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Glasses have existed for over 700 years — but for most of that time, nobody could explain why a piece of curved glass fixed someone's blurry world. A short-sighted person sees a sharp image of everything that doesn't exist yet (the rays haven't converged), while a long-sighted person forms a sharp image behind their retina (where there's no screen). One curved piece of glass, costing a few pence, bends light by exactly the right amount to shift that image onto the right spot. How does curvature alone know how much to bend?
Glasses have existed for over 700 years — but for most of that time, nobody could explain why a piece of curved glass fixed someone's blurry world. A short-sighted person sees a sharp image of everything that doesn't exist yet (the rays haven't converged), while a long-sighted person forms a sharp image behind their retina (where there's no screen). One curved piece of glass, costing a few pence, bends light by exactly the right amount to shift that image onto the right spot. How does curvature alone know how much to bend?
A lens refracts light at two curved surfaces. Converging (convex) lenses bring parallel rays together at a single focal point; diverging (concave) lenses spread them apart. Whether the refracted rays actually meet — and on which side — determines every property of the image: its size, orientation, and whether it can be projected on a screen. This one idea explains cameras, microscopes, telescopes, the human eye, and every pair of spectacles ever made.
Every thin lens has a focal length f: the distance at which it focuses parallel rays. Converging lenses have positive f; diverging lenses have negative f. Three image types exist: Real (rays actually converge on the far side — can be projected on a screen, always inverted), Virtual (rays appear to diverge from a point — cannot be projected, usually upright), and No image (object at the focal point sends rays out parallel — image at infinity). The thin lens equation relates object distance u, image distance v, and focal length f.
Sign convention (real-is-positive): real objects have positive u, real images have positive v, virtual images have negative v. For a distant object (u very large), 1/u ≈ 0 so v ≈ f — the image forms almost exactly at the focal point. This is why a camera sensor sits at the focal plane. The human eye adjusts its focal length by squeezing or relaxing the crystalline lens (accommodation). Short-sightedness (myopia): the eye's focal length is too short, so a diverging correction lens is needed. Long-sightedness (hyperopia): focal length too long, corrected with a converging lens. Lens power P = 1/f measured in dioptres (D) — a −2 D lens is diverging with f = −0.5 m.