# 3 Focus Distance

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.