Graphics Reference
In-Depth Information
arriving light, and absorbs the rest. If we carry out the preceding analysis for such
a mirror, we find that its BRDF
also
takes on an infinite value for mirror reflection,
but we have the sense that it's a “different infinity” that's only half as big. We'll
circumvent these problems by splitting any BRDF into two parts: a “diffuse” part
and an “impulse” part, where the latter is a representation of things like mirror
reflection and Snell's law refraction.
If we ignore, for a moment, the problem of the infinite values, we can still
look at the formula for
f
r
and consider the cosine that appears there. What happens
when we swap the roles of
v
o
for the perfect mirror? Since the incoming
and outgoing radiances are equal, the only possible change is in the cosine. But for
a mirror reflection, the incoming and outgoing angles are equal; this means that to
the degree that
f
r
makes any sense at all, it seems to satisfy Helmholtz reciprocity.
On the other hand, when it comes to measuring
f
s
for a material like glass, the
transmissive part of the computation involves two
different
angles, determined by
Snell's law. It's evident that in this case, even if we can make sense of the infinite
value of
f
s
, it will not satisfy the reciprocity law.
v
i
and
The function
L
is defined on
R
R
3
S
2
R
+
. Thus, it makes sense to write
×
×
×
S
2
and
P
R
3
; it makes equally good
expressions like
L
(
t
,
P
,
,
λ
)
, where
v
∈
∈
v
sense to write
L
(
t
,
x
,
y
,
z
,
)
, where
x
,
y
, and
z
are real numbers. In a com-
puter program, of course, assuming that “point in 3-space” is a class of objects,
we'd have to use an overloaded function, with arguments of type
real
*
point3
*
spherepoint
*
real
or
real
*
real
*
real
*
real
*
spherepoint
*
real
, but the distinction between the two would be so tiny that it wouldn't matter.
But the
spherepoint
class is trickier. One
can
choose to characterize each
point on the unit sphere by its
(
,
λ
v
)
coordinates, but those two numbers are not
the same thing as the point on the sphere. One could also represent the point by its
(
x
,
y
,
z
)
coordinates, although in a practical sense it's very hard to find computer-
based real numbers satisfying
x
2
+
y
2
+
z
2
=
1 exactly. In engineering and physics
it's common to gloss over these difficulties, and, for a function
U
defined on
S
2
,
write things like
θ
,
φ
U
(
θ
,
φ
)=
U
(
x
,
y
,
z
)
,
(26.83)
where
x
=cos
θ
sin
φ
,
(26.84)
y
=cos
φ
, and
(26.85)
z
=sin
θ
sin
φ
”.
(26.86)
Unfortunately, this sort of overloading, although it can be used in computer pro-
grams sometimes, makes little sense in mathematics. The symbol
U
can mean
only one thing.
For this reason, we'll carefully reserve the symbol
L
to denote the function
whose domain is
R
R
+
; when we need one of the closely related
functions (e.g., defined in terms of polar angles in Chapter 27), we'll give it a
new name to distinguish it from the original. Later when we discuss rendering
R
3
S
2
×
×
×
