Graphics Reference
In-Depth Information
In the case of real-time graphics, we are interested in finding the integral of
L
o
(
p,ω
o
) over the whole hemisphere
H
set around point
p
with surface normal
vector
n
. Therefore, we are looking for
L
o
(
p,ω
o
)=
H
2
f
r
(
p,ω
o
,ω
i
)
Ldω
i
.
Using the definition of radiance
L
,
L
o
(
p,ω
o
)=
H
2
f
r
(
p,ω
o
,ω
i
)
L
i
(
p,ω
i
)cos
θ
i
dω
i
.
(1.1)
During rendering we evaluate a finite number of lights. Therefore, we are inter-
ested in expressing integrals over area. In the case of irradiance over area
A
,we
can define
E
(
p,n
)=
A
L
i
(
p,ω
i
)cos
θ
i
dω
i
.
In the simplified case of
n
contributing lights, we can express the integral
from equation (1.1) as a sum of integrals over all area lights that are visible from
point
p
:
L
o
(
p,ω
o
)=
1
..n
f
r
(
p,ω
o
,ω
i
)
L
i
(
p,ω
i
)cos
θ
i
dω
i
(
n
)
.
(1.2)
A
(
n
)
Equation (1.2), our main lighting equation, will be the basis of all derivations.
For simplicity we can assume that the light source has uniform light flux
distribution so
L
(
p,w
) is constant and denoted
L
; therefore,
L
o
(
p,ω
o
)=
1
..n
L
n
A
(
n
)
f
r
(
p,ω
o
,ω
i
)cos
θ
i
dω
i
(
n
)
.
The differential solid angle is also in relation with the differential area.
In the case of a light source defined on a quadrilateral, the differential solid
angle can be expressed as a function of differential area of light:
dω
=
dAcosθ
o
r
2
,
(1.3)
where
r
is the distance from point
p
on the surface to point
p
on
dA
and
θ
o
is
the angle between surface normal
dA
at point
p
and
−
p
p
(see Figure 1.4).
It is worth noting that the solid angle is well defined for multiple primitives
that can be used as a light source shape, such as a disk and a rectangle.
To finalize, radiance at a point per single light is defined as
L
o
(
p,ω
o
)=
1
..n
L
n
A
(
n
)
f
r
(
p,ω
o
,ω
i
)
cosθ
i
dAcosθ
o
r
2
.
(1.4)
