The coordinate sign convention

Mirrors, lenses, refracting surfaces — one formula each, and every one of them is written in the same convention. Learn to read a diagram as signed coordinates and you never again memorise "real is minus, virtual is plus": the algebra does it for you. Drag the point below and watch its sign flip.

01 — the whole idea

Everything is a signed coordinate from the pole

Put the origin at the pole P (the mirror's vertex, or a lens's optical centre). Point the +x axis along the incident light. Now every distance is just a coordinate — with a sign baked in. Which way the light travels is your choice: send it either way and watch only the signs change, never the distances.

stand on the:

✋ drag the yellow point across the axis and above/below it

ruler: {{ axMag }} units

→

x = {{ axXtxt }}

y = {{ axYtxt }}

Why can you flip the light?

A sign convention is a choice of reference, not a law of nature. Walk to the other side of the table and the same experiment sends its light the opposite way. Every length you'd read off a ruler is identical — but +x now points the other way, so every sign flips. That is all the signs on u, v and f ever encode: which side of the pole a thing sits on, relative to the light — never its raw distance.

02 — the four quantities you'll actually plug in

u, v, f and R — where their signs come from

03 — the sign students trip on most

Why concave is f < 0 and convex is f > 0

Send in a beam parallel to the axis. Where it focuses decides the sign. A concave mirror focuses the light in front — against the incident direction — so f is negative. A convex mirror only appears to focus behind it, along the incident direction — so f is positive.

|f| small |f| large

live signs · {{ mirrorLabel }}

focal length f{{ fSignTxt }}

radius R = 2f{{ rSignTxt }}

focus is{{ focusNature }}

04 — the payoff

One formula, every case — let the signs work

Worked: concave mirror, real object

Object 30 cm in front, focal length 20 cm concave. Put in the signs — u = −30, f = −20 — and solve the mirror equation.

1v= 1−20− 1−30

This gives v = −60 cm. The minus sign tells you the image is real and in front — you never had to decide that yourself. Magnification m = −v/u = −(−60)/(−30) = −2: inverted, twice as tall.

The sign, decoded

05 — quick check

Call the sign

Six one-tap questions. Don't compute anything — just read the geometry and name the sign. {{ quizSolved }}/6 right.

Q{{ q.n }}. {{ q.q }}

Recap card — the convention

◦ Origin at the pole / optical centre.

◦ +x = direction of incident light.

◦ Against the light → negative.

◦ Above axis +, below −.

◦ Real object u < 0; concave f < 0, convex f > 0.

◦ m = −v/u; negative m ⇒ inverted.

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