth of field and related concepts, we also need to have a look at the aperture of the lens.