PBR: Cook-Torrance BRDF Explained

1. Why a microfacet model?

No real surface is perfectly smooth at the microscopic scale. Under a microscope, even polished metal is a landscape of tiny facets, each acting like a perfect mirror pointed in a slightly different direction. Microfacet theory treats a surface point as a statistical distribution of these facets rather than modelling each one individually — which is exactly what makes it tractable in a real-time fragment shader.

Only facets whose normal h (the half-vector between the view direction v and the light direction l) happens to align with the true geometric normal can reflect light straight into the camera. A rough surface has facet normals scattered widely → broad, dim highlight. A smooth surface has them tightly clustered → a small, bright highlight. This single idea explains why roughness controls both the size and the brightness of a specular highlight.

2. The Cook-Torrance formula

The specular term of the Cook-Torrance BRDF is a fraction with three factors on top and a normalisation term on the bottom:

• D — the Normal Distribution Function: what fraction of microfacets are oriented exactly along the half-vector h?
• G — the Geometry term: what fraction of those facets are not shadowed by neighbouring facets on the way in, or masked on the way out?
• F — the Fresnel term: what fraction of light is reflected (rather than refracted/absorbed) at this viewing angle?
• 4·(n·l)·(n·v) — a normalisation factor that corrects for the change of measure between the microfacet-normal space and the outgoing-direction space.

The final shaded colour also needs a diffuse (Lambertian) term for the light that refracts into the surface, scatters, and exits again — see section 7.

3. D — Normal Distribution Function (GGX)

Cook and Torrance originally used a Beckmann distribution; modern engines almost universally use GGX (also called Trowbridge-Reitz), because it has longer "tails" — it fades out more slowly at grazing angles, matching measured real-world materials far better and producing the soft, glowing highlight edges characteristic of physically based renders.

float distributionGGX(vec3 N, vec3 H, float roughness) {
  float a      = roughness * roughness;
  float a2     = a * a;
  float NdotH  = max(dot(N, H), 0.0);
  float denom  = (NdotH * NdotH * (a2 - 1.0) + 1.0);
  denom = PI * denom * denom;
  return a2 / max(denom, 0.0000001);
}

4. G — Geometry (Smith Shadowing-Masking)

Even a facet aligned to reflect light towards the camera might be blocked: shadowing stops light reaching it from the light source, and masking stops reflected light reaching the camera. The Smith model factors this into two independent terms multiplied together, each using the same "Schlick-GGX" building block:

float geometrySchlickGGX(float NdotX, float k) {
  return NdotX / (NdotX * (1.0 - k) + k);
}

float geometrySmith(vec3 N, vec3 V, vec3 L, float roughness) {
  float r  = (roughness + 1.0);
  float k  = (r * r) / 8.0;
  float ggx1 = geometrySchlickGGX(max(dot(N, V), 0.0), k);
  float ggx2 = geometrySchlickGGX(max(dot(N, L), 0.0), k);
  return ggx1 * ggx2;
}

At grazing angles both dot products shrink towards zero, so G correctly darkens the highlight — this is what stops rough spheres from looking like glowing discs at their silhouette.

5. F — Fresnel (Schlick approximation)

The Fresnel effect says every material becomes more reflective at grazing angles — this is why a still lake looks like a mirror near the horizon but transparent looking straight down. The exact Fresnel equations involve the material's index of refraction and are expensive; Christophe Schlick's 1994 approximation is accurate to within a fraction of a percent and costs one pow():

F₀ is the reflectance at normal incidence (looking straight at the surface). This is where the metalness workflow plugs in: dielectrics (plastic, wood, skin) have a low, achromatic F₀ around 0.04, while metals have a high, tinted F₀ equal to their albedo colour (gold ≈ (1.0, 0.71, 0.29)).

vec3 fresnelSchlick(float cosTheta, vec3 F0) {
  return F0 + (vec3(1.0) - F0) * pow(clamp(1.0 - cosTheta, 0.0, 1.0), 5.0);
}

6. Putting it together in GLSL

A minimal single-light Cook-Torrance fragment shader combining all three terms plus a Lambertian diffuse component:

vec3 cookTorrance(vec3 N, vec3 V, vec3 L,
                     vec3 albedo, float roughness, float metalness,
                     vec3 radiance) {
  vec3  H  = normalize(V + L);
  vec3  F0 = mix(vec3(0.04), albedo, metalness);

  float D  = distributionGGX(N, H, roughness);
  float G  = geometrySmith(N, V, L, roughness);
  vec3  F  = fresnelSchlick(max(dot(H, V), 0.0), F0);

  vec3  kS = F;                       // specular contribution ratio
  vec3  kD = (vec3(1.0) - kS) * (1.0 - metalness); // metals have no diffuse

  float NdotL = max(dot(N, L), 0.0);
  float NdotV = max(dot(N, V), 0.0);
  vec3  numerator   = D * G * F;
  float denominator = 4.0 * NdotV * NdotL + 0.0001;
  vec3  specular = numerator / denominator;

  return (kD * albedo / PI + specular) * radiance * NdotL;
}

7. Energy conservation and the diffuse term

A physically plausible material must not reflect more light than it received. The code above enforces this with kD = (1 - kS) * (1 - metalness): whatever fraction of light isn't reflected specularly (kS = F) is available for diffuse scattering, and metals get zero diffuse because all the light that enters a conductor is absorbed by free electrons almost immediately — metals have no subsurface scattering.

The Lambertian albedo / π diffuse term is itself an approximation — real diffuse reflectance is not perfectly uniform in all directions (Oren-Nayar is a rougher-surface alternative) — but for most PBR use cases the error is imperceptible next to the specular term's contribution.