Graphics Reference
In-Depth Information
rather than one; so does topaz. (That's because the speed of light is different in
different directions through these materials!)
Form of Fresnel's Equations
While Fresnel's laws describe transmitted and reflected power, in graphics we're
mostly concerned with radiance, which we'll define in the next few sections.
Because radiance involves an angle measure in its definition, and the angle
between two light beams refracted by Snell's law is different before and after
refraction, the ratio between outgoing and incoming
radiance
involves an extra
factor of
sin
2
=
n
2
n
1
θ
i
.
(26.16)
sin
2
θ
t
The derivation of this factor is given in the web materials for this chapter.
Although we've observed that light, after reflection, tends to be increasingly
polarized, it's common in graphics to treat light as
unpolarized,
that is, to assume
that the polarization of incident light is, on average, zero. With that assump-
tion, the Fresnel equations can be simplified to a single factor, called the Fresnel
reflectance, which is
R
F
=
1
2
(
R
s
+
R
p
)
.
(26.17)
The energy reflected is
R
F
times the incident energy. And the energy transmitted
is
(
1
R
F
)
times the incident energy. This means that the reflected and transmitted
radiance values can be computed as
−
L
(
P
,
v
r
)=
R
F
L
(
P
,
−
v
i
)
and
(26.18)
R
F
)
n
2
n
1
L
(
P
,
v
t
)=(
1
−
L
(
P
,
−
v
i
)
.
(26.19)
Note that
R
F
here depends implicitly on
θ
i
,
n
1
, and
n
2
which, together with Snell's
law, lets us compute
θ
t
.
80
Imagine standing at a crossroads, looking north. You count the cars that come to
the crossroads from the north, and observe that 60 cars arrive in the course of
an hour. You report that the arrival rate for cars is 60 per hour, and this lets you
guess that in 10 minutes, about 10 cars will arrive; in 5 minutes 5 cars will arrive,
etc. Of course, at an actual crossroads, cars arrive irregularly, so your “5 cars in 5
minutes” claim is probably not exactly correct. Nonetheless, if you counted cars
for each hour over the course of the day, you could make a graph like the one
shown in Figure 26.16, where we've connected the dots with straight lines, but
could have used a smooth curve. Later you could say something like “the arrival
rate at 9:30 was about 65 cars per hour.”
70
60
50
40
30
20
10
0
600
700
800
900 1000 1100 1200 1300
Time
Figure 26.16: The number of cars
arriving
In making such a statement, you are treating the arrival rate as something that
makes sense
at a particular instant;
you are treating this problem as if it were
from
the
north
at
an
intersection, at each hour.
