Graphics Reference
In-Depth Information
3. Say that point lights are in fact small spheres of a known radius,
r
, but that
we'll restrict our scene to be sure that the distance from a point luminaire
to any surface is much greater than
r
so that the diffusely reflected light
can be very well approximated by treating the sources as point sources,
but mirror-reflected light must be computed using
r
.
4. Observe that when we include both point lights and mirror reflections,
both of which are approximations of physical phenomena made for con-
venience, the mathematics becomes intractable, and hence we'll abandon
one or both.
Each of these approaches has its merits, although we prefer approach 3 to
approach 2 (even though they both may result in identical programs). In Chap-
ter 32, we choose the first option: We simply ignore mirror-reflected point lights.
It's natural to ask, having derived the rendering equation, how to solve it. That
is to say, if we know the scene geometry and materials and illumination, how
can we compute
L
(
P
,
)
for any point
P
and any vector
, or for every
P
and
v
v
every
? We've already given you a taste of the answer in discussing a path tracer.
But the general topic is the subject of the next three chapters. Because integration
is central to the rendering equation, the first discusses probability and Monte Carlo
integration. The second describes the ideas behind several techniques for solving
the rendering equation. The third gives an implementation of two renderers. One
of the shocking things is how very short the two programs are, given the length of
this chapter and the next three. That brevity is partly due to the use of libraries for
things like visibility testing, basic linear algebra, and material representation. But
as you'll see, it's also due to the simplicity of basic Monte Carlo integration.
v
In the course of the Monte Carlo integration used to estimate solutions to the ren-
dering equation, we'll break up the integrals of the rendering equation into differ-
ent parts—sometimes by breaking up the domain of integration into the categories
of “directions in which we see luminaires” and “other directions,” and sometimes
by breaking up the integrand into a sum of a finite part and an impulse part, usu-
ally as a means of distinguishing between things like point luminaire and area
luminaires, or between mirror reflections and diffuse reflections.
Because of this distinction in treatment, it's useful to be able to discuss the
path that light took in getting from the luminaire to the receiver (a
light path
), or
its reverse, the path we traced from the eye to eventually reach the light source (an
eye path
). Heckbert [Hec90] used a notation that's now universally accepted:
1
L
denotes a luminaire,
E
the “eye,”
S
a specular reflection, and
D
a diffuse reflection.
Thus,
LDE
represents light that left the source, scattered from a diffuse surface,
1. Hanrahan [JAF
+
01] attributes this notation to Shirley, who claims [Shi10] that he's
uncertain who first used it.
