Pinhole camera

The most basic form of camera is the pinhole camera: The ray diagram above determines the basic perspective equation that pinhole camera images follow: $$m=\frac{v}{u}$$ where \$m\$ is the image magnification (i.e. \$m=\frac{\text{image size}}{\text{object size}}\$), \$u\$ is the object distance and \$v\$ is the image distance.

What is the lens equation?

If the pinhole is replaced by a thin lens then the diagram remains unchanged and the equation for \$m\$ remains true. However, the object and image distances (\$u\$ and \$v\$) can no longer be chosen independently. For the object to be in focus, they must satisfy the lens equation: $$\frac{1}{u}+\frac{1}{v}=\frac{1}{f}$$ where \$f\$ is the focal length of the lens. The lens equation describes the mathematical relationship between the object and image distances (and the focal length) when the image of the object is in sharp focus.

So, for example, if we know the object distance and the focal length, the lens equation enables us to calculate the image distance needed to bring the object into focus: $$v=\frac{uf}{u-f}$$

Alternative forms of the lens equation

Let \$x=u-f\$ and \$y=v-f\$. Then if we rewrite the lens equation in terms of \$x\$ and \$y\$, it can be reduced to the form: $$xy=f^2$$ which is particularly simple and easy to work with. The distances \$x\$ and \$y\$ are now the object and image distances measured from the front and rear focal points of the lens, respectively. This form of the lens equation was popular in the early days of photography. Other forms of the lens equation exist as well, but are of less general interest and not included here.

It is fairly easy to derive other useful equations from those already given: $$x=\frac{f}{m},\space\space y=mf,\space\space\frac{y}{x}=m^2$$

If you know the magnification you want, these equations can be used to calculate the required object and image distances very simply.

Thick lenses

Most real camera lenses are nowhere near being "thin" lenses. Fortunately, the lens equations are still true provided that the object and image distances are measured from the appropriate points in the lens.

The distances \$x\$ and \$y\$ are the same as before (measured from the focal points). However, the distances \$u\$ and \$v\$ must be measured from what are called the front and rear principal points respectively. The easiest way to find the principal points of a lens is to find the focal points of the lens and then measure a distance \$f\$ from each to get the principal points.

Some common uses of the lens equation are described briefly below.

Working out the lens extension needed for a given image magnification

The equation \$y=mf\$ can be used to calculate the lens extension needed to achieve a given magnification. The image distance, \$y\$, is the distance between the image and the focal point (which is the position of the image when the object is at infinity). So \$y=mf\$ is the extension required (for a lens fixed at infinity focus) to achieve a magnification of \$m\$.

For example, a magnification of 0.5 requires an extension of 0.5 times the focal length and a magnification of 3 requires an extension of 3 times the focal length.

Measuring the focal length of an unknown lens

The equation \$x=\frac{f}{m}\$ can be rearranged to \$f=mx\$ and used to calculate the focal length from \$m\$ (the image magnification) and \$x\$ (the object distance from the front focal point).

This is a good method for measuring the focal length of an unknown lens. Use an object of known size at a known distance, \$x\$, from the front focal point of the lens. Measure the size of the image (when it is in focus) and calculate the magnification \$m=\frac{\text{image size}}{\text{object size}}\$. Then calculate the focal length of the lens \$f=mx\$. If the measurements are done at fairly large object distances, then it doesn't matter if the position of the front focal point is not known accurately.

Calculating the image magnification at a known object distance

The equation \$x=\frac{f}{m}\$ can be rearranged to \$m=\frac{f}{x}\$ and used to calculate the magnification from the object distance and focal length.

For example, at 100 m distance from a 50mm lens, the image magnification will be \$\frac{0.050}{100}=0.0005\$. Strictly speaking, the object distance should be measured from the front focal point of the lens, but at large distances the error in measuring from somewhere else on the lens or camera is insignificant.

For lenses that focus by moving the whole lens assembly, the equation \$m=\frac{f}{x}\$ is correct for all object distances, including macro photography.

However, for internal focussing lenses (as are most modern auto-focus lenses) the focal length of the lens often changes significantly during focussing and the position of the front focal point may change also. Unfortunately lens manufacturers rarely provide information on how these parameters change with focussing.