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where we've replaced the dot product with
cos
φ
. Finally, the finite part of the
0.5
π
sr
−
1
, so the reflected radiance becomes
BSDF is the constant
v
o
)=
π/
2
−π/
2
π/
2
0. 5
π
L
ref, 0
(
P
,
6
cos
φ
sin
φ
d
φ
d
θ
(29.21)
0
π/
2
π/
2
=
3
π
cos
φ
sin
φ
d
φ
d
θ
(29.22)
−π/
2
0
π/
2
3
π
=
π
cos
φ
sin
φ
d
φ
(29.23)
0
=
3
2
, so that
(29.24)
v
o
)=
3
L
ref
(
P
,
2
+
I
Wm
−
2
sr
−
1
,
(29.25)
where
I
is the impulsively reflected (i.e., mirror-reflected) radiance. Since the
surface is 30
%
mirror-reflective, the incoming radiance of 6 W m
−
2
sr
−
1
is
multiplied by the magnitude 0. 3 to get the outgoing mirror-reflected radiance,
1.8 W m
−
2
sr
−
1
. Thus, the total reflected radiance is
3
2
+
1. 8 W m
−
2
sr
−
1
=
3. 3 W m
−
2
sr
−
1
.
As a result, we've converted the handling of an impulse in the scattering func-
tion from an integral to a simple multiplication by a constant, the impulse magni-
tude.
Now, as we look at the second situation, with illumination provided by a 10 W
radiating small sphere, we'll see how each term (the diffuse and the impulse)
behaves when there's an “impulse” in the incoming light field (i.e., a point light),
by seeing what happens as the radius of the sphere approaches zero.
As we showed in Section 26.7.3, a uniformly radiating sphere of radius
r
and
total power
Φ
produces radiance
Φ
4
π
(
π
r
2
)
along every outgoing ray, and subtends
a solid angle approximately
π
r
2
R
2
at a point at distance
R
, with the approximation
growing better and better as
R
increases or
r
decreases.
The integral to compute the Lambertian-reflected light from this small spheri-
cal source is essentially the same as the one above, except that instead of integrat-
ing over all directions
v
i
=
xyz
T
with
x
0, we now must integrate over
just the small solid angle
Ω
subtended at
P
by the small spherical source. So
≥
v
o
)=
L
r
(
P
,
f
s
(
P
,
v
o
,
v
i
)
L
(
P
,
−
v
i
)(
v
i
·
n
(
P
))
d
v
i
+
I
,
(29.26)
Ω
where
I
as before represents the impulse-reflected radiance and
f
s
again is the
sr
−
1
. Furthermore, for
constant function 0. 5
/π
v
i
in the solid angle subtended by
the luminaire,
n
(
P
)
,
w
here
u
is the unit vector
from
P
to the center
Q
of the radiating sphere, that is,
√
2
1
1
0
T
. Since the
v
i
·
n
(
P
)
is well approximated by
u
·
normal
n
(
P
)
points in the
y
-direction, this dot product is just
√
2
/
2. Hence,
√
2
2
0. 5
4
L
r
(
P
,
v
o
)
≈
L
(
P
,
−
v
i
)
d
v
i
+
I
.
(29.27)
π
Ω
Φ
The radiance along each ray is the constant
4
π
(
π
r
2
)
, so this becomes
