Law of Reflection

Equivalent forms

\angle i = \angle r

The simplest optical law — symmetry of incidence and reflection — emerges from Fermat's principle of least time.

Symbol	Name	SI unit	Dimension	Range
$theta_i$	Angle of incidence Angle between the incoming ray and the surface normal	rad	1	0 – 1.5708
$theta_r$	Angle of reflectionoutput Angle between the reflected ray and the surface normal	rad	1	0 – 1.5708

Unit systems

Symbol	SI	CGS	Imperial
$theta_i$	rad	rad	deg
$theta_r$	rad	rad	deg

Where it holds

Valid for specular (smooth) surfaces. Breaks down for rough surfaces where diffuse reflection dominates (surface roughness >> wavelength).

Dimensional analysis

\theta _i \to [1]

(radians, dimensionless)

\theta _r \to [1]

(radians, dimensionless)

Both sides are dimensionless angles.

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Discovery

Euclid · -300

Described in Euclid's Catoptrics (~300 BC). Hero of Alexandria later proved it follows from the principle of shortest path.

Try this

Where should you aim a laser pointer to hit a target via a mirror?

A laser hits a flat mirror at 35 degrees from the normal. Find the reflection angle and trace the reflected beam path.

Research status: stable

Real-world applications

Periscopes and kaleidoscopes: multiple reflections redirect images.
Laser alignment: mirrors steer beams precisely in optical systems.
Architectural acoustics: sound reflection off walls follows the same law.
Retroreflectors on the Moon: Apollo mission ranging experiments.

Common misconceptions

Angle is measured from the surface — it is always measured from the normal.
Rough surfaces don't reflect — they do, but diffusely (each micro-facet obeys the law locally).
Only mirrors reflect — all surfaces reflect; mirrors are simply smooth enough for specular reflection.

Experimental verification

Verified since antiquity using flat mirrors and protractors. Modern optical benches confirm to arc-second precision. Laser interferometry validates the law for surfaces with roughness << wavelength.

Derivation

From Fermat's principle: light travels the path of least time from source to observer via the mirror.

Minimizing the total path length d1 + d2 (same medium, same speed) with the constraint that the bounce point lies on the mirror surface yields

\theta _i = \theta _r

.

Equivalently, from Huygens' principle, each point on the mirror acts as a wavelet source, and constructive interference of reflected wavelets produces a wavefront at the reflection angle equal to the incidence angle.

Limiting cases

\theta _i \to 0

⟶

\theta _r \to 0

Normal incidence reflects straight back.

\theta _i \to 90^{\circ}

⟶

\theta _r \to 90^{\circ}

Grazing incidence: the ray barely skims the surface.

What if…

What if the mirror is curved?

The law still holds locally at each point — the normal changes along the curve, creating focusing (concave) or diverging (convex) reflections.

What if the surface is rough?

Each micro-facet obeys $\theta i = \theta r$ , but random orientations scatter light in all directions — diffuse reflection.

What if light hits at

0^{\circ}

(normal incidence)?

It reflects straight back: $\theta i = \theta r = 0^{\circ}$ . This is the principle behind retroreflectors.

1

Laser hitting a mirror at 35°

Given ·

theta i:: 0.6109

Find · theta_r

Steps

Apply the law of reflection: $\theta _r = \theta _i$
$\theta _r = 0.6109\,\mathrm{rad}$
Convert: $0.6109 \times (180/\pi ) = 35.0^{\circ}$

Result ·

\theta _r = \theta _i = 0.6109\,\mathrm{rad} = 35.0^{\circ}

2

Two-mirror periscope angle

Given ·

theta i:: 0.7854

Find · theta_r

Steps

First mirror: $\theta _r_{1} = \theta _i_{1} = 45^{\circ}$
Beam redirected $90^{\circ}$ downward
Second mirror: $\theta _r_{2} = \theta _i_{2} = 45^{\circ}$
Beam redirected another $90^{\circ} to$ horizontal — total $180^{\circ}$ path shift

Result · Each mirror reflects at

45^{\circ}

. Two

45^{\circ}

reflections redirect the beam by

90^{\circ}

total, enabling a periscope to see over obstacles.

Snell's Law→Brewster's Angle→