A *converging lens* is characterized by collecting parallel light rays in a focal point **F**. As a consequence, light emitted from a point in an *object plane* (parallel to the lens, at a finite distance) is collected in a single point in a corresponding *image plane* (also parallel to the lens). For a *thin* converging lens where its thickness along the optical axis (drawn as a horizontal blue line) is small compared to its focal length, the situation looks as shown in fig. 1.

The lens equation describes the relation between focal length **f** of the lens, *object distance* **g** (between object and lens) and *image distance* **h** (between lens and image sensor or film) when the image of the object appears sharp. It can be derived as follows.

By similar triangles along the purple ray passing through **F** left of the lens, we have

G / (g - f) | = | H / f | (L1) |

Likewise, by similar triangles along the purple ray through **F** right of the lens, we get

G / f | = | H / (h - f) | (L2) |

Solving both equations for G / H results in

(g - f) / f | = | f / (h - f) | (L3) |

After some simple algebraic transformations, the equation looks like this

g h | = | f h + f g | (L4) |

After division by f g h, we get the well-known form

1 / f | = | 1 / g + 1 / h | (L5) |

This equation is known as the *lens equation* or *thin lens formula*.

When the lens is focused, it is actually moved back or forth such that the lens equation is satisfied, i.e. the distances **g** and **h** are adjusted. However, the *focusing distance* **d** which you can read and set on your lens measures the full distance between object and image plane, i.e.

d | = | g + h | (L6) |

To calculate **g** and **h** for a given distance **d**, we solve equation (L6) for **h** and substitute it into the lens equation (L5)

1 / f | = | 1 / g + 1 / (d - g) | (L7) |

With some algebra, we end up with a quadratic equation

g² - g d + f d | = | 0 | (L8) |

This equation has (at most) two different real solutions, given by

g | = | d / 2 ± sqrt (d² / 4 - f d) |
(L9) |

where *sqrt* denotes the square root. With equation (L6), we also get

h | = | d / 2 ± sqrt (d² / 4 - f d) |
(L10) |

For simplicity, we define the root term as

r | = | sqrt (d² / 4 - f d) |
(L11) |

If we add **r** in equation (L9), we must subtract it in equation (L10), and vice versa. In practice, the vast majority of lenses are constructed such that the distance **g** between object and lens can be much larger than the distance **h** between lens and image sensor. Otherwise, you could not focus at infinity (I am only aware of one such lens, the dedicated macro lens Canon MP-E 65mm). Thus, we set

g | = | d / 2 + r | (L12) |

h | = | d / 2 - r | (L13) |

These equations are surprisingly powerful to calculate e.g. the magnification or depth of field of a lens.