Concept: Fresnel Effect

• Parent: PathTracer Learning - Phase 2 - CPU Ray Tracing

What Is the Fresnel Effect?

• At any interface between two media, light is partially reflected and partially transmitted
• The ratio depends on the angle of incidence
• At grazing angles (θ → 90°): almost all light is reflected
• At normal incidence (θ = 0°): only a small fraction is reflected
• This is why water looks like a mirror when viewed at a shallow angle

Exact Fresnel Equations (Schlick’s Approximation)

• Full Fresnel equations (from Maxwell’s equations) are expensive
• Schlick (1994) approximation — used in virtually all real-time renderers
• F(θ) = F0 + (1 - F0) * (1 - cos(θ))^5
• cos(θ) = dot(V, H) — angle between view direction and half-vector
• F0 — reflectance at normal incidence (0° angle)

F0 Values

• Dielectrics (non-metals: plastic, glass, skin, wood)
  • F0 ≈ 0.04 (4%) for most dielectrics
  • Range: 0.02 (water) to 0.08 (gemstones)
  • Formula from IOR: F0 = ((n - 1) / (n + 1))²
  • Glass (n=1.5): F0 = ((1.5-1)/(1.5+1))² = 0.04
• Metals (conductors: gold, silver, copper, iron)
  • F0 = base color of the metal (RGB values)
  • Gold: F0 ≈ (1.0, 0.71, 0.29)
  • Silver: F0 ≈ (0.95, 0.93, 0.88)
  • Copper: F0 ≈ (0.95, 0.64, 0.54)
  • Metals absorb transmitted light — no diffuse component

Metallic Workflow (PBR)

• metallic parameter in [0, 1]
• F0 = lerp(vec3(0.04), baseColor, metallic)
• diffuse_color = baseColor * (1 - metallic) — metals have no diffuse
• This is the Disney/Unreal/Godot PBR convention
• Why: real materials are either dielectric (F0=0.04) or metal (F0=baseColor)
• Values between 0 and 1 are for blending (e.g., dirty metal, oxidized surface)

Energy Conservation with Fresnel

• Fresnel tells us how much light is reflected vs transmitted
• For opaque surfaces: transmitted = 1 - F(θ) → absorbed or diffusely scattered
• Specular contribution: F(θ) * f_specular
• Diffuse contribution: (1 - F(θ)) * f_diffuse
• This ensures total reflectance ≤ 1
• In code:

  vec3 F = schlick(F0, dot(V, H));
  vec3 kS = F;                    // specular fraction
  vec3 kD = (1.0 - kS) * (1.0 - metallic);  // diffuse fraction (metals have no diffuse)
  vec3 brdf = kD * diffuse + kS * specular;

Exact Fresnel (for Reference)

• For dielectrics (real IOR):
  • rs = (n1*cos(θi) - n2*cos(θt)) / (n1*cos(θi) + n2*cos(θt))
  • rp = (n2*cos(θi) - n1*cos(θt)) / (n2*cos(θi) + n1*cos(θt))
  • F = (rs² + rp²) / 2
  • cos(θt) from Snell’s law: cos(θt) = sqrt(1 - (n1/n2)² * (1 - cos²(θi)))
• For conductors (complex IOR n + ik):
  • More complex formula involving absorption coefficient k
  • Schlick with measured F0 is a good approximation

Total Internal Reflection

• When light travels from dense to less dense medium (e.g., glass to air)
• At angles beyond the critical angle: all light is reflected, none transmitted
• Critical angle: θ_c = arcsin(n2/n1) where n1 > n2
• In GLSL refract: returns vec3(0) when TIR occurs
• Check: 1 - eta² * (1 - NdotI²) < 0 → TIR

Related

• PathTracer Learning - Concept - BRDF
• PathTracer Learning - Concept - Microfacet Theory
• PathTracer Learning - Phase 2 - CPU Ray Tracing