GEOMETRIC MODELING: A First Course
Copyright © 1995-2000 by Aristides A. G.
Permission is hereby granted to copy this document for individual student use at USC,
provided that this notice is included in each copy.
Dazzling cinematic special effects.
.. Realistic simulations of machines in motion.
Analyses of the stresses in aerospace structures during flight.
.. Programs that automatically
drive robots along collision-free paths in cluttered environments.
.. Rapid prototyping
machines that behave like three-dimensional printers.
.. Underlying these and many other
of the objects under study. Geometric models are
computational (symbol) structures that capture the spatial aspects of the objects of interest
for an application. This course is primarily concerned with geometric models for three-
dimensional objects, and with the associated computer algorithms for constructing and
querying the models.
We are interested in modeling both real (i.e., physical) and virtual objects. Virtual objects
often correspond to physical objects that are being designed but have not yet been built. But
they may also correspond to objects that do not obey the laws of Physics, and therefore are
purely imaginary. In addition, some of the objects we encounter in the applications are
themselves mathematical abstractions—for example, the set of accessible directions along
which a touch probe may approach a given surface.
Geometric information is pervasive in many engineering and scientific fields, such as (i)
VLSI layout, (ii) geographic information systems, (iii) electronic packaging, (iv) computer
graphics and visualization, (v) computer vision, (vi) architectural and structural design, and
(vii) design and manufacture of electromechanical products, to name a few. The first two
examples just cited are primarily two dimensional (2-D), whereas the last two are
intrinsically 3-D; examples (iii) through (v) may be either 2-D or 3-D. This course
emphasizes modeling of 3-D objects, and attempts to strike a balance between applications
in two areas: graphics and multimedia, and robotics and automation.
3-D computer graphics is becoming ubiquitous. Most desktop computers systems are
expected to bundle support for 3-D applications in the near future, and the use of 3-D in
multimedia and the World Wide Web is burgeoning. Display techniques are relatively well-
developed and are implemented in hardware accelerators and in software browsers.
Construction of the geometric models of the objects to be displayed, however, is becoming
a bottleneck for 3-D graphics. In this text we attempt to complement, rather than compete
with, Computer Graphics textbooks. We focus on modeling, and de-emphasize display and
visualization, because they are well covered in the graphics texts—e.g. [Foley
Models for free-form curves and surfaces are also treated briefly, because they constitute a