Graphics Reference
In-Depth Information
the differential area
dA
corresponds to an area on the sphere that is smaller than
dA
, according to an effect known as
projective foreshortening
. The ratio of this
foreshortening is the cosine of the incident angle
θ
(
Figure 1.2(b)
)
. Consequently,
the irradiance at
x
is
)=
Φ
cos
θ
E
(
x
.
(1.4)
r
2
4
π
This cosine factor often appears in radiative transfer equations. It comes about
because incident light is spread over an increasingly large area as the angle of
incidence increases. This is one reason sunlight is stronger when the sun is high
in the sky than when it is low on the horizon. It is also the principal cause of the
Earth's seasons: in the winter, the sun is lower in the sky and therefore produces
less irradiance.
While irradiance
E
is the radiant power received at a point
x
, the radiant
power
leaving
a surface at
x
is the known as the
radiant exitance
or
radiosity
,
denoted by
M
(
x
)
. The definition is the same as that of irradiance, flux per
area, but the flux is regarded as leaving the surface rather than arriving. The term
“radiant exitance” has become the preferred term in recent years, to avoid confu-
sion with the “radiosity method” for computing global illumination. (described in
Chapter 2).
(
x
)
or
B
(
x
)
1.1.3 Radiance
Just as flux does not depend on surface position, irradiance and radiant exitance
do not depend on direction. But light emission and reflection clearly do depend
on direction: the color and strength of light perceived at a photoreceptor in the
human eye is dependent on the particular direction. The radiant power exiting
from (or incident on) a point
x
in a direction ω
.
2
Radiance is essentially a directional restriction of irradiance or radiant exi-
tance: it is the flux per area in a differential cone of directions (
Figure 1.3
)
. As
a matter of convenience, the surface through which the flux is measured need not
be perpendicular to the direction ω
,
ω
)
is called the
radiance L
(
x
.If
θ
is the angle the surface normal makes
with the direction ω
,then
dA
cos
θ
is the
projected differential area
—the cos
θ
factor accounts for the projective foreshortening. If
d
ω
denotes the differential
solid angle
3
about the direction ω
, as illustrated in Figure 1.3, radiance can be
2
Although directions are often represented by 3D vectors, it is also convenient to represent di-
rections from a point
x
as points on the unit sphere centered at
x
. Each point on the unit sphere
corresponds to a unique direction. By convention, directions are often denoted by ω
3
A set of directions corresponds to a subset of the unit sphere; the
solid angle
of the set is the area
of the corresponding spherical subset. The set of all directions therefore has solid angle 4π, the area
of the unit sphere. The differential solid angle
d
ω of a differential cone is the differential area of the
unit sphere subtended by the cone.
