The directional diffuse component spreads reflected light out to the
entire hemisphere above a reflecting surface. The component has a
very strongly-directional, nonuniform character. It is responsible
for highlights of light sources and blurry reflection images on a
rough surface. It arises from diffraction and scattering by the
surface roughness. The reflected radiance can be expressed as a
product:

where G is a geometric factor, and D is a distribution function.
The |F|^{2} and S terms are the Fresnel and shadowing terms previously
discussed.

Note that the incident solid angle
dω_{i}
appears in Equation
(7). In the interactive figures that have a
dω_{i}
the coefficient of
dω_{i}
is approximated using a midpoint rule,
and
dω_{i}
is evaluated assuming a circular cross-section. The
values of solid angle shown for “dOmega” on the control panels have
been multiplied by a scale factor of ten. The actual solid angles are
one-tenth of the values shown. The maximum slider value of unity
corresponds to a maximum incident cone angle of 20.4°. This
represents an incident solid angle of 0.1 steradians, or 1/20π = 1.6%
of the incident hemisphere.

The illustration below demonstrates how the geometric factor G varies
with incidence angle θ_{i}.
The incidence direction is shown
with the red line. The semicircle is the unit hemisphere. The
function G arises from geometric projections of the incident and
reflected directions onto the surface and is given in
[HE91].

Figure 11: Interactive display of geometric factor G

The distribution function D is given by the following infinite
summation:

where h is
a function depending on the surface parameter τ, wavelength
λ, and incidence and reflection angles. It is given by:

The illustration below shows how the distribution function D varies
with θ_{i}, σ, τ,
and λ.
The semicircle is the unit
hemisphere. The incident direction is shown with the red line on the
left; the dashed line to the right indicates the specular direction
with respect to the mean plane of the surface.

Figure 12: Interactive display of the distribution function D

Finally, let's see how the entire directional diffuse component
dI_{r,dd}
varies with the parameters
θ_{i}, σ, τ, λ,
and material. This is shown in Figure 13, where the incident
light cone is on the left and the specular-ray direction for the mean
surface is indicated on the right (dashed line). The dashed
semicircle is added for reference, and corresponds to Lambertian,
ideal-diffuse, 100% reflection of the incident beam. For such a
reference reflector, the reflected radiance is
where dEr is the reflected energy flux. The π-factor
arises when converting energy flux to radiance for a Lambertian
surface. For a 100% reflector,
dE_{r} = dE_{i},
with dE_{i} given by Equation (2). Thus,

and we see that the radius of the reference hemisphere varies with
θ_{i} and dω_{i}.

Figure 13: Interactive display of entire directional diffuse term