Graphics Reference
In-Depth Information
representation, which CSG representation of the object does one want? Which
is the “correct” one? Ad hoc answers are not good enough. See [Shap91] and
[GHSV93].
(3) A theoretical foundation for operations on objects is lacking or not well inte-
grated into the representation scheme concept.
From a very global point of view however, a basic bottleneck is computer power.
The fact is that (at least in some important aspects) there is a great body of known
mathematics about geometry and topology that is central to doing geometric model-
ing “right” and which is simply waiting for the time that computers are fast enough
to make dealing with it feasible. It is easy to see this relationship between progress
in geometric modeling and the development of speedier computers. One simply has
to look at the fantastic advances in rendering photorealistic images. This is not to say
that no innovations are needed. The use of computers has brought along with it a host
of additional problems that have to be solved while at the same time creating oppor-
tunities for finding new ways to understanding. An example of the former is the fact
that computers do not have infinite precision arithmetic so that algorithms that are
mathematically simple become very tricky to implement when one has to worry about
round-off errors. An example of the latter is the ability to use computers to visualize
data in ways that was not possible before. This by itself can lead to advances in knowl-
edge. Coming up with good user interfaces is also a great challenge. Nevertheless, if
computers could just deal with all the mathematics related to geometric modeling
that is known
right now
we would be a giant step further along. Along these lines,
two features that modeling systems should support but do not because the algorithms
are too expensive computationally are:
(1) The ability to differentiate between objects topologically.
(One would need to implement the basic machinery of algebraic topology.)
(2) The ability to represent space and objects intrinsically.
Another aspect of this is that one should represent not only objects but the
space in which they are imbedded. With the rise of volume rendering we are
beginning to see some movement on that front.
These issues will be addressed again in Chapter 16. Given the certainty of rapid
advances in hardware, geometric modeling has an exciting future.
Finally, here are some “philosophical” comments, alluded to before, having to do
with the way that one should approach the subject of geometric modeling ideally. One
of the things one learns from mathematics is that whenever one introduces a certain
structure, be it that of a group, vector space, or whatever, it has always been fruitful
to define maps that preserve that structure, like homomorphisms, linear transforma-
tions, and so on, respectively. The sets and maps are in a sense interchangeable. One
could start with a class of maps and define a structure in terms of that which is left
invariant by the maps. Furthermore, new structures and maps are usually studied
by associating simpler invariants to them. A perfect example of this is the field of alge-
braic topology. See Chapter 7 in [AgoM05] for some simple examples of this (for
example, the functor from the category of simplicial complexes and simplicial maps
to the category of chain complexes and chain maps). In computer science one looks
for “representations” and everything boils down to finding suitable representations
