Graphics Reference
In-Depth Information
CHAPTER 5
Approaches to Geometric
Modeling
Prerequisites: Section 4.2 (topology of R n ) and Chapter 7 (cell complexes, Euler char-
acteristic) in [AgoM05]
5.1
Introduction
The last four chapters covered the basic mathematics and computer graphics algo-
rithms needed to display two- or three-dimensional points and segments. As limited
as this may sound, this is actually enough to develop a quite decent modeling system
that would handle complex linear three-dimensional objects as long as we represent
them only in terms of their edges (“wireframe” mode). Such a system might be ade-
quate in many situations. On the other hand, one would certainly not get any eye-
catching displays in this way. To generate such displays, we need to represent objects
more completely. Their surfaces, not just their edges, must be represented. After that,
there is the problem of determining which parts of a surface are visible and finally
the problem of how to shade those visible parts.
Recall the general geometric modeling pipeline shown in Figure 5.1. Of interest
are the last three boxes and maps between them. This chapter presents a survey of
the various approaches that have been used to deal with that part of the pipeline.
First, one has to understand the “pure” mathematical objects and maps. The next task
is to represent these in a finite way so that a computer can handle them. Finally, the
finite (discrete) representations have to be implemented with specific data structures.
By in large, users of current CAD systems did not require the systems to have much
understanding of the “geometry” of the objects. That is not to say that no fancy math-
ematics is involved. The popular spline curves and surfaces involve very intricate
mathematics, but the emphasis is on “local” properties. So far, there has not been any
real need for understanding global and intrinsic properties, the kind studied in topol-
ogy for example.
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