This page assumes you know nothing. Before you can read the parent note Sign convention, you must own every symbol it throws at you. We build them one at a time, each on top of the last.
Everything in optics is measured along ONE horizontal line called the principal axis. Think of it exactly like the x-axis you drew in school, but lying flat through the centre of the mirror or lens.
Figure s01 — A horizontal arrow (the principal axis) with the red pole marked as the origin x=0. Tick marks to the left read −3,−1; ticks to the right read +1,+3. It shows that "distance" in optics is really a signed position on one shared ruler.
Why do we need it? Because "distance" alone is ambiguous — 30 cm which way? By fixing an axis and an origin, "30 cm to the left" and "30 cm to the right" become two different numbers. That difference is the whole game.
Now we must pick which direction counts as +. The convention picks the direction the incident light is travelling, which we always draw left → right.
Figure s02 — Three red arrows (the incident light) travelling left→right toward a black mirror/lens line. The object dot sits upstream on the left (u<0); the region behind, downstream on the right, is labelled positive. Read it as: "the direction the light heads is +; behind the object is −."
Why this and not "positive = to the right of me"? Because for a mirror the light bounces back, so "downstream" is actually behind the mirror. Tying + to the light, not to a fixed left/right, is what lets one rule cover mirrors and lenses.
The beauty: the sign of v tells you the nature of the image for free.
Figure s03 — Two side-by-side panels. LEFT (mirror): object arrow on the left, its real inverted image (red) also on the left, so v<0. RIGHT (lens): object arrow on the left, its real inverted image (red) on the right, so v>0. The picture shows why the same word "real" gives opposite signs for a mirror and a lens — because a mirror reflects light back, a lens sends it through.
Mirror: rays reflect back to the left. Image on the left (real) ⇒v<0; image behind the mirror (virtual) ⇒v>0.
Lens: light passes through to the right. Image on the right (real) ⇒v>0; image on the left (virtual, same side as object) ⇒v<0.
For the language of real vs virtual images, see Real and virtual images.
A curved mirror/lens is a slice of a sphere. The centre of that sphere is the centre of curvature C; the distance from pole to C is the radius of curvature R.
Figure s04 — A concave mirror with centre of curvature C and pole P. One red ray comes in parallel to the axis, hits the mirror at point M, and reflects through the focus F. The line CM is a radius (a "normal"), and the two marked equal angles show why the reflected ray crosses the axis exactly halfway between P and C — that is, PF=21PC, i.e. f=R/2.
Because F and C are also positions on the same number line, they too get signs:
Figure s05 — A concave mirror with the object arrow (height h, up) at distance u on the left and the inverted image arrow (height h′, down, red) at distance v. The chief ray runs from the object tip to the pole P and reflects. The two shaded right triangles — object-tip–axis–pole and image-tip–axis–pole — are similar (they share the angle at P and both have a right angle). This similarity is what produces the ratio, and the flip of the image below the axis is what produces the minus sign.
