Graphics Reference
In-Depth Information
algorithms, we'll be making a steady-state assumption (that
L
(
t
,
...
)
is indepen-
dent of the parameter
t
), and the wavelength parameter
λ
will never enter into any
formulas in any significant way, so we'll write
L
(
P
,
)
instead of
L
(
t
,
P
,
,
λ
)
.
v
v
We've described how light energy is measured, and given some details of the
microscopic characteristics of light and their relationship to the characteris-
tics of the matter with which light interacts, but proper treatment of these
belongs in the realm of physics. A superb first reference is Crawford's book on
waves [Cra68]. One should also understand some electromagnetic theory [Pur11].
Anyone who studies graphics will also benefit from even a partial reading of New-
ton's
Opticks
[New18]. It teaches not only the ideas of optics, but how a brilliant
observer and experimenter works. For the application of these ideas in practical
algorithms, see the topic by Pharr and Humphreys [PH10].
We've discussed the
radiometric
terms used in measuring light, but there are
also
photometric
terms, which attempt to capture the human perception of light.
In particular, light of different wavelengths can be perceived as equally bright, and
so by summing up the radiance at many different wavelengths, each multiplied by
a factor
y
(
)
representing the perceived brightness of a certain amount of light
energy at wavelength
λ
, you can get a single number (called
luminance
) that
represents total brightness. The function
y
is called the
luminous efficiency;
it
also is used in the CIE color system, which we discuss in Chapter 28. Luminance
is measured in
lumens.
Such a single number for brightness of a light makes sense
in contexts where most light is broad-spectrum, and most reflectors reflect light
across much of the spectrum. But because light energy and reflectivity are spectral
quantities, it's possible to have a light of high luminance (e.g., a bright red laser)
and a surface with high reflectivity (where this reflectivity is an average over all
wavelengths of the spectral reflectivity), and yet have the reflected light be low
luminance, if the surface, for example, happens to absorb rather than reflect light
at the wavelength of the laser. Because of this, such photometric terms see little use
in graphics, but they are of considerable importance in illumination engineering.
λ
Exercise 26.1:
We claimed in the chapter that the horizontal projection
P
to the
unit sphere from the vertical cylinder enclosing it was an area-preserving map.
Compute the derivative of this map at a point
(
x
,
y
,0
)
of the cylinder, and verify
that it
is
area-preserving. Why is it sufficient to carry out this computation at a
point where
z
=
0?
Exercise 26.2:
We computed the solid angle subtended by a spherical cap of
radius
r
cos(
r
))
.
(a) What is the solid angle subtended by a spherical cap of radius
r
on a sphere of
radius
R
?
(b) What is the actual
area
of a spherical cap of radius
r
on a sphere of radius
R
?
You should be able to do this problem almost by inspection, without any integrals
at all.
Exercise 26.3:
We computed the radiance emitted from each point of a spher-
ical uniform source of radius
r
and total power
Φ
in each outward direction
<π
on the unit sphere as 2
π
(
1
−
v
