Graphics Reference
In-Depth Information
Φ
Light source
→
(Normal direction)
θ
r
dA
cos
θ
dA
dA
x
d
Φ =
Φ
−
x
4
πr
2
dA
(a)
(b)
Figure 1.2
Irradiance caused by a light source. (a) The flux at a distance
r
from the source is spread
out over a sphere of radius
r
. (b) When the flux hits a surface at an angle, the differential
surface area
dA
gets the flux of a smaller differential area on the sphere: the irradiance is
reduced by the cosine of the angle of incidence.
per area, the
irradiance
at the area. The irradiance
E
at a point
x
on a surface
is the differential flux on a differential area
dA
at
x
divided by that area:
(
x
)
d
dA
,
E
(
x
)=
(1.2)
where
dA
is perpendicular to the surface normal direction at
x
, i.e., is parallel to
the surface at
x
. Irradiance is thus a measure of the radiant energy per area per
time. The function
E
is formalized as a function of surface position; it is the
limit of the flux in a small region containing
x
divided by the area of the region,
as the region shrinks uniformly to
x
. Irradiance is also known as
flux density
,asit
measures the local density of flux at the surface point.
Radiant emission of a sufficiently small light source can be regarded as a col-
lection of expanding concentric spheres centered at the light source, each having
constant flux
(
x
)
. When one of these spheres hits a surface point
x
, its radius
r
is
the distance from
x
to the source (
Figure 1.2(a)
)
. If the light source lies directly
above the surface, i.e., in the direction of the surface normal, then the differen-
tial flux
d
Φ
Φ
incident on the differential area
dA
at
x
is the total flux
Φ
times the
r
2
. The irradiance at
x
is
/
fraction of the sphere area that hits
dA
,whichis
dA
4
π
therefore
dA
4
Φ
dA
=
d
dA
=
Φ
r
2
π
E
(
x
)=
r
2
,
(1.3)
4
π
which is the familiar inverse square law of illumination.
Of course, light sources do not generally lie directly above the illuminated
surface; the incident flux sphere usually hits the surface at an angle
θ
. In this case
