How this instrument works
A curved mirror does its work by geometry alone. Every ray leaving one point of an object, striking the surface, and obeying equal angles either side of the normal, returns to a single other point — and that second point is the image. Reciprocals appear in the relation because what a mirror really alters is vergence, the rate at which a wavefront converges or spreads, counted in reciprocal metres. A concave surface adds a fixed 1 ⁄ f of convergence to whatever arrives; the bookkeeping of that addition is the entire equation.
Ibn al-Haytham gave spherical mirrors whole sections of his Book of Optics, finished in Cairo around 1021, among them a puzzle awkward enough to still carry his Latin name: Alhazen's problem, locating the point on a sphere that reflects a ray between two given positions, which collapses into a quartic. Isaac Barrow taught the conjugate-distance relation at Cambridge in his Lectiones Opticae of 1669, Edmond Halley published a general focus formula covering both mirrors and lenses in the Philosophical Transactions of 1693, and Gauss folded all of it into a systematic paraxial framework in Dioptrische Untersuchungen, 1841 — which is why this arrangement is still labelled the Gaussian form. Useful magnitudes: a shaving mirror runs f near 0.5 m, a passenger-side car mirror roughly −0.6 m, a shop security dome about −0.15 m, and Hubble's 2.4 m primary 5.52 m.
Paraxial is the word doing quiet work here. This relation assumes rays hugging the axis at shallow angles, where sin θ may be swapped for θ without penalty. Throw a wide cone onto a spherical surface instead and outer zones focus short, smearing a point into a caustic — spherical aberration, and the reason Newton's 1668 reflector and every serious telescope since prefers a paraboloid, which gathers genuinely parallel rays with no approximation whatsoever. Two further assumptions hide inside: your object sits on or very near the axis, and vertex plus centre of curvature capture all geometry that matters. Stray off-axis and coma and astigmatism arrive, at which point no single image distance honestly describes what you are looking at.
- Enter Focal length. Concave takes a positive value, convex a negative one, and f is half the radius of curvature — a 200 mm sphere gives 100 mm.
- Enter Object distance, measured from the mirror's vertex out to whatever you are imaging. Millimetres, centimetres and metres all work; conversion happens before any reciprocal is taken.
- Read Image distance. Positive means a real image, forming in front where a card will catch it. Negative means a virtual one, apparently sitting behind the surface.
- Read Magnification. Its sign gives orientation — negative is inverted — and its size gives scale, so −0.5 describes half height, upside down.
- Object distance equal to Focal length is refused. Rays leave parallel there, so no image forms at any finite distance.
Worked example — a 100 mm mirror on the bench
Clamp a concave mirror of 100 mm focal length at one end of an optical bench, the sort of thing school kits ship by the dozen; its curvature radius is 200 mm. Stand a lit arrow filament 300 mm away, then slide a white card until that arrow snaps into focus. Enter 0.1 into Focal length and 0.3 into Object distance. Image distance returns 0.15 m, Magnification −0.5. Follow the arithmetic yourself: 1 ⁄ 0.1 is 10, 1 ⁄ 0.3 is 3.3333, their difference 6.6667 reciprocal metres, and one over that lands on 0.15 exactly.
Everything those figures claim shows up on the card. Light converges 150 mm out, half of 300, so your arrow measures half its original height — and points downward. Inversion is the signature of a real image, which is precisely why M carries a minus sign instead of being quoted bare. Walk the filament inward to 200 mm, twice the focal length, and both distances equalise at 200 mm with M = −1, full size. Walk it inside 100 mm and Image distance flips negative: nothing lands on any card, because what you now have is virtual, upright and apparently behind the glass, which is how a shaving mirror earns its keep.
Questions
Why did I get a negative image distance?
Because your image is virtual — it forms behind the surface, where no light actually travels, so no card or sensor can capture it. That happens whenever an object sits closer than the focal length on a concave mirror, and always on a convex one. Your eye still sees it perfectly well, since diverging rays entering a pupil get traced back to an apparent source. Magnification turns positive at the same instant, meaning upright.
How do I find the focal length of a mirror I already own?
Aim it at something effectively infinitely far — sunlight, or a distant streetlamp — and catch the bright spot on paper. Vertex to sharpest point is f, because 1 ⁄ dₒ vanishes for an infinite object distance. On a concave mirror that takes ten seconds and a clear sky, though never try it with anything bigger than a hand mirror, since a large one is a genuine fire-starter. Convex surfaces are easier gauged against curvature templates, then halved.
What sign convention does this sheet use?
Real-is-positive. Object distance counts positive for anything genuinely in front, image distance positive for real images also in front, focal length positive for concave and negative for convex. Magnification is −dᵢ ⁄ dₒ, so negative reads as inverted. Plenty of textbooks instead run a full Cartesian scheme with a signed axis and a chosen direction of travel, where object distances go negative. Identical physics, different placement of minus signs — and mixing both inside one problem is the quickest route to nonsense.
Does the same equation work for lenses?
Arithmetically yes: the thin-lens relation is 1 ⁄ f = 1 ⁄ dₒ + 1 ⁄ dᵢ, same symbols, same magnification rule. What changes is which side counts as positive, because light passes through a lens and bounces off a mirror. Real images from a lens land on the far side; real images from a mirror land on the same side as your object. Keep that geometry straight and one formula serves both instruments.
Why doesn't 1 ⁄ f = 1 ⁄ dₒ + 1 ⁄ dᵢ mean f = dₒ + dᵢ?
Because reciprocals refuse to distribute across addition, and this is far and away the commonest slip with curved mirrors. Take f = 0.1 m with dₒ = 0.3 m: the answer is 0.15 m, not 0.2 m, and that gap widens as an object creeps toward focus. Invert each term first, subtract in reciprocal metres, then invert once at the end. Optometrists sidestep the whole trap by working in dioptres, where it becomes plain subtraction — 10 D less 3.33 D is 6.67 D, and 6.67 D is 0.15 m.
What happens with the object exactly at the focal point?
Nothing focuses. Reflected rays leave parallel to one another, the image goes to infinity, and the equation hands you a division by zero rather than a number, so this sheet stops and says so. Far from being a defect, that is a working principle: searchlights, car headlamps and lighthouses all park their source at the focus to throw a collimated beam. Run the same geometry backwards and it collects starlight onto a detector.