The thin lens equation ties together where an object sits, where its image forms, and the focal length of the lens. The same equation describes curved mirrors, with the sign conventions adjusted.
Sign conventions
Almost every mistake here is a sign error. Using the standard convention: focal length is positive for a converging lens and negative for a diverging one. Image distance is positive when the image forms on the far side of the lens, which makes it real and projectable onto a screen; negative when it forms on the same side as the object, which makes it virtual. Negative magnification means the image is inverted.
The five cases for a converging lens
- Beyond 2f: real, inverted, reduced. This is a camera.
- At 2f: real, inverted, same size.
- Between f and 2f: real, inverted, enlarged. This is a projector.
- At f: no image; rays leave parallel. This is a collimator.
- Inside f: virtual, upright, enlarged. This is a magnifying glass.
What “thin” leaves out
The model assumes the lens has negligible thickness and that rays stay close to the axis. Real lenses suffer spherical aberration, where edge rays focus at a different point from central ones, and chromatic aberration, where different colours focus differently because the refractive index varies with wavelength. Camera lenses use several elements of different glasses precisely to cancel these effects.