 Solution

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.