Graphics Reference
In-Depth Information
√
2
4
Φ
L
r
(
P
,
v
o
)
≈
d
v
i
+
I
(29.28)
π
4
π
(
π
r
2
)
Ω
=
√
2
4
Φ
r
2
)
π
r
2
R
2
+
I
(29.29)
π
4
π
(
π
=
√
2
4
Φ
R
2
+
I
.
(29.30)
π
4
π
Since
R
=
√
2 and
Φ=
10, we get that the reflected radiance is
5
√
2
16
π
2
+
I
Wm
−
2
sr
−
1
. Notice that this approximation of Equation 29.28 is independent
of
r
so that even as the spherical luminaire shrinks to a point, the reflected radiance
remains the same.
The constancy of the reflected radiance depends on two things:
the
assumption that the dot product
n
(
P
)
, and the
fact that the finite portion of the scattering function is constant. The first of these
is justified because we're letting
r
approach zero. The second is
not
true for a gen-
eral BSDF. But if
f
s
is
continuous,
as it is in all the BSDFs that we consider, then
the mean value theorem for integrals tells us that the integral we wish to compute
is equal to
v
i
·
n
(
P
)
can be approximated by
u
·
f
s
(
P
,
v
i
)
L
(
P
,
−
v
i
)(
v
i
·
m
(Ω)
·
v
o
,
n
(
P
))
(29.31)
v
i
for some
in
Ω
. As the area of
Ω
goes to zero, this vector
v
i
∗
must approach
u
,
so the integral approaches
R
)
2
f
s
(
P
,
π
(
r
/
v
o
,
u
)
L
(
P
,
−
u
)(
u
·
n
(
P
))
.
(29.32)
To summarize the preceding argument, a point luminaire of power
Φ
, in direc-
tion
v
i
from
P
, at distance
R
, produces reflected radiance (from the nonimpulse
portion of scattering) in direction
v
o
in the amount
Φ
f
s
(
P
,
v
i
,
v
o
)
v
i
·
n
P
,
for
v
i
·
n
P
>
0 and
v
o
·
n
P
>
0.
(29.33)
4
π
R
2
Finally, let's consider the impulse reflection of the very small spherical source.
The radiance leaving each point of the luminaire is again
Φ
4
π
(
π
r
2
)
. Because
v
i
Φ
points to a point of the luminaire, the radiance arriving in direction
−
v
i
is
4
π
(
π
r
2
)
;
to get the outgoing radiance in the mirror-reflected direction
v
o
, we multiply by
the impulse value 0.3 so that the outgoing radiance due to mirror reflection of the
very small spherical source is 0.3
Φ
4
π
(
π
r
2
)
.
Notice that this depends on the value of
r!
As the size of the luminaire decreases, the radiance emitted must increase to
keep the total power the same, with the result that the mirror-reflected radiance
also increases without bound. If we try to take a limit, we end up with an answer
that includes
, which is unsatisfactory.
There are several possible ways to address this problem.
∞
1. Assert that no eye ray that we trace will ever “just happen” to hit a point
luminaire, so this is a probability-zero event, and we can ignore the infinity
that would arise if this event happened.
2. Say that for the sake of reflecting from diffuse surfaces, point lights are
points, but for the sake of specular reflections, they have a nonzero radius
r
, which must be chosen by the user. Note that this makes the world in
which we're trying to simulate light transport internally contradictory.
