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# ch5 - GMCh5 5-1 GEOMETRIC MODELING A First Course Copyright...

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GMCh5 3/2/99 5-1 GEOMETRIC MODELING: A First Course Copyright © 1995-1999 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 3-D 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 3-D 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. Computationally-useful 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 3-D, without dangling faces or edges. We saw earlier that this requirement is met by regular sets in 3-space. 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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GMCh5 3/2/99 5-2 Second, subtractive and additive manufacturing-process models are guaranteed to produce solids. Finite describability
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ch5 - GMCh5 5-1 GEOMETRIC MODELING A First Course Copyright...

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