Graphics Reference
In-Depth Information
the area. The cosine term also varies from point to point for an area source but is
constant for the omni-light's point source; one might say that this is another way
that size “matters.” Regardless, the conclusion is that we can use exactly the same
formula for the biradiance from area and point sources.
The biradiance at
P
due to the omni-light at
Q
(see Figure 14.31) is given by
Q
v
P
Φ
M
i
(
Q
,
P
)=
2
sr
,
(14.39)
4
π ||
Q
−
P
||
Figure 14.31: Points and direc-
tions in point-light equations.
if there is no scene point on the open line segment from
Q
to
P
; it is 0 if there is
an occlusion.
Note that we could say that the effective radiance is
Φ
L
(
P
,
−
v
i
)=
2
,
(14.40)
π
||
−
||
4
sr
Q
P
that is, it is proportional to the total power of the luminaire and falls off with the
square of distance from the luminaire. In fact, if we were to insert that expression
into a renderer it would yield the desired image so long as
was “suf-
ficiently large.” However, this is not actually a true radiance expression because
it is not conserved along a ray—it falls off with distance because our omni-light
in fact has the physically impossible zero surface area, which leads it to create a
physically impossible radiance field in space. Note that both the radiance and the
biradiance approach infinity as
||
Q
−
P
||
0. It is common practice to clamp the
maximum biradiance from an omni-light, since when that distance is small our
original assumption that the distance to the luminaire is much greater than the size
of the luminaire is violated and the resultant estimated light intensity is greater
than intended. A less efficient but more accurate correction would be to actually
model the luminaire as an object with nonzero surface area (such as a sphere)
when the distance is less than some threshold.
There is no illumination from the omni-light at locations that do not have
an unobstructed line of sight. These regions form the shadows in an image. The
boundary of the shadows from an omni-light will be “hard,” with a distinct curve
across a surface that distinguishes lit from shadowed. This is unlike the area
sources, which produce “soft” shadows with blurry silhouettes. A lighting algo-
rithm such as shadow mapping that evaluates light visibility at lower precision
than the radiance magnitude can appear to produce soft shadows from a point
light. This is in fact an artifact of reconstruction from an aliased set of samples.
Nonetheless, it may be visually pleasing in practice.
||
Q
−
P
||→
For an omnidirectional point light that is far from all locations in the scene,
v
i
and
L
(
P
,
−
v
i
)
due to the light vary little across the scene. A
directional light
is an
omni-light with the further simplifying approximation that it is so far from the rest
of the scene that
v
i
and
L
can be treated as constant throughout the scene. This
eliminates some of the precision and modeling challenges of placing a point light
very far away, while giving a reasonable model for a distant light source such as
the sun.
We could model the total power of the distant point light, but it is typically
enormous and “distant” is ambiguous, so it is easier to model the (constant) inci-
dent radiance at points in the scene by simply letting
L
(
P
,
)=
L
0
for
v
i
directed
v
