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In-Depth Information
more like a series of nested cones than a cylinder. The reflection and scattering
is computed according to the cylinder, but the reflection angles are given a small
longitudinal shift to account for the scaled surface. The angle of this shift de-
pends on the angle
α
the surfaces of the scales deviate from the cylinder surface
(
Figure 8.52
)
.
Because the model is based on an elliptical cylinder rather than a circular
cylinder, the scattering is not symmetric about the axis of the hair fiber. Fig-
ure 8.53 illustrates the coordinate system for reflection. Incoming direction vector
ω
i
and the axis of the cylinder are contained in a unique
plane of incidence
.An
incoming direction vector
ω
i
is represented by two angles: the angle
θ
i
from the
surface normal, and the angle
φ
r
the plane of incidence makes with the long axis
of the elliptical cross section (see
Figure 8.53
)
. The angles
φ
r
are measured
similarly. The scattering model is the sum of the R, TT, and TRT components,
each of which can be expressed as the product of a function of
θ
r
and
θ
i
and
θ
r
,anda
function of
θ
r
−
θ
i
,
φ
i
,and
φ
r
:
S
(
θ
i
,
θ
r
,
φ
i
,
φ
r
)=
M
R
(
θ
i
,
θ
r
)
N
R
(
θ
,
φ
i
,
φ
r
)
+
M
TT
(
θ
i
,
θ
r
)
N
TT
(
θ
,
φ
i
,
φ
r
)
+
M
TRT
(
θ
i
,
θ
r
)
N
TRT
(
θ
,
φ
i
,
φ
r
)
.
(8.15)
The functions
M
are called
longitudinal
functions, as they depend on the longi-
tudinal angles
θ
r
, both of which are measured according to the long axis
of the cylinder. The functions
N
are
azimuthal
functions. Equation (8.15) thus
expresses the four-parameter reflectance function as a sum of products of two-
and three-parameter functions. If the model is simplified to have a circular cross
θ
i
and
A
(Incidence Plane)
w
ө
i
B
(Normal Plane)
Φ
i
ө
r
Φ
r
v
Figure 8.53
Angles used in the Marschner hair reflectance model. The plane of incidence contains
the incident ray and is parallel to the axis of the cylinder and contains the incident light
direction; the normal plane is perpendicular to the axis.
