Graphics Reference
In-Depth Information
to really work with things in a reasonable way, you'll want to be familiar with the
following:
• Trigonometry
• Operations on small vectors and matrices (which we already discussed in
this chapter)
• Integrals and derivatives
• And some geometric and topological notions, like continuity, the geometry
of surfaces in three dimensions, and curvature
All of this is made easier by a working knowledge of basic linear algebra, which
we assume throughout the topic.
The most fully developed area of computer graphics is
photorealistic
rendering
—producing an image from some model of a scene and the lights in
it. Each pixel of a rendered image can be thought of as representing a
measure-
ment
of the light passing along certain rays in the scene, just as each pixel of a
digital photograph is a measurement of all the light that hit one small region of
the photo sensor in the camera. This can be seen as an integral of the incoming
light energy over that region. Since it's impractical to evaluate most such integrals
exactly, we end up using approximations (e.g., the rule we mentioned earlier that
states that the “integral is approximately the value at the center of the region, mul-
tiplied by the area of the region”). In doing this, we've replaced the desired value
by a value computed from a single sample; we could have used more samples,
but in practice, we'll always be using a finite number of samples, and using these
to estimate some integral. Thus, the process of sampling, and of approximating
integrals through samples, is central to rendering.
Every measurement in science is an act of statistics: Our measuring device
may function differently from day to day; the thing we measure may be just one
of many possible, nearly equivalent, measurements (think of measuring the tem-
perature in a beaker of water; you only really measure it in one part of the beaker).
In the case of rendering, the statistic is some integral; the random variable is the
set of samples that we use to evaluate it, and the result is that a given rendering of a
scene usually depends on some random number generator: Multiple renderings of
the same scene with the same software will produce different values for any par-
ticular pixel. This distribution of values will typically cluster around some mean
value, which one hopes is correct, and will have some variance. If the variance of
adjacent pixels is uncorrelated, it may appear in the output as speckle, or visual
noise. If it's correlated, it may appear as
jaggies
—a staircaselike representation of
what should be a smooth diagonal line. This means that assessing the quality of
an algorithm also entails statistical measurements.
The subject matter of computer graphics is no longer linearizable in any reason-
able way. Each topic ends up so intertwined with all others that there's no way
to decide which one to discuss first, and any presentation ends up with succes-
sive disclosures: a first description, a later correction, a further improvement,
etc. Readers, naturally, like topics to be
organized;
when you want to review
