Reflection

The law of reflection

When a ray of light hits a smooth surface:

1. The angle of incidence (θ_i, between incident ray and surface normal) equals the angle of reflection (θ_r): θ_i = θ_r.
2. The incident ray, normal, and reflected ray all lie in the same plane.

Both angles measured FROM THE NORMAL (perpendicular to surface), not from the surface itself.

Specular vs diffuse

Type	Surface	Behavior	Examples
Specular (mirror)	Smooth (Δh < λ)	Parallel rays reflect parallel	Mirror, polished metal, calm water
Diffuse	Rough (Δh > λ)	Scattered in many directions	Paper, drywall, fabric, dirt
Mixed (most surfaces)	Partially smooth	Both effects	Glossy paint, varnished wood

For visible light (λ ≈ 500 nm), surface roughness needs to be much less than 500 nm for specular reflection. Most polished metals achieve this; paper does not.

Reflectivity values

Surface	Visible light reflectivity
Silver (clean)	~95-98%
Aluminum	~85-92%
Gold (in IR)	~98% IR; ~70% visible
Glass at normal incidence	~4% (per surface; 8% for double surface)
Water at 60° incidence	~6%
Water at 80° incidence	~30%
Snow (fresh)	~80-90%
Asphalt (dry)	~5-10%
White paint	~70-80%
Black paint	~3-5%
Black hole	0% (everything absorbed)

Real-world reflection

System	Reflection type
Bathroom mirror	Specular; second-surface mirror with silver backing
Camera lens	Specular at glass surfaces (~4% loss per surface)
Anti-reflection coatings	Reduce surface reflection from 4% to <0.5%
Telescope mirror	First-surface; ~95% reflectivity
Solar panel	Anti-reflective top to maximize absorption
Periscope (submarine)	Two prisms; total internal reflection
Optical fiber	Total internal reflection at core/cladding boundary
Bicycle reflector	Corner-cube design — sends light back to source (retro-reflection)
Glittering ocean	Specular reflection of sun off water surface

JavaScript — reflection geometry

// Reflection of a ray off a surface with given normal
function reflect(rayDir, normal) {
  // r = d - 2·(d·n)·n where d is incident, n is normal
  const dot = rayDir[0]*normal[0] + rayDir[1]*normal[1] + rayDir[2]*normal[2];
  return [
    rayDir[0] - 2*dot*normal[0],
    rayDir[1] - 2*dot*normal[1],
    rayDir[2] - 2*dot*normal[2]
  ];
}

// Test: ray hitting a horizontal surface (normal = up)
const rayDown = [0, -1, 0];  // straight down
const surfaceUp = [0, 1, 0];
console.log(reflect(rayDown, surfaceUp));  // [0, 1, 0] — straight up

const rayDiagonal = [0.7071, -0.7071, 0];  // 45° down to right
console.log(reflect(rayDiagonal, surfaceUp));  // [0.7071, 0.7071, 0]

// Angle from normal
function angleFromNormal(rayDir, normal) {
  const dot = -rayDir[0]*normal[0] - rayDir[1]*normal[1] - rayDir[2]*normal[2];
  return Math.acos(dot) * 180 / Math.PI;
}

console.log(`Angle of incidence: ${angleFromNormal(rayDiagonal, surfaceUp).toFixed(1)}°`);

// Fresnel reflection coefficient (s-polarization, simple form)
function fresnelReflectivity(theta_i_deg, n1, n2) {
  const theta_i = theta_i_deg * Math.PI / 180;
  const sin_t = (n1 / n2) * Math.sin(theta_i);
  if (Math.abs(sin_t) > 1) return 1;  // total internal reflection
  const theta_t = Math.asin(sin_t);
  const r_s = (n1 * Math.cos(theta_i) - n2 * Math.cos(theta_t)) / (n1 * Math.cos(theta_i) + n2 * Math.cos(theta_t));
  return r_s * r_s;  // intensity ratio
}

console.log(`Air to glass at normal: ${(fresnelReflectivity(0, 1, 1.5) * 100).toFixed(1)}%`);  // ~4%
console.log(`Air to water at 30°: ${(fresnelReflectivity(30, 1, 1.33) * 100).toFixed(1)}%`);
console.log(`Air to water at 80°: ${(fresnelReflectivity(80, 1, 1.33) * 100).toFixed(1)}%`);  // ~30%

// Field of view of a flat mirror (mirror image)
function mirrorFieldOfView(viewerDistance, mirrorWidth) {
  // FOV = 2·atan(width/(2·distance))
  return 2 * Math.atan(mirrorWidth / (2 * viewerDistance)) * 180 / Math.PI;
}

console.log(`Bathroom mirror (viewer 1m, mirror 0.5m): ${mirrorFieldOfView(1, 0.5).toFixed(1)}°`);

Where reflection shows up

• Optics — mirrors, telescopes. Astronomical telescopes use mirrors (parabolic for primary). Laser cavities use mirrors at both ends.
• Photography. Anti-reflection coatings on lenses prevent flare; mirrors in viewfinders, periscopes.
• Medical imaging. Endoscopes use light reflection (and fiber optics for transmission).
• Communications. Fiber optics rely on total internal reflection.
• Solar energy. Reflective surfaces concentrate sunlight on collectors. Heliostat farms.
• Architecture. Reflective glass for energy efficiency. Mirrored buildings reflect surroundings.
• Safety. Reflective vests, retro-reflectors on bicycles and roads — make people visible at night.

Common mistakes

• Measuring angles from surface, not normal. θ_i and θ_r are from the normal, not from the surface itself. (i.e., 0° = perpendicular incidence, 90° = grazing).
• Forgetting plane of reflection. Reflection happens IN the plane defined by incident ray and normal. Doesn't bend out of plane.
• Treating mirror image as real. Plane mirror image is VIRTUAL — light doesn't actually go to image location. It just appears so to your eye.
• Confusing reflectivity with absorbtivity. Reflectivity + absorbtivity + transmittivity = 1 (energy conservation). For opaque material, transmittivity = 0; reflectivity + absorbtivity = 1.
• Ignoring polarization. Reflection coefficients differ for s- and p-polarization. At Brewster's angle, p-polarized light reflects 0% (used for polarization filters).
• Forgetting wavelength dependence. Most "reflective" objects only reflect specific wavelengths. Mirrors are usually broadband but not perfect.