Graphics Reference
In-Depth Information
as
Φ
4
π
(
π
r
2
)
. Now consider some distant point
P
, lying on a surface that faces the
center
C
of the spherical source, and that the distance from
C
to
P
is
R
r
.We
can compute the irradiance at
P
by computing the solid angle subtended by the
spherical source, and the radiance arriving at
P
along each direction of that solid
angle, etc.
(a) Do this computation to determine the irradiance at
P
.
(b) Now suppose that the total power
H
remains constant, while the radius
r
of the
source shrinks. What is the limit of the expression you found in part (a) as
r
>>
0?
Exercise 26.4:
It would be nice to imagine a “beam” of light arriving at a
point
P
of a surface in direction
→
v
i
, and being reflected out in many directions; we
could then look at how much light goes in each direction and talk about the “scat-
tering” from the surface. The problem is that light arriving at a single point cannot
carry any energy, and light arriving in a single direction cannot carry any energy.
Instead, we can imagine light arriving in directions
η
with
|η−
v
i
|≤
(i.e., direc-
tions very nearly parallel to
v
i
), and arriving at points
Q
with
Q
−
P
<
r
,for
some small
r
, with constant radiance
. In this problem, we'll examine the irradi-
ance due to this “beam.” If the BRDF is continuous as a function of position and
incoming direction, then the outgoing radiance in direction
v
o
will vary smoothly
as a function of
r
and
.
(a) What's the solid angle of the incoming rays as a function of
?What'sasim-
plified expression for small
?
(b) How should we adjust the radiance
along incoming rays, as we reduce
r
and
toward zero, to make certain that the incoming power is constant? When
we adjust the incoming radiance in this way, and take a limit, we can speak of a
beam of light in direction
v
i
having a certain irradiance; the resultant radiance in
direction
v
o
can then be measured (theoretically). The ratio
L
o
(
r
,
)
|
n
·
v
i
|π
2
π
r
2
has a limit as
, and the limit is exactly
the BRDF; this is the justification for defining the BRDF as “the ratio of the out-
going radiance in direction
and
r
go to zero because of the form of
v
o
to the incoming radiance of a beam in direction
v
i
.” Therefore, the integral of the BRDF, over all outgoing directions, multiplied
by
, tells how much of the power arriving in the beam gets reflected in any
direction at all, and is therefore called the
directional hemispherical reflectance.
Exercise 26.5:
Latex wall paint is designed to be
Lambertian,
that is, it's
designed so that when illuminated by light from any direction, it has the same
apparent brightness regardless of the direction from which it's viewed (i.e., the
radiance along every outgoing ray is the same). Furthermore, the outgoing radi-
ance should be independent of the incoming direction of the illumination, so long
as the power arriving at a fixed region of the painted surface is constant. Good latex
paint very nearly achieves this goal, although at grazing angles the reflectance
varies from the ideal. If it
were
such an ideal reflector, what would its BRDF look
like?
Exercise 26.6:
A planar surface
S
sits in a room that's bathed in light so that
the radiance along every ray arriving at
S
is the same constant, 10 watts per stera-
dian per square meter. What's the irradiance at a point
P
of the surface?
Exercise 26.7:
Two incandescent bulbs emit the same total power in the vis-
ible spectrum; one has a filament at 4000 K, the other at 6500 K. Because of the
v
o
·
n
