Graphics Reference
In-Depth Information
CHAPTER 16
Intrinsic Geometric Modeling
Prerequisites:
Chapters 7-9 in [AgoM05] (for Sections 16.3 and 16.4)
16.1
Introduction
The modeling that we have discussed so far in this topic dealt with objects imbedded
in
R
3
(or
R
n
more generally). Euclidean space was ever present, even if implicitly, when
we talked about spaces whose points were tuples of real numbers. The curves and sur-
faces that we discussed were always presented as subspaces of Euclidean space. This
is of course not surprising because most people when they talk about geometric
objects, even when they talk about them in abstract terms, usually do so in the context
of a particular imbedding of them in
R
3
rather than thinking of them intrinsically.
However, computers have become powerful enough to start doing more of the latter.
The motivation for this is not just theoretical. For example, to understand our universe
we need to try to represent space in a global way. Consider the analogy with a hypo-
thetical two-dimensional world that is actually a sphere or a torus. Even though the
view locally would make it look like
R
2
, to understand it in its entirety we need to think
of it as a
collection
of “flat” patches. This world exists on its own without any “outside”
that a three-dimensional person could give it using some imbedding of it in
R
3
. When
we come to our own three-dimensional universe, is it like
R
3
or like the three-
dimensional unit sphere in
R
4
? We do not hypothesize any “outside” surrounding space
in which our universe is imbedded when we think about the world around us.
In this chapter we want to suggest that we have come to a point in time where
there is a place for modeling programs that representing spaces intrinsically and do
graphics
inside
a space (manifold) rather than pretend that everything we want to
look at and understand lies in a single view. Although the idea of doing this is not
entirely new, mathematicians have developed a few programs of this type for research
purposes, such a goal has not received much attention in the broader computer graph-
ics community up to now. Virtual reality programs only partially accomplish what we
have in mind. Also, note that although we have used the word “global” in previous
chapters, as in the discussion of local and global illumination models in Chapter 9
and in the title of Chapter 14, we are taking the meaning of “global” a step further
