Graphics Reference
In-Depth Information
the points and lines with WPF's 2D rendering tools do we get a picture. Trans-
formations like this into intermediate representations occur elsewhere in graphics
as well. In some expressive rendering algorithms, input images are transformed
into an intermediate representation that consists of edges detected by some image-
processing algorithms and regions bounded by those edges, for instance. Choosing
a good intermediate representation can make the difference between success and
failure.
The discovery of perspective in Western art, and the working out of the asso-
ciated mathematics, is a fascinating subject. When lines that are parallel in the
world appear to converge in the rendered picture, the eye is drawn to the point
of convergence, or
vanishing point.
Almost as fascinating as the development of
perspective is the clever use by artists of this property of vanishing points; some-
times artists created different vanishing points for different regions of a picture,
drawing the viewer's eye to multiple things in the scene (
The Resurrection,
by
Piero della Francesca, is said to have this property.) It's not known whether this
was in fact deliberate. There are many systematic ways of creating proper per-
spective renderings using so-called “vanishing points” and ideas from projective
geometry. These too can be considered basic rendering engines like the one shown
in the Dürer woodcut.
Also intriguing is the understanding one can gain of non-Western art. Rock's
book on perception [Roc95], for instance, explains that the view seen in some
Chinese scroll paintings, which appears odd to the Western eye, is actually a
perspective-correct view for a view from a very high point, with the projection
plane perpendicular to the ground plane.
Perspective projections do not preserve relative lengths (think of a picture of
railroad tracks disappearing into the distance—the equal distances between adja-
cent railroad ties become unequal distances in the picture), but they do preserve
some other properties. The study of projections and the transformations of space
that are associated to them became the field of
projective geometry;
for the math-
ematically curious, the topics of Hartshorne [Har09]and Samuel [SL88] are excel-
lent introductions.
Representations of polyhedral models by arrays of vertices together with
arrays of faces described by vertex indices are sometimes called
indexed face
sets;
a large repository of models stored in this way has been collected in the
Brown Mesh Set [McG]. While not particularly compact, it is an easy-to-parse
format. Similarly simple is the
PLY2
format. Many example models are available
on the Internet in both forms. More complex model formats, using more compact
binary representations, abound. One fairly common format was
3DS
, developed
for use with the 3D Studio Max software (now known as 3ds Max), and widely
adopted or imported by other tools. 3ds Max now uses the
.max
format, which is
proprietary, but many
3DS
models are still available. And Maya, another popular
shape-modeling program, has its own proprietary format,
.mb
. Both are essen-
tially meta-formats, which specify the plugin (shared library) that should be used
to parse each subpiece of a model; in practice, it's impractical to reverse-engineer
such a format, and as a result, for hobbyist and classroom use people continue to
use older and simpler formats.
What we called the “view region” in this chapter—the portion of the world
that's visible in the picture we're making—is an instance of the more general
notion of
view volume;
the difference is that a view volume, rather than being an
infinite pyramid, may be truncated so that objects farther away than some distance
