Graphics Reference
In-Depth Information
is used to smooth out automotive finishes and to polish sharp knives. Thus, mate-
rials whose roughness is between that of a highly polished metal and a piece of
newsprint are likely to contain scratches that are on the scale of a few wavelengths
of light. Despite this, the geometric optics approach seems to make good predic-
tions in practice at this scale. We'll describe the main ideas of the geometric optics
approach in the next several sections.
We do so with a caution, however: Given the scale disagreements (facets
must be large compared to the wavelength of light, but in practice are quite close
to it), these models are at best weak approximations of the underlying phenom-
ena. Recent careful measurement work has shown the weakness of the approxi-
mations [Lei10].
The underlying physics of reflection from a flat surface depend on the electrical
properties of the atomic structure of the material, some of which we described in
Chapter 26. In particular, metals, which tend to make the best mirror reflectors,
have many free electrons that float about the surface, creating an almost perfectly
planar sheet of constant electrical potential with which the electromagnetic light-
wave interacts. The Fresnel equations determine the degree of reflection for light
of varying polarizations; in graphics, we typically assume unpolarized light, and
thus average the perpendicular and parallel terms of the Fresnel equations. We'll
review these equations, and describe how they're applied in practice, since they
are part of all the physically based scattering models.
As we said, the assumption, in the physical computations that support these
reflectance models, is that the reflecting surface is large compared to the wave-
length of the arriving light, or else diffraction will start to dominate. The mirror-
plane model can also be used to compute the reflection from a mirror surface that's
nonplanar, provided that its curvature is not too great; to compute the reflection
at a point
Q
, we use the normal vector
n
(
Q
)
to compute the mirror direction just
as before. If the curvature is too large (i.e., if the normal vector changes too fast),
then again the “sheet of constant potential” model fails, and diffraction starts to
dominate. A radius of curvature (in any surface direction) that's near or lower than
the wavelength indicates a place where the mirror model is no longer appropriate.
It's worth noting that in polygonal models, the curvature at every point of every
edge is infinite. This is typically ignored in ray tracers, where a ray hits either one
facet or the other, and proximity of the ray to an edge is ignored. If the ray actually
hits an edge precisely, it may be ignored or treated as lying on one of the two adja-
cent surfaces. The results look correct enough that they have not generally been a
point of concern.
In Chapter 26, we saw that at a surface between dielectric materials such as water
and air, or glass and air, the amount of light reflected and transmitted depended
strongly on the angle of the arriving light. Under the assumption that the arriving
light was unpolarized, the fraction of light energy reflected is a function of the
refractive indices of the two materials and the incident angle
θ
i
=cos
−
1
(
v
i
·
n
)
,
and the angle of the transmitted ray is
θ
t
. The Fresnel reflectance
R
F
is the average
