Playground

Static SVG showing incident and reflected rays at equal angles from the normal on a flat mirror surface.

Variables

SymbolNameSIDimensionRange
thetaitheta_iAngle of incidence
Angle between the incoming ray and the surface normal
rad10 – 1.5708
thetartheta_rAngle of reflectionoutput
Angle between the reflected ray and the surface normal
rad10 – 1.5708

Deep dive

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.
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.
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.
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.

Worked examples

Laser hitting a mirror at 35°

Given:
theta_i:
0.6109
Find: theta_r
Solution

theta_r = theta_i = 0.6109 rad = 35.0°

Two-mirror periscope angle

Given:
theta_i:
0.7854
Find: theta_r
Solution

Each mirror reflects at 45°. Two 45° reflections redirect the beam by 90° total, enabling a periscope to see over obstacles.

Scenarios

What if…
  • scenario:
    What if the mirror is curved?
    answer:
    The law still holds locally at each point — the normal changes along the curve, creating focusing (concave) or diverging (convex) reflections.
  • scenario:
    What if the surface is rough?
    answer:
    Each micro-facet obeys θi = θr, but random orientations scatter light in all directions — diffuse reflection.
  • scenario:
    What if light hits at 0° (normal incidence)?
    answer:
    It reflects straight back: θi = θr = 0°. This is the principle behind retroreflectors.
Limiting cases
  • condition:
    theta_i → 0
    result:
    theta_r → 0
    explanation:
    Normal incidence reflects straight back.
  • condition:
    theta_i → 90°
    result:
    theta_r → 90°
    explanation:
    Grazing incidence: the ray barely skims the surface.

Context

Euclid · -300

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

Hook

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.

Dimensions:
lhs:
θ_i → [1] (radians, dimensionless)
rhs:
θ_r → [1] (radians, dimensionless)
check:
Both sides are dimensionless angles. ✓
Validity: Valid for specular (smooth) surfaces. Breaks down for rough surfaces where diffuse reflection dominates (surface roughness >> wavelength).

Related formulas

Snell's LawBrewster's Angle