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Computer Graphics and Geometric Modeling GOSPR 5/5/2005 5:47 PM Page Iii GOSPR 5/5/2005 5:47 PM Page i Computer Graphics and Geometric Modeling GOSPR 5/5/2005 5:47 PM Page iii Max K. Agoston Computer Graphics and Geometric Modeling Implementation and Algorithms GOSPR 5/5/2005 5:47 PM Page iv 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 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as per- mitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in 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 Springer is part of Springer Science+Business Media springeronline.com © Springer-Verlag London Limited 2005 Printed in the United States of America The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Typesetting: SNP Best-set Typesetter Ltd., Hong Kong 34/3830-543210 Printed on acid-free paper SPIN 10971451 GOSPR 5/5/2005 5:47 PM Page v Preface 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 GOSPR 5/5/2005 5:47 PM Page vi vi Preface 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 GOSPR 5/5/2005 5:47 PM Page vii Preface vii emphasize understanding what is going on, there is a natural emphasis on theory and not on tricks of the trade. The reader will also not find any beautiful glossy pictures. Clearly, no one book can cover all that falls under the general heading of geo- metric modeling. As usual, the topics that are in fact covered and the degree to which they are covered reflect my own bias. In a large field, there are many special topics and it should not be surprising that some are not discussed at all and others only briefly in an overview. On the other hand, one would expect to see a discussion of principles and issues that are basic to the field as a whole.
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