10
5
.5
.1
1
10
–30º 30º
–60º
5
.5
.1
1
10
5
.5
.1
1
10
5
.5
.1
1
10
5
.5
.1
1
10
5
.5
.1
1
TIPROOT TIPROOT TIPROOT
Black hair
Synthetic
Blond hair
θ
r
θ
r
θ
r
θ
r
θ
r
θ
r
Figure 4: Measurements of scattering in the incidence plane: scattering as a function of scattering angle with illumination at 45
◦
from the tip
and root ends. Black and blond hair are shown, along with a synthetic fiber from a wig.
R highlight, since it is surface reflection, is white, whereas the TRT
highlight, which is formed by light that passes through the interior
of the fiber, is colored.
2.2 Scattering
Before getting into more detail, let us establish the notation we will
use throughout the paper to describe the scattering geometry (Fig-
ure 3). The tangent to the hair is u, pointing in the direction from
the root toward the tip; the vectors v and w complete a right-handed
orthonormal basis, and if the cross section is elliptical v is the ma-
jor axis and w is the minor axis. We refer to the v–w plane as the
normal plane. The direction of illumination is ω
i
, and the direction
in which scattered light is being computed or measured is ω
r
; both
directions point away from the center. We express ω
i
and ω
r
in
spherical coordinates. The inclinations with respect to the normal
plane are denoted θ
i
and θ
r
(measured so that 0
◦
is perpendicular to
the hair, 90
◦
is u, and −90
◦
is −u). The azimuths around the hair
are denoted φ
i
and φ
r
(measured so that v is 0
◦
and w is +90
◦
).
We also use several derived angles. The difference angle (θ
r
−
θ
i
)/2 is denoted θ
d
. The relative azimuth φ
r
− φ
i
is denoted simply
φ. The averages θ
h
= (θ
i
+ θ
r
)/2 and φ
h
= (φ
i
+ φ
r
)/2 are called
half angles.
The bidirectional scattering function S for a fiber is different
from the bidirectional reflection distribution function f
r
for a sur-
face, although it shares the same physical units. For the incident
and reflected light we use curve irradiance
¯
E, or power per unit
length, and curve intensity
¯
L, or intensity per unit length, respec-
tively. These units are analogous to irradiance (power per unit area)
and radiance (intensity per unit area) on a surface.
S(ω
i
,ω
r
) =
d
¯
L
r
(ω
r
)
d
¯
E
i
(ω
i
)
,
where
¯
L
r
is the curve intensity scattered from an infinitesimal length
of fiber, and
¯
E
i
is the curve irradiance on that portion of the fiber.
This irradiance is proportional to incoming radiance:
d
¯
E
i
(ω
i
) = DL
i
(ω
i
)cosθ
i
dω
i
,
where D is the diameter of the fiber (which depends on φ
i
for an
elliptical fiber). Note that the area over which the irradiance is mea-
sured is Ddl where dl is an infinitesimal arc length along the fiber.
Given this definition, the scattering integral is written as
¯
L
r
(ω
r
) = D
Z
S(ω
i
,ω
r
)L
i
(ω
i
)cosθ
i
dω
i
(1)
Note that, unlike a surface where the integral extends over the up-
per hemisphere, this integral extends over the entire sphere. The
presence of D in this equation indicates that a thick fiber intercepts
more light, and therefore appears brighter from a distance, than a
thin fiber.
3 Scattering measurements
The experimental component of our study of hair was intended to
provide a qualitative and quantitative understanding of the phenom-
ena that need to be explained by a scattering model for hair. In this
paper, we briefly outline these measurements; a full description of
our experiments will be provided in a future paper.
In our experiments we illuminated individual hairs with a narrow
beam and measured the scattered light in various directions using a
setup based on a four-axis goniometer that positioned a light source
and a CCD camera at arbitrary directions from the sample. We
used a focused beam to illuminate only a small length of hair, on
the order of 1 to 2 cm, and this illuminated segment defined the
length of hair being measured. This served to reduce the effects of
any variations in properties along the length of the hair.
3.1 Incidence plane
Stamm et al.’s and Bustard and Smith’s measurements were in the
incidence plane, meaning that they observed only the 2D slice of the
scattering function for which the source and detector are coplanar
with the fiber. To verify these earlier results, we made the same
kind of measurements on samples of several different types of hair
and on a synthetic fiber from a wig (Figure 4).
For each sample, we set the angle of incidence to 45
◦
and mea-
sured scattering for varying outgoing angles. We performed this
experiment twice, once with the illumination from the direction of
the root (with θ
i
= −45
◦
), and again with the illumination from the
direction of the tip (with θ
i
= 45
◦
). Each measurement was made in
three color bands across the visible wavelength range. Note that the
small gap near the incidence angle in each plot is due to the camera
occluding the light source.
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