Computer Graphics and Geometric Modeling
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Max K. Agoston
Computer Graphics and
Geometric Modeling
Implementation and Algorithms
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Max K. Agoston, MA, MS, PhD
Cupertino, CA 95014, USA
British Library Cataloguing in Publication Data
Agoston, Max K.
Computer graphics and geometric modeling:implementation & algorithms
1. Computer graphics 2. Geometry—Data processing 3. Computer-aided design
4. Computer graphics—Mathematics I. Title
006.6
ISBN 1852338180
Library of Congress Cataloging-in-Publication Data
Agoston, Max K.
Computer graphics & geometric modeling/Max K. Agoston.
p. cm.
Includes bibliographical references and index.
Contents: Implementation & algorithms
ISBN 1-85233-818-0 (v. 1 : alk. paper)
1. Computer graphics. 2. Geometry—Data processing. 3. Mathematical models. 4. CAD/CAM
systems. I. Title.
T385.A395 2004
006.6—dc22 2004049155
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the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright
Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the
publishers.
ISBN 1-85233-818-0
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GOSPR 5/5/2005 5:47 PM Page iv
This book and [AgoM05] grew out of notes used to teach various types of computer
graphics courses over a period of about 20 years. Having retired after a lifetime of
teaching and research in mathematics and computer science, I finally had the time to
finish these books. The two books together present a comprehensive overview of com-
puter graphics as seen in the context of geometric modeling and the mathematics that
is required to understand the material. Computer graphics itself is a multifaceted
subject, but it has grown up. It is no longer necessary that a book on graphics demon-
strate the diversity of the subject with a long list of “fun” projects at the expense of
the mathematics. From movies, television, and other areas of everyday life, readers
have already seen what graphics is about and what it can do. It follows that one should
be able to present the geometric modeling aspect of the subject in a systematic
fashion. Unfortunately, the sheer amount of material that I wanted to cover meant
that it had to be divided into two parts. This book contains the practical stuff and
describes the various algorithms and implementation issues that one runs into when
writing a geometric modeling program. The book [AgoM05] provides the mathemat-
ical background for the underlying theory. Although each book can be read by itself
without reading the other, one will get the most benefit from them if they are read in
parallel.
The intended audience of this book (and the combined two volumes especially) is
quite broad. It can be used in a variety of computer graphics courses or by those who
are trying to learn about graphics and geometric modeling on their own. In particu-
lar, it is for those who are getting involved in what is referred to as computer-aided
design (CAD) or computer-aided geometric design (CAGD), but it is also for mathe-
maticians who might want to use computers to study geometry and topology. Both
modeling and rendering issues are covered, but the emphasis is on the former. The
basic prerequisites are that the reader has had an upper division data structure course,
minimally three semesters of calculus, and a course on linear algebra. An additional
course on advanced calculus and modern algebra would be ideal for some of the more
advanced topics. On the companion CD there is a geometric modeling program (GM)
that implements many of the algorithms discussed in the text and is intended to
provide a programming environment both for further experimentation and applica-
tion development. Another program (SPACE) on the CD is an application that uses
some of the more advanced geometric modeling concepts to display the intrinsic
Preface
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geometry of two- and three-dimensional manifolds. Both programs were written using
the Microsoft Visual C++ compiler (and OpenGL) and run under Microsoft Windows
98 or later. Their source code and documentation are included on the CD. The ReadMe
file on the CD lists what all is on the CD and also contains instructions for how to use
what is there.
As I began to develop this book on geometric modeling, one concern obviously
was to do a good job in presenting a thorough overview of the practical side of the
subject, that is, the algorithms and their implementation details. However, there were
two other goals that were important from the very beginning. One was to thoroughly
explain the mathematics and the other, to make the material as self-contained as pos-
sible. In other words, pretty much every technical term or concept that is used should
be defined and explained. The reason for putting all the computer graphics–related
material into one book and all the mathematics into the other rather than inter-
weaving the material was to keep the structure of the implementation of a modeling
program as clear as possible. Furthermore, by separating out the mathematics it is
easier for readers to skip those mathematical topics that they are already familiar with
and concentrate on those with which they are not. In general, though, and in partic-
ular as far as instructors using this book are concerned, the intent is that the mate-
rial in the two books be covered in parallel. This is certainly how I always taught my
courses. An added motivation for the given division was that the applied part of geo-
metric modeling was often a moving target because, largely due to improvements in
hardware (faster CPUs, more memory, more hard disk space, better display devices),
the way that one deals with it is changing and will continue to change in the future.
This is in contrast to the supporting mathematics. There may be new mathematics
relevant to computer graphics in the future but it will be a long time before the math-
ematics I do discuss will lose its relevance. A lot of it, in fact, is only now starting
to be used as hardware becomes capable of dealing with computationally expensive
algorithms.
One property that sets this book apart from others on geometric modeling is
its breadth of coverage, but there is another. The combined books, this one and
[AgoM05], differ from other books on computer graphics or geometric modeling in
an important way. Some books concentrate on implementation and basically add the
relevant mathematics by tossing in appropriate formulas or mathematical algorithms.
Others concentrate on some of the mathematical aspects. I attempt to be as compre-
hensive on both the implementation and theory side. In [AgoM05] I provide a com-
plete reference for all the relevant mathematics, but not in a cookbook fashion. A
fundamental guiding principle was to present the mathematics in such a way that the
reader will see the motivation for it and understand it. I was aiming at those indi-
viduals who will want to take the subject further in the future and this is not possi-
ble without such understanding. Just learning a few formulas is not good enough. I
have always been frustrated by books that throw the reader some formulas without
explaining them. Furthermore, the more mathematics that one knows, the less likely
it is that one will end up reinventing something. There are instances (such as in the
case of the term “geometric continuity”) where unfamiliarity with what was known
caused the introduction of confusing or redundant terminology. The success or failure
of the two books should be judged on how much understanding of the mathematics
and algorithms the reader gets. In the case of this book by itself, it is a question of
whether or not the major topics were covered adequately. In any case, because I
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