Graphics Reference
In-Depth Information
z
→
Hemisphere
Ω
θ
d
ω
x
y
dA
Φ
x
Figure 1.4
Radiance at a surface point. Directions above the surface correspond to points on the
hemisphere Ω; the differential area in the direction ω is smaller than differential surface
area by the cos θ factor.
sufficiently distant light source. The formula depends only on the flux (intensity)
of the light source, not its size, and therefore remains valid if the source is shrunk
to a single point. Such a light source is known as a
point source
. When a point
source is far enough away, neither the direction of nor the relative distance to the
point change appreciably across a surface. In this case, the point source becomes
a
directional source
.
Point and directional sources were used almost exclusively in the early days
of graphics, and remain in use for many applications. However, they have limited
applicability in photorealistic rendering, because real light sources have some
effective nonnegligible surface area. Such sources are called
area sources
. While
the radiance from a point or directional source comes from a single direction, an
area source subtends a nonzero solid angle from a surface point. The irradiance
due to an area source is computed by integrating the radiance over the set of
directions to the source.
In real environments, incident radiance comes not only from light sources,
but also from light reflected off other objects. This is known as
indirect illumina-
tion
. To account for this, the irradiance at a surface point must be computed by
integrating the incoming radiance in all directions above the surface, not just the
directions to the light source. More precisely, irradiance
E
at a surface point
x
comes from integrating the cosine-weighted incident radiance
L
i
(
(
x
)
x
,
ω
)
function
over the hemisphere
Ω
above the surface at
x
(
Figure 1.4
)
:
,
ω
)(
·
ω
)
ω
.
E
(
x
)=
L
i
(
x
n
d
(1.6)
Ω
·
ω
=
θ
,where
θ
the incident angle. Mul-
Here
n
is the unit normal vector;
n
cos
θ
essentially undoes the projective foreshortening and thereby
accounts for the spreading out of flux coming in at an angle.
tiplication by cos
