Real-is-positive thin-lens convention: f > 0 converging; f < 0 diverging. A real object has u > 0; a virtual object has u < 0. A real image has v > 0; a virtual image has v < 0.
Thin-lens equation: 1/f = 1/u + 1/v. Lateral magnification m = −v/u, with negative m indicating an inverted image.
Cartesian radii: Light travels left to right. Radius R is positive when the centre of curvature lies on the outgoing side of its surface, and negative when on the incoming side. A plane face has zero curvature and is represented by R = null.
Lensmaker: 1/f = (n_lens/n_medium − 1)(1/R₁ − 1/R₂), for an ideal thin lens in a uniform surrounding medium. Positive resulting f means converging.
Optical power: P = 1/f in metres, or P = 100/f when f is in centimetres. Power is measured in dioptres (D).
Two thin lenses: The first image becomes the second object. u₂ = d − v₁. A negative u₂ is a virtual object. Total magnification is the product of the individual signed magnifications.
Infinity regime: When the original object is at the first lens's focal point, rays entering the second lens are collimated; the second image forms at f₂. A final image at infinity has no finite image position.
Dispersion: Different wavelengths can have different refractive indices. The longitudinal chromatic aberration here is Δf = f_red − f_blue.
Assumptions: Geometrical optics, thin lenses, paraxial rays, homogeneous isotropic materials and monochromatic light for each individual ray calculation. Lens thickness and reflections are neglected.