# ch7 - GMCh7 7-1 GEOMETRIC MODELING A First Course Copyright...

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5/10/99 7-1 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. 7. Application Algorithms Unambiguous geometric models are potentially capable of supporting fully automatic algorithms for any applications that involve object geometry. However, only a few application algorithms have reached maturity and are routinely used in industry. This chapter discusses currently understood applications in graphics and simulation; mass- property calculation (i.e., volume, moments of inertia); and interference (i.e., collision) analysis. It also touches upon other applications such as planning for inspection and robotics, which are emerging from the research labs. Generally, analysis algorithms are well-developed, whereas synthesis algorithms, which require geometric reasoning , are not. 7.1 Graphics and Kinematic Simulation 7.1.1 Overview Most of the rendering software in use today operates on BReps. Simple images are produced swiftly, but photo-realistic renderings, with texture, shadows, and so on, may take several minutes or even hours per image. If objects are modeled by using Boolean operations, the BRep must be evaluated before rendering, and this is a time consuming procedure. Thus, although rendering itself is fast, the entire process suffers from what is sometimes called “the Boolean bottleneck”. Rendering methods that do not require boundary evaluation, and therefore avoid the Boolean bottleneck, are attractive for modeling systems that define objects primarily through Booleans. Graphic algorithms for BReps are covered in standard graphics texts, and are not discussed extensively here. We focus on algorithms that do not require explicit boundary information. Graphic displays of solids and surfaces are typically produced in three styles: Wireframes. Line drawings with hidden lines removed. Shaded displays. Wireframes – All the edges of the object are displayed, regardless of whether they are truly visible or not. For curved objects, which have few edges, displays often contain additional curves, usually called generators . These may be computed by intersecting the object with a set of parallel planes, or, more commonly, by tracing curves of constant parameter value in parametric surfaces. Figure 7.1.1.1 provides an example. Given parametric representations for the curves to be drawn, the display process consists of (i) stepping along the curve through suitable parameter increments, (ii) generating a piecewise linear approximation on the fly, and (iii) projecting the line segments on the screen, as discussed in Chapter 2. Parameter increments may be constant, or they may be smaller in regions of high curvature,

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ch7 - GMCh7 7-1 GEOMETRIC MODELING A First Course Copyright...

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