Graphics Reference
In-Depth Information
operator as an integral to be approximated by naive sampling—we'll essentially
never sample the direction of the mirror-reflected ray. When we combine the two,
things are even worse.
Consider a scene consisting of a smooth mirrored ball illuminated by a point
light. If we ray-trace from the eye through the pixel centers, we'll almost certainly
miss the point light; if we ray-trace from the light, we'll miss the pixel centers.
But if we suppose that the point light is in the scene as a proxy for a spherical light
of some small radius
r
, then we know that we should see a highlight on the
mirrored ball.
Losing that highlight is perceptually significant, even though the highlight
might appear at only a single pixel of the image. We have three choices: We can
abandon the convenient fiction of a point light, we can adjust the BRDF to com-
pensate for the abstraction, or we can choose some other method for estimating the
radiance arriving at the eye from that location. In an ideal world, with infinite ren-
dering resources, we'd choose to use tiny point lights and cast a great many rays.
Within the context of ray tracing, we can clamp the maximum shininess (i.e., the
specular exponent) when we are combining a BRDF with a direct luminaire in
the reflection operator. This ensures that with sufficiently fine sampling, the point
light will produce a highlight. Of course, it also slightly blurs the reflection of
every other object in the scene. The difference in appearance between a specu-
lar exponent of 10,000 and
tends to be unnoticeable in general, so this is an
acceptable compromise. On the other hand, if the specular exponent is 10,000, a
very fine sampling around the highlight direction is required, or else we'll get high
variance in our image. This leads us to the third alternative. It may make sense to
separate out the impulse reflection of point lights (or even small lights) into a sep-
arate computation to avoid these sampling demands, but we will not pursue this
approach here.
∞
In our theoretical discussion, we treated light as being defined by the radiance
field
(
P
,
)
→
L
(
P
,
)
: At any point
P
, in any direction
,
L
(
P
,
)
represented the
v
v
v
v
radiance along the ray through
P
in direction
, measured with respect to a surface
v
at
P
perpendicular to
. When
P
is in empty space, this is a good abstraction.
When
P
is a point exactly on a surface, there are two problems.
v
v
L
(
P,
v
)?
Transparent surface
P
1. The precise relationship between geometric modeling and physics has
been left undefined. We haven't said whether a solid is open (i.e., does
not contain its boundary points, like an open interval) or closed; equiv-
alently, we haven't said whether a ray leaving from a surface point of a
closed surface intersects that surface or not.
L
(
P,
v
)?
Figure 32.4: Is L
(
P
,
v
)
the radi-
ance along the solid red or the
solid green arrow?
2. When
P
is a point on the surface of a
transparent
solid, like a glass sphere,
and
points into the solid (see Figure 32.4), there are two possible mean-
ings for
L
(
P
,
v
)
: the light arriving at
P
from distant sources, or the light
traveling from
P
into the interior of the surface. Because of Snell's law,
material opacity, and internal reflection, these two are almost never the
same.
v
We addressed the second problem in Chapter 26, by defining an incoming and out-
going radiance for each pair
(
P
,
)
, where
P
was a surface point and
was a unit
v
v
