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,