The rendering equation, and why PBR feels heavier

Rays, triangles and depth decide what is visible, but not the radiance leaving each visible surface. In 1986, James Kajiya published an influential rendering equation that formalized how emission and reflected light contribute; it remains a foundation, not a guarantee that a rendered scene is physically correct.

L_o(x, ω_o) = L_e(x, ω_o)                       // emitted
              + ∫_Ω  f_r(x, ω_i, ω_o)                // how the surface scatters
                     · L_i(x, ω_i)                   // light arriving from ω_i
                     · (n · ω_i)  dω_i               // cosine: tilt matters

//  L_o  outgoing radiance
//  L_e  emitted (a lamp, not a wall)
//  f_r  BRDF — the material
//  L_i  incoming light; it may include light reflected from other surfaces
//  n·ω  cosine factor; Ω is the hemisphere above the surface

Read the terms as a model for outgoing radiance. Common rendering effects are approximations or special cases of this model:

· Direct lighting evaluates the contribution from emitters; an ideal point light reduces this part to an explicit direction, while an area light still needs integration or sampling.
· The `n · ω_i` cosine factor (over the upper hemisphere) makes grazing light contribute less for a diffuse surface.
· The full equation is recursive when incoming radiance includes light reflected from other surfaces; solving those bounces is global illumination.
· Different BRDF models describe diffuse and glossy scattering; roughness changes the angular distribution rather than replacing one universal function.
· An ideal mirror is represented by a delta-like BRDF concentrated at the reflected direction, not an ordinary smooth function.

The outer per-pixel rendering loop can stay the same, but its shading step now evaluates or estimates a physical scattering model. This costs more and may be noisy. Physical plausibility still depends on the accuracy of the geometry, materials, lighting, camera and numerical estimate.