GMCh5
3/2/99
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GEOMETRIC MODELING: A First Course
Copyright © 19951999 by Aristides A. G.
Requicha
Permission is hereby granted to copy this document for individual student use at USC,
provided that this notice is included in each copy.
5. Solids
This chapter discusses 3D solids. First we address mathematical modeling issues, and
then representations. We consider only single objects that are rigid and made of
homogeneous materials. Assemblies of several components and inhomogeneous objects
raise additional issues, which are briefly discussed in the section on Further Explorations.
Solids may be represented by using the various methods discussed in Chapter 3. Solidity
does not raise significant new issues for most of these methods. However, BReps of solids
deserve a more detailed treatment than that provided in Chapter 3, and are the main focus of
our representational discussion in this chapter.
5.1 Mathematical Models for Rigid and Homogeneous Solids
Most of the physical objects we encounter in real life are solids. Sometimes they can be
modeled as surfaces or curves. For example, in stress analysis, thin shells are usually
analyzed as if their thickness was effectively zero. But full 3D models are required for
many applications. In addition, we often need to model not only solid objects but also
operations on them. For example, fabrication processes such as machining or welding are
important in CAD/CAM. The geometrical aspects of machining may be modeled as the
(regularized) difference between the initial state of the workpiece and the volume swept by
the cutter in its motion. Welding and other additive processes may be modeled by set
union.
Computationallyuseful mathematical models for rigid solids should exhibit the following
properties.
Rigidity
– This is easily achieved since the distances and angles among points of a set in
Euclidean space are fixed. Rigid motions preserve distances and angles. Therefore all the
instances of a set obtained from one another by rigid motions can be used to model a rigid
object in all its poses.
Finiteness
– A physical object should have a finite extent. To ensure finiteness all we need
is to require that our sets be
bounded
.
Solidity
– A model for a solid should be homogeneously 3D, without dangling faces or
edges. We saw earlier that this requirement is met by
regular
sets in 3space.
Closure under Boolean operations
– Boolean operations applied to solids should produce
other solids. This has two important advantages. First, the results of a Boolean operation
can be used as inputs to other Booleans, and a solid model can therefore be incrementally
constructed by successive Boolean operations (perhaps interspersed with other operations).
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Second, subtractive and additive manufacturingprocess models are guaranteed to produce
solids.
Finite describability
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 Spring '09
 REQUICHA
 Topology, Manifold, Small Faces, BReps

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