Graphics Reference
In-Depth Information
of the parallel- and perpendicular-polarized terms,
R
s
and
R
p
, which are given by
r
p
=
n
2
cos
θ
i
−
n
1
cos
θ
t
(27.26)
1
n
2
cos
θ
i
+
n
1
cos
θ
t
R
p
=
r
p
(27.27)
r
s
=
n
1
cos
θ
i
−
n
2
cos
θ
t
(27.28)
n
1
cos
θ
i
+
n
2
cos
θ
t
0.5
R
s
=
r
s
.
(27.29)
Recall that
θ
i
and
θ
t
are related by Snell's law:
0
θ
t
)=
n
1
0 0 0 0 0
sin(
n
2
sin
θ
i
.
(27.30)
u
i
in degrees
For an air-water interface, we have
n
1
, the refractive index of air, is approxi-
mately 1.0, while that of water is approximately 1.33. The plot of
R
F
as a function
of
Figure 27.12: The Fresnel
reflectance F
R
as a function of
θ
i
,
for an air-to-water interface
θ
i
is shown in Figure 27.12.
As you'll observe, the function is nearly constant until we approach grazing,
at which point it rises rapidly. If you plot
F
R
for some other ratios of refractive
indices (see Exercise 27.5), you'll see that this characterization is quite general:
nearly constant for small angles, a sudden rise near grazing angles.
For metallic surfaces, the formula for
R
F
is somewhat more complex, but it
exhibits the same general characteristics.
Schlick [SCH94] observed that for
metallic
surfaces
R
F
could be well approx-
imated by a simple expression, and others have observed that the approximation
works reasonably well even for nonmetallic materials. The approximation is
θ
i
)
5
,
R
F
(
θ
i
)=
R
F
(
0
)+(
1
−
R
F
(
0
))(
1
−
cos
(27.31)
where
R
F
(
0
)
is the reflection at normal incidence (
n
)
is the angle of incidence. When the cosine is 1, we get
R
F
(
0
)
; when the cosine is
0, we get 1.0.
θ
i
=
0) and
θ
i
=cos
−
1
(
v
i
·
Inline Exercise 27.4:
There's usually no reason to compute
θ
explicitly, since
θ
θ
many formulas involve the cosine or sine of
rather than
itself. Rewrite
v
i
and
n
rather than
θ
.
Schlick's approximation in terms of
For insulators,
R
F
(
0
)
tends to be small, so there's large variation in
R
F
with
angle
θ
i
, leading to a pronounced Fresnel effect. For conductors,
R
F
(
0
)
tends to
be large (typically greater than 0.5), so the Fresnel effect is less pronounced.
Note that the index of refraction and the coefficient of extinction depend on
wavelength (although they have not been tabulated for many materials, which is
a problem); this means that the Fresnel reflectance is also a function of wave-
length. For many metals, this dependence is considerable. For gold, for instance,
the extinction coefficient drops substantially above about 500 nm, while the index
of refraction rises steadily above about 500 nm, which together give gold its char-
acteristic yellow appearance. For insulators, the refractive index is nearly constant
with respect to frequency, causing highlights on insulators to be the color of the
incoming light.
