Graphics Reference
In-Depth Information
We need one more definition: We let
Ω
jk
denote the solid angle of directions
from patch
j
to patch
k
(see Figure 31.13), assuming that they are mutually visible;
if they're not, then
Ω
jk
is defined to be the empty set.
C
k
k
Now let's use these assumptions to simplify the rendering equation. Let's start
with a point
P
in some patch
j
. The rendering equation says that for a direction
v
o
V
jk
with
v
o
·
n
j
>
0, that is, an outgoing direction from patch
j
,
C
j
v
o
)+
v
o
)=
L
e
(
P
,
L
(
P
,
f
r
(
P
,
v
i
,
v
o
)
L
(
P
,
−
v
i
)(
v
i
·
n
j
)
d
v
i
.
(31.33)
j
v
i
∈
S
+
(
n
j
)
Figure 31.13: Patch k is visible
from patch j; when it's projected
onto the hemisphere around C
j
,
we get a solid angle called
Ω
jk
.
We now introduce some factors of
π
to simplify the equation a bit. We let
B
j
=
L
(
P
,
v
o
)
/π
. Since
L
(
P
,
v
o
)
is assumed independent of the outgoing direc-
tion
v
o
, the number
B
j
does not have
v
o
as a parameter. Similarly, we define
E
j
=
L
e
(
P
,
v
o
)
/π
. And substituting
f
r
(
P
,
v
i
,
v
o
)=
ρ
j
/π
, we get
E
j
+
ρ
j
π
π
B
j
=
π
L
(
P
,
−
v
i
)(
v
i
·
n
j
)
d
v
i
.
(31.34)
v
i
∈
S
+
(
n
j
)
The inner integral, over all directions in the positive hemisphere, can be broken
into a sum over directions in each
Ω
jk
, since light arriving at patch
j
must arrive
from some patch
k
. The equation thus becomes
E
j
+
ρ
j
π
π
B
j
=
π
L
(
P
,
−
v
i
)(
v
i
·
n
j
)
d
v
i
.
(31.35)
v
i
∈
Ω
jk
k
The radiance in the integral is radiance leaving patch
k
, and is therefore just
π
B
k
. Substituting, and rearranging the constant factors of
π
a little, we get
v
i
∈
Ω
jk
π
E
j
+
ρ
j
π
π
B
j
=
π
B
k
(
v
i
·
n
j
)
d
(31.36)
v
i
k
B
k
E
j
+
ρ
j
=
π
π
π
(
v
i
·
n
j
)
d
v
i
(31.37)
v
i
∈
Ω
jk
k
1
π
B
k
.
=
π
E
j
+
ρ
j
π
(
v
i
·
n
j
)
d
(31.38)
v
i
v
i
∈
Ω
jk
k
π
Dividing through by
, we get
1
π
B
k
.
ρ
j
k
B
j
=
E
j
+
(
v
i
·
n
j
)
d
(31.39)
v
i
v
i
∈
Ω
jk
The coefficient of
B
k
inside the summation is called the
form factor
f
jk
for
patches
j
and
k
. So the equation becomes
ρ
j
k
B
j
=
E
j
+
f
jk
B
k
,
(31.40)
which is called the
radiosity equation.
Before we try to solve it, let's look at the
form factor more carefully. For patches
j
and
k
,itis
f
jk
=
1
π
(
v
i
·
n
j
)
d
v
i
.
(31.41)
v
i
∈
Ω
jk
