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 the 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)|
Finally, after division by f g h, we get the well-known form
|1 / f||=||1 / g + 1 / h||(L5)|