Graphics Reference
In-Depth Information
This simple model of mirror reflection ignores the Fresnel equations that arise
from the polarization of light. There is no physical surface that actually reflects
like a perfect mirror (or one with constant reflectivity 0
<ρ<
1) for all angles
of incoming and outgoing light. We'll return to a more sophisticated version of
mirror scattering when we discuss physically based models.
A Lambertian surface has the property that when it's illuminated, the outgo-
ing radiance in every (reflected) direction is the same (there's no transmission).
Furthermore, this outgoing radiance varies linearly with irradiance: Whether we
reduce the incoming illumination or make it arrive at a grazing angle, the outgo-
ing radiance varies in the same way. Thus, the BRDF is constant; we usually write
L
(
P
,
V
v
o
)=
ρ/π
, where
ρ
is a constant indicating what fraction of the arriving
light energy is scattered.
Let's now check for which values of
R
this reflector will be energy-conserving.
Imagine (as shown in Figure 27.11) that light arrives at a small, rectangular region,
R
, of material with area
A
, from a source like the sun: All incoming rays are in
some small, solid angle
Ω
, and the radiance along each ray in a direction from
Ω
is the same constant
ρ
Figure 27.11: Light arrives at a
small, rectangular sample R from
directions
v
∈
Ω
, a small solid
angle; the radiance of the incom-
ing light is independent of posi-
tion and angle.
. Then the total rate of energy arrival at the surface region
R
is the integral over
R
and
Ω
of the arriving radiance, multiplied by the dot product
of the incoming direction and surface normal:
Power
=
Energy arrival rate
(27.4)
=
L
(
P
,
−
v
i
)(
v
i
·
n
)
d
v
i
dP
(27.5)
P
∈
R
v
i
∈
Ω
=
v
i
∈
Ω
(
v
i
·
n
)
d
v
i
dP
(27.6)
P
∈
R
A
v
i
∈
Ω
=
(
v
i
·
n
)
d
(27.7)
v
i
A
≈
cos(
θ
)
d
(27.8)
v
i
v
i
∈
Ω
=
Am
(Ω)cos(
θ
)
,
(27.9)
where
m
(Ω)
denotes the measure of the solid angle
Ω
, and
θ
is the angle between
a typical vector
Ω
and the (constant) surface normal
n
.As
Ω
gets small, the
approximation of the dot product by a single central dot product gets increasingly
accurate.
v
∈
Inline Exercise 27.1:
Use the reflectance equation to show that the radiance of
a ray emitted from the region
R
is well approximated by
(
ρ/π
)cos(
θ
)
m
(Ω)
.
To compute the rate at which light energy is emitted, let us surround the sam-
ple by a very large, black, completely absorptive hemisphere, and determine the
rate of energy arrival at that sphere. Just as in Chapter 26, we'll make the sphere
so large that the ray from the point
Q
to any point on the emitter
R
always has
essentially the same direction, independent of the emitter point.
The density
d
(
Q
)
of light energy (per second) arriving at a point
Q
of the
hemisphere is the integral, over all directions, of the radiance arriving at
Q
. Since
