If the scattering model is provided in terms of the Bi-Directional Scatter Distribution Function (BSDF), then the integrals necessary to relate the random numbers to the scatter angles include an extra cosq factor:

I = ∫∫ BSDF(q,f) sinq cosq dq df

The cosine factor arises from the fact that BSDF is defined in terms of the irradiance falling on the surface, and thus requires use of the projected surface area (A_p = A*cosq). Otherwise, the procedure for determining the scatter angles q and f is identical to the procedure described in the previous section.

Let’s look at a simple example – the case of a Lambertian scattering source. For such a source, the BSDF is constant for all angles, equal to 1/p. I is then given by:

I = ∫∫ BSDF(q,f) sinq cosq dq df 
  = (1/p)*[∫ sinq cosq dq]*[∫ df]
  = sin²q*(f/2p)

The total integrated scatter is calculated by setting the limits of integration from 0 to p/2 in q and from 0 to 2p in f:

TIS = [sin²q (0:p/2)]*[f/2p (0:2p)] = [1-0]*[1-0] = 1

Therefore I is already normalized. Since the BSDF is separable in q and f in this case, it is straightforward to determine the scatter angles from the random numbers:

RN1 = sin²q; q = sin^-1(√RN1)
RN2 = f/2p; f = 2p*RN2

The trajectory of the scattered ray s with respect to the surface normal n is given by:

s_n = cosq
s_p = sinq*cosf
s_q = sinq*sinf

where p and q represent the coordinate axes on the plane perpendicular to the surface normal. The trajectory of the scattered ray in (x,y,z) space is easily determined from the known components of (n,p,q) in this coordinate system:

s_x = s_n*n_x + s_p*p_x + s_q*q_x
s_y = s_n*n_y + s_p*p_y + s_q*q_y
s_z = s_n*n_z + s_p*p_z + s_q*q_z