Mirror ray diagrams: move the object, watch the image form

How to use this simulation

1. Pick a concave, convex or plane mirror under "Mirror type". The pole P, principal focus F and centre of curvature C are marked on the principal axis.
2. Drag the blue object arrow along the axis with a mouse or finger, or use the "Object distance" slider.
3. For the concave mirror, the "Place the object" buttons jump to the six textbook positions: very far, beyond C, at C, between F and C, at F and between F and P.
4. Move the "Focal length" slider and the radius of curvature follows as R = 2f, so F and C slide together.
5. Orange lines are incident rays and red lines are reflected rays. Where reflected rays really cross in front of the mirror, the green image is drawn solid (real); where only their dashed extensions meet behind the mirror, it is dashed (virtual).
6. Read u, v, f, R, the magnification m and the nature of the image in the readings panel, and compare them with the solved problems below.

A spoon, a bus mirror and a torch: curved mirrors are everywhere

Pick up a shiny steel spoon and look into its hollow side. Your face is upside down. Turn the spoon over and look at the back: now your face is the right way up, but smaller and a little stretched. It is the same spoon, yet the two sides give completely different pictures, because the hollow side behaves like a concave mirror and the back behaves like a convex mirror.

Now bring the hollow side slowly towards your eye. At some point the upside-down face turns into a blur, and when the spoon is very close you see a big, upright picture instead. Changing only the distance flipped the image from inverted to erect. That single observation is what this whole page explains, and the simulation above lets you repeat it step by step.

Curved mirrors are all around you. The mirror on a bus or a motorbike that shows the road behind is convex. A dentist's little mirror, a shaving or make-up mirror, and the shiny bowl behind a torch bulb are concave. Once you know a few simple rules for how light bounces, you can predict with a ruler and a pencil where any of these mirrors puts its image, whether it is upright or upside down, and how big it is.

Starting from zero: the two laws of reflection

When light hits a smooth, shiny surface and bounces back into the same medium, we call it reflection. A household mirror is a sheet of glass with a thin coat of silver or aluminium on the back; the coat stops the light and sends it back.

Reflection follows two laws, and curved mirrors obey them exactly like flat ones. First, the incident ray, the reflected ray and the normal at the point of incidence all lie in the same plane. Second, the angle of incidence equals the angle of reflection, i = r.

The only extra difficulty with a curved mirror is that the normal points in a different direction at every point. Here geometry helps: on a sphere, the normal at any point always passes through the centre of the sphere. So a line drawn from the centre of curvature to any point on the mirror is the normal at that point. Every rule on this page grows out of that one fact.

Imagine cutting a piece out of a hollow rubber ball. If the inside of the piece is polished, you have a concave mirror; if the outside is polished, you have a convex mirror. A concave mirror makes parallel rays meet at a point, so it is called a converging mirror. A convex mirror makes parallel rays spread out, so it is called a diverging mirror.

Key words for spherical mirrors

Before drawing a single ray, get the vocabulary straight. Most one-mark definition questions come straight from this table, and every mark on the simulation uses these names.

Why the focal length is half the radius: f = R/2

Take a concave mirror and a ray travelling parallel to the principal axis that strikes the mirror at a point M. The line CM is the normal at M, because a radius is always perpendicular to the surface of a sphere. Call the angle of incidence θ.

The incident ray is parallel to the axis and CM cuts across both, so the alternate angle ∠MCP is also θ. By the law of reflection the reflected ray also makes θ with CM, and it crosses the axis at F. Triangle FMC therefore has two equal angles, so it is isosceles and FM = FC.

For a mirror with a small aperture, M is very close to the pole, so FM is almost equal to FP. Then FP = FC, which puts F exactly halfway between P and C. So PF = PC/2, or f = R/2. The same argument works for a convex mirror, with F and C both behind the mirror.

The four principal rays

Countless rays leave the tip of the object, but any two of them are enough to find its image: wherever two reflected rays cross, or appear to cross, the image of the tip sits. These four rays are the easy ones, because we know in advance exactly where each goes.

Concave mirror ray diagram: all six cases

This table is the most-asked table in the chapter. The last column is computed from the mirror formula for a concave mirror with f = -10 cm (R = -20 cm) in the New Cartesian sign convention. Press the "Place the object" buttons in the simulation to check each row.

Convex and plane mirrors: always the same answer

Put the object anywhere in front of a convex mirror and the image is always behind the mirror, between P and F, virtual, erect and diminished. The further the object goes, the smaller the image and the closer it creeps to F, but it never gets past F. The table below is computed for a convex mirror with f = +15 cm, in the Cartesian convention.

That is why vehicles use convex mirrors as rear-view and wing mirrors: a smaller image means a much wider field of view fits in a small mirror, and the image is always upright, so the driver is never confused. The catch is that things look further away than they are, which is why many wing mirrors carry the warning "objects in mirror are closer than they appear".

A plane mirror can be treated as a spherical mirror whose radius of curvature is infinite, so its focal length is infinite too. Putting 1/f = 0 into the mirror formula gives v = −u: the image is exactly as far behind the mirror as the object is in front, and m = +1, so it is virtual, erect and the same size. Choose "Plane mirror" in the simulation and watch the dashed extensions meet at the object's mirror twin.

The mirror formula: 1/v + 1/u = 1/f

A ray diagram tells you roughly where the image is; the mirror formula tells you exactly. Here u is the object distance, v the image distance and f the focal length, all measured from the pole.

This page uses the New Cartesian sign convention, the one in NCERT, most international textbooks and JEE/NEET. The pole is the origin and the incident light travels in the positive direction (left to right in the simulation). Distances measured along the incident light are positive, against it negative; heights above the axis are positive, below negative.

Because the object always sits in front of the mirror, against the incident light, u is always negative. The focus of a concave mirror is in front, so f is negative; the focus of a convex mirror is behind, so f is positive. A real image in front of the mirror has negative v, and a virtual image behind it has positive v.

Magnification is m = h′/h = −v/u. A negative m means a real, inverted image; a positive m means a virtual, erect image. If |m| > 1 the image is magnified, if |m| < 1 it is diminished. The readings panel above prints exactly these signs.

Things to try in the simulation

Do these one at a time, and after each one ask yourself "why did that happen?"

Solved problems (New Cartesian convention)

In each problem, write down what is given with signs, then the formula, then substitute. Problem 1 is the simulation's default state, so the panel shows the same answer when the page loads.

Concave mirror vs convex mirror

For every "state the differences" question, here is the whole comparison:

Common mistakes

Examiners see these again and again. Check your own answers for them.

• Forgetting the sign of u. In the Cartesian convention the object distance is always negative, because the object is in front of the mirror.
• Giving a concave mirror a positive focal length in Cartesian problems. Concave f is negative, convex f is positive.
• Mixing up R and f: if a question gives the radius of curvature, halve it first.
• Drawing the part behind the mirror with solid lines. Every virtual ray extension and every virtual image is dashed.
• Leaving arrowheads off the rays, or pointing them the wrong way. Light travels from the object to the mirror and then away from it.
• Saying the image is "at F" when the object is at F. The reflected rays are parallel and the image is at infinity.
• Writing m = v/u in a Cartesian problem. For mirrors the Cartesian magnification is m = −v/u, unlike a lens.

Spherical mirrors in real life

Shaving and make-up mirrors: concave mirrors with a long focal length. Your face is inside F, so you see an upright, enlarged image where every small detail stands out.

Dentist's mirror: a small concave mirror on a handle, held close to a tooth, gives an erect, magnified view of it. The round mirror an ENT doctor wears on the forehead is concave too; it gathers light onto a spot to light up the inside of an ear or throat.

Torches, car headlights and searchlights: the bulb sits at the principal focus of a concave reflector. Rays from F come back parallel to the axis (Ray 2), so the light leaves as a strong, narrow beam that reaches a long way instead of spreading out.

Solar cookers and solar furnaces: large concave mirrors collect parallel sunlight at the focus, where a black pot gets hot enough to cook.

Rear-view and wing mirrors: convex mirrors give an upright, smaller image, so a small mirror shows a wide stretch of road behind.

Road-bend and shop security mirrors: big convex mirrors at blind corners on hill roads and in the corners of supermarkets let you see round the bend or the whole shop floor at once.

Spherical aberration and parabolic mirrors

The rule f = R/2 is only exact for mirrors with a small aperture. On a wide spherical mirror, parallel rays that strike near the edge cross the axis a little in front of F, closer to the mirror. The rays no longer meet at one point but smear over a small region, and the image blurs. This defect is called spherical aberration.

To avoid it, torches, headlights, satellite dishes and large telescopes use parabolic mirrors instead of spherical ones. A parabola has the special property that every ray parallel to its axis, near the edge or near the middle, reflects through exactly the same focus. Like the textbooks, this simulation assumes a small aperture, so its rays meet perfectly.

Exam corner

Spherical mirrors appear every year in Grade 10 physics under "Light: reflection and refraction", return in Grade 12 ray optics, and turn up in JEE and NEET as quick sign-convention numericals. The questions follow a few patterns:

• Definitions: pole, centre of curvature, radius of curvature, principal focus, focal length, aperture, magnification.
• Proofs: show that f = R/2 for a mirror of small aperture, using the law of reflection and an isosceles triangle.
• Ray diagrams: draw the image for an object at a given position, usually between F and C or between F and P, and state its nature.
• Numericals: given u and f (or R), find v, m and the image height with correct signs, as in Problems 1 to 8.
• Applications: which mirror is used in headlights, as a shaving mirror or as a rear-view mirror, and why.

Quick revision

If you only read one section the night before the exam, read this one.

• A concave mirror converges light; a convex mirror diverges it; a plane mirror has infinite focal length.
• f = R/2: the focal length is half the radius of curvature.
• Four rays: parallel → through F; through F → parallel; through C → straight back; to P → equal angle on the other side.
• Mirror formula 1/v + 1/u = 1/f (Cartesian, u negative); magnification m = h′/h = −v/u.
• Concave mirror: object beyond F → real and inverted; between F and P → virtual, erect, magnified.
• Object at C → same-size image at C; object at F → image at infinity.
• Convex mirror → image always virtual, erect, diminished, between P and F.
• Simulation default: f = -10 cm, u = -30 cm → v = -15 cm, m = -0.50.

Frequently asked questions