Thin-Lens Equation + Magnification
Find image distance and magnification from focal length and object distance for a single thin lens.
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The engineering
The thin-lens equation ties focal length to the object and image conjugate distances, assuming lens thickness is negligible next to those distances. Solve it to place the sensor, size the image, and tell whether the image is real (projectable) or virtual (only viewable through the lens).
Sign convention here: distances measured from the lens, real images on the far side give positive sᵢ, and negative magnification means inverted. A rule of thumb: an object at 2f images at 2f at 1:1 (m = −1), and moving the object toward f pushes the image toward infinity — which is why macro work needs so much bellows draw.
Where this math comes from
Lensmakers ground working spectacles and telescope objectives long before anyone had the equation — Renaissance craftsmen worked by trial and feel. The Gaussian conjugate relation came together as geometric optics matured, with Carl Friedrich Gauss's 1841 Dioptrische Untersuchungen formalizing the treatment of optical systems by principal planes and cardinal points.
The thin-lens simplification — collapsing a real element to a single plane — is the first approximation every optics student meets, and Eugene Hecht's textbook Optics made the sign conventions and magnification bookkeeping standard for generations of engineers designing everything from camera lenses to laser relay optics.
- 1611Johannes KeplerDescribes real-image formation and the converging-lens telescope in Dioptrice.
- 1841Carl Friedrich GaussFormalizes paraxial optics and cardinal points in Dioptrische Untersuchungen.
- 1974Eugene HechtOptics standardizes the thin-lens sign conventions used on this card.
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