Laws of reflection of light

Look into any mirror and light is obeying two strict rules on its way back to your eye. Every time a ray strikes a smooth surface, it bounces off at exactly the angle it arrived, and the whole event — incoming ray, outgoing ray, and the perpendicular line at the point of contact — stays flat, in one plane. These are the laws of reflection, and they are not a special property of mirrors alone: the same two rules govern a torch beam off a steel almirah, a laser off a spoon, and every curved mirror you will meet later in this chapter. Once you can measure angles from the right reference line — the normal, never the surface — the laws turn into a simple, reliable tool for predicting exactly where a reflected ray will go, in one mirror or in a system of several.

The core insight

At any smooth surface the angle of incidence equals the angle of reflection, and the incident ray, reflected ray and normal all lie in one plane.

The core idea

When a ray of light meets a smooth, polished surface, part of it bounces back into the same medium — that is reflection. Every ray diagram of reflection needs four labelled parts: the normal (the line drawn perpendicular to the surface at the point of incidence), the incident ray (the ray arriving), the reflected ray (the ray leaving), and the two angles measured from the normal — the angle of incidence $i$ and the angle of reflection $r$ . The two laws of reflection are:

i = r

where: $i$ is the angle between the incident ray and the normal, and $r$ is the angle between the reflected ray and the normal, both measured at the point of incidence, on the same side of the surface. The second law has no equation but is just as essential: the incident ray, the reflected ray and the normal at the point of incidence all lie in the same plane. Skip this condition in an answer and a reflection description is incomplete even if $i = r$ is stated correctly. Both laws hold at every point of a smooth surface, whether that surface is flat or curved. For a plane mirror the normal points the same way everywhere; for a spherical mirror the normal is different at each point, because it runs along the radius of curvature drawn to that particular point. This is why the laws of reflection, learned here for a flat surface, carry forward unchanged into the curved-mirror ray diagrams later in the chapter. Reflection from a smooth surface — a mirror, still water, polished metal — is regular (specular) reflection: parallel incident rays stay parallel after bouncing off, so a sharp image forms. Reflection from a rough surface — paper, cloth, a plastered wall — is diffuse reflection: the laws of reflection still hold at every microscopic patch of the surface, but because those patches face in many different directions, the reflected rays scatter every which way and no clear image forms. Diffuse reflection is why an ordinary room lit by one bulb is visible from every corner, even though you cannot see the filament's image on the wall.

Figure

A ray diagram showing an incident ray and reflected ray making equal angles with the normal to a plane mirror

Deviation of the reflected ray

Derivation ADeviation of a ray reflected once from a plane mirror

The goal is to find a single formula, using only the two laws of reflection and the geometry of a straight line, for the angle of deviation — the angle through which a ray is turned when it reflects off a plane mirror once — expressed purely in terms of the angle of incidence $i$ , so that later, when two mirrors are combined, the total turning of the ray can be built up mirror by mirror instead of redrawn from scratch each time.

Step 1. Draw the incident ray, the normal, and the reflected ray at the point of incidence $O$ . By the first law of reflection, the angle of incidence and the angle of reflection are equal: $i = r$ . The angle between the incident ray and the reflected ray, measured going the short way round through the normal, is therefore $i + r = 2i$ .

Step 2. The deviation $\delta$ of a ray is defined as the angle between the original direction of travel (the incident ray extended straight through $O$ , as if it had not been reflected at all) and the actual direction of travel after reflection (the reflected ray). The extended incident ray and the actual incident ray lie along the same straight line, so the angle between the extended ray and the mirror surface is the same as the angle between the incident ray and the surface, namely $90° - i$ .

Step 3. The reflected ray makes an angle $90° - r = 90° - i$ with the mirror surface. The incident ray extended straight through $O$ makes the same angle $90° - i$ with the mirror surface, but on the opposite side of the point of incidence. The deviation $\delta$ — the angle between the extended incident ray and the reflected ray, both measured up from the mirror surface on the outgoing side — is the sum of these two equal angles:

\delta = (90° - i) + (90° - i) = 180° - 2i

Step 4. Read the result. The deviation produced by a single reflection is $\delta = 180° - 2i$ . Check the limiting cases: at normal incidence $i = 0°$ the ray is sent straight back on itself, so $\delta = 180°$ ✓; at grazing incidence $i \to 90°$ the ray barely bends, so $\delta \to 0°$ ✓. (The bare angle between the incident and reflected rays is $2i$ from Step 1; the deviation is its supplement, $180° - 2i$ , because deviation is measured from the incident ray's forward direction of travel.) This is the building block used below for a system of two mirrors, where the deviations at each mirror add.

Worked example

Two plane mirrors are placed facing each other at right angles ( $90°$ ), like the corner mirrors of a simple periscope. A ray of light strikes the first mirror at an angle of incidence $i_1 = 40°$ , reflects, and then strikes the second mirror. Find the deviation produced at each mirror and the total deviation of the ray after both reflections.

Step 1. At mirror 1, the ray strikes at $i_1 = 40°$ , so by the law of reflection $r_1 = 40°$ . Using the deviation formula from the derivation, the ray is turned through $\delta_1 = 180° - 2i_1 = 180° - 2 \times 40° = 100°$ at the first mirror.

Step 2. Because the two mirrors are perpendicular, the geometry of the triangle formed by the ray path and the two normals fixes the angle of incidence at the second mirror as $i_2 = 90° - i_1 = 90° - 40° = 50°$ . By the law of reflection, $r_2 = i_2 = 50°$ , so the deviation at mirror 2 is $\delta_2 = 180° - 2i_2 = 180° - 2 \times 50° = 80°$ .

Step 3. The total deviation after both reflections is $\delta_1 + \delta_2 = 100° + 80° = 180°$ . A ray turned through exactly $180°$ emerges travelling antiparallel to (exactly opposite, but parallel in line to) the original incident ray — the emerging ray is parallel to the incoming ray but reversed in direction. This confirms that two mirrors at right angles always send the ray back parallel-reversed, whatever the entry angle $i_1$ — the working principle of a corner reflector and a simple periscope.

Competitive stretch

If the two mirrors were placed at a general angle $\theta$ instead of $90°$ , the same two-step deviation argument gives a total deviation of $\delta_1 + \delta_2 = (180° - 2i_1) + (180° - 2i_2) = 360° - 2(i_1 + i_2) = 360° - 2\theta$ , since $i_1 + i_2 = \theta$ for mirrors inclined at $\theta$ : the total turning depends only on the angle between the mirrors. Setting $\theta = 90°$ recovers the $180°$ found above.

Common traps

Measuring $i$ or $r$ from the mirror surface instead of the normal. The laws of reflection are defined strictly relative to the normal; always draw the normal first before marking any angle.
Thinking diffuse reflection breaks the laws of reflection. It does not — every tiny facet of a rough surface still obeys $i = r$ at its own local normal; only the many differing facet orientations scatter the rays.
Dropping the "same plane" condition. Stating only $i = r$ is an incomplete answer; the coplanarity of incident ray, reflected ray and normal is the second law and is often asked for separately.
Forgetting the deviation formula only applies per mirror. $\delta = 180° - 2i$ is the turning at one reflecting surface; for multiple mirrors, add the deviations at each surface, as in the worked example.

Ask the tutor: if the geometry of multi-mirror setups or converting a surface-angle to an incidence-angle keeps tripping you up, ask the AI tutor to walk through two or three more ray diagrams step by step.

Why do we always measure angles from the normal and not from the mirror surface itself?

The normal is chosen as the reference because it is the one line that is always well defined at the point of incidence, even on a curved mirror where the surface itself is tilted differently at every point. Measuring from the surface would give a different-looking rule for every shape of mirror; measuring from the normal keeps the law $i = r$ identical for a flat mirror, a concave mirror, or a convex mirror. If a question gives you an angle from the surface, first subtract it from $90°$ to get the angle from the normal before applying the law.

Does the law i = r apply to every single ray hitting a rough, non-shiny object, or only to mirrors?

It applies to every ray at every point, on any smooth patch of any surface, mirror or not. What differs between a mirror and a sheet of paper is not the law but the surface's texture. A mirror is smooth on the scale of the wavelength of light, so all its facets share one orientation and the reflected rays stay parallel, giving a clear image. Paper is rough on that scale, so countless tiny facets face in countless directions; each one obeys $i = r$ locally, but the rays leave in all directions, which is diffuse reflection, not a violation of the law.

In the two-mirror problem, why is the angle of incidence at the second mirror equal to 90 degrees minus the angle at the first mirror, and not just the same angle?

It comes from the geometry of the triangle formed by the ray's path between the two mirrors and the two normals, combined with the fact that the mirrors themselves are perpendicular. The normal to mirror 2 is rotated $90°$ from the normal to mirror 1, so the same ray makes a different angle with each normal. Working through the triangle's angle sum ( $180°$ ) shows the two angles of incidence, $i_1$ and $i_2$ , must add up to $90°$ whenever the mirrors are perpendicular — so $i_2 = 90° - i_1$ . This is a geometry consequence of the mirror arrangement, not a new law of reflection.