Several boundary features can be used to scatter rays. These include diffuse and isotropic reflection from a Wall, and perturbations due to surface slope error with an Illuminated Surface. These scattering models can be used in both transmission and/or reflection with the Scattering Boundary feature.

Diffuse (or Lambertian) scattering follows the cosine law. That is, the flux dn of rays across a surface element A whose directions lie within a small solid angle dω is proportional to the cosine of the polar angle θ:

where φ is the azimuthal angle.

In 3D, the normalized probability distribution functions f(θ, φ) for diffuse and isotropic scattering are given by

where δn is the component in the direction of the surface normal (ns) and where δt1 and δt2 are the components in the two directions t1 and t2 orthogonal to the surface normal. That is, after scattering the reflected and transmitted ray directions are

Following Ref. 20, rays will deviate upon reflection or refraction from a surface with a nonzero slope error. It is assumed that the tangential deviation is negligible and that the radial distribution is Gaussian. Therefore, given an angular slope error σ, the probability distribution function for the polar angle is a Rayleigh distribution of the form

which can be used to get the direction components δn, δt1, and δt2, as shown in the previous section. The effect of applying a surface slope error is to perturb the surface normal ns to a value given by