Thin Lens Equation

1/f = 1/d_o + 1/d_i - relates the focal length of a thin lens to the object and image distances. Converging (convex) lenses have a positive focal length; diverging (concave) lenses have a negative focal length.

Sign convention

The standard sign convention for thin lenses (real-is-positive or Cartesian):

d_o > 0: object is on the incoming-light side of the lens (real object - the usual case).
d_i > 0: image forms on the outgoing-light side - a real image where light rays actually converge (can be projected onto a screen).
d_i < 0: image appears on the same side as the object - a virtual image where rays only appear to diverge from a point (cannot be projected). A magnifying glass always produces a virtual image.

Magnification formula

m = −d_i / d_o

m < 0 (negative): image is inverted - typical of real images formed by converging lenses (cameras, projectors).
m > 0 (positive): image is upright - typical of virtual images (magnifying glass, eyepiece lenses).
|m| > 1: image is enlarged; |m| < 1: image is reduced.

Common applications

Application	Condition	Result
Camera lens	d_o ≫ f	d_i ≈ f - small, inverted real image on sensor
Magnifying glass	d_o < f	Virtual, upright, enlarged image
Projector	d_o slightly > f	Large, inverted real image on screen (d_i ≫ d_o)
Telescope objective	d_o = ∞ (distant star)	Real image at focal point (d_i = f)

Lensmaker's equation

For a lens made of material with refractive index n and radii of curvature R₁ and R₂:

1/f = (n − 1)(1/R₁ − 1/R₂)

R₁ is the radius of the surface facing the incoming light; R₂ is the radius of the exit surface. By convention, a surface curving toward the incoming light has a positive radius. This equation explains why a biconvex lens (R₁ > 0, R₂ < 0) always has a positive focal length.