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A bidirectional reflectance distribution function, or BRDF, describes how a surface scatters incident illumination into each outgoing direction. At a fixed surface point $p$ and wavelength, its inputs are two directions above the surface, ${\omega }_i$ toward the incident source and ${\omega }_o$ toward the observer; equivalently these are two pairs of angular coordinates. Its output is the ratio of differential reflected radiance to differential incident irradiance:Position and wavelength may be additional arguments when material properties vary. A colour implementation can return one value per colour channel. The BRDF is a material response, not the final outgoing light value: it must be combined with the incident illumination and the projected-angle factor.^[ref]

For incident radiance $L_i$, the reflection part of the rendering equation isEmission is added separately if the surface itself emits. This integral explains how a BRDF replaces the assumed material response of a local lighting model. For Lambertian reflectance, $f_r={\rho }_d/\pi$, so the cosine factor yields the diffuse term from part (a). A directional specular lobe supplies a highlight. The empirical Phong reflection model is an approximate angular-response model of this kind; its original direct specular term is not automatically a normalized, energy-conserving BRDF. The ambient term approximates missing illumination and is not itself a third direction-independent material scattering mechanism. More general BRDFs can reproduce measured surface behaviour, while illumination from other surfaces supplies indirect light in the rendering equation.

Path tracing estimates the rendering equation by a Monte Carlo method. Launch camera rays; at each surface hit sample an outgoing continuation direction according to a chosen probability density function, trace to the next hit, and continue through successive reflections. Accumulate emitted light or environment light with the product of scattering, cosine and inverse-density factors along the path. A bounce sampled with density $p(\omega )$ updates the path weight byAverage many independently sampled paths per pixel. Multiple bounces account for indirect illumination, including colour bleeding between surfaces, instead of approximating all of it by one ambient constant. Termination can use Russian roulette: continue with probability $q>0$ and divide surviving weights by $q$, preserving the expected contribution. A fixed bounce cutoff is simpler but truncates transport.

The elementary strategy that obtains illumination only when a randomly continued path hits an emitter fails for ideal point lights. From a surface point, an ideal point source occupies zero solid angle; sampling a direction from an ordinary continuous probability density function hits that one direction with probability zero. Thus this strategy almost surely never discovers the light, regardless of the number of samples. An area emitter occupies nonzero solid angle and can be hit.

The qualification matters: practical path tracing can use point lights by explicitly sampling a source and connecting the current surface to it with a shadow ray, a technique called next-event estimation. For a point source the connection direction is known exactly, rather than discovered by a random hemisphere hit. Thus the restriction applies to naive emitter-hit path tracing, not to all path tracing algorithms.^[ref]