Deriving the Lens Equations

The Derivation:

We can derive equations for locating images formed by converging lenses in much the same way as we did for converging mirrors.

Ray Diagram for a Converging Lens - only 2 rays are shown

In the ray diagram above, the two shaded triangles are similar, so corresponding sides are proportional. Therefore,

Solving for the image height gives:

Another ray diagram for the same situation

In the ray diagram above (the "ray through the focus" in the first diagram has been replaced by the "ray parallel to the axis"), the shaded triangles are similar, so:

Combining this equation with the first one on this page (both right sides equal H_i/H_o) gives:

Solving for the image distance gives:

Summary:

To determine the size and position of an image formed by a converging lens, you may use the equations:

Isn't it strange that these are the same equations as we derived for locating images in converging mirrors!

Especially since the images are formed by different processes - mirrors form images by reflection, but lenses form images by refraction...

last update November 26, 1997 by JL Stanbrough (jstanbro@venus.net)