Agoston M.K. Computer Graphics and Geometric Modelling: Mathematics
Подождите немного. Документ загружается.
Computer Graphics and Geometric Modeling
Max K. Agoston
Computer Graphics and
Geometric Modeling
Mathematics
Max K. Agoston, MA, MS, PhD
Cupertino, CA 95014, USA
British Library Cataloguing in Publication Data
Agoston, Max K.
Computer graphics and geometric modeling mathematics
1. Computer graphics 2. Geometry – Data processing 3. Computer-aided design
4. Computer graphics – Mathematics
I. Title
006.6
ISBN 1852338172
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: Mathematics.
ISBN 1-85233-817-2 (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
permitted 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 1-85233-817-2
Springer is part of Springer Science+Business Media
springeronline.com
© 2005 Springer-Verlag London Ltd.
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 10971444
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 goal of these books was to present a comprehensive overview
of computer graphics as seen in the context of geometric modeling and the mathe-
matics that is required to understand the material. The reason for two books is that
there was too much material for one. The practical stuff and a description of the
various algorithms and implementation issues that one runs into when writing a geo-
metric modeling program ended up in [AgoM05], and the mathematical background
for the underlying theory ended up here. I have always felt that understanding the
mathematics behind computer graphics was just as important as the standard algo-
rithms and implementation details that one gets to in such courses and included a
fair amount of mathematics in my computer graphics courses.
Given the genesis of this book, the primary intended audience is readers who are
interested in computer graphics or geometric modeling. The large amount of mathe-
matics that is covered is explained by the fact that I wanted to provide a complete
reference for all the mathematics relevant to geometric modeling. Although computer
scientists may find sections of the book very abstract, everything that was included
satisfied at least one of two criteria:
(1) It was important for some aspect of a geometric modeling program, or
(2) It provided helpful background material for something that might be used in
such a program.
On the other hand, because the book contains only mathematics and is so broad in
its coverage (it covers the basic definitions and main results from many fields in math-
ematics), it can also serve as a reference book for mathematics in general. It could in
fact be used as an introduction to various topics in mathematics, such as topology
(general, combinatorial, algebraic, and differential) and algebraic geometry.
Two goals were very important to me while writing this book. One was to thor-
oughly explain the mathematics and avoid a cookbook approach. The other was to
make the material as self-contained as possible and to define and explain pretty much
every technical term or concept that is used. With regard to the first goal, I have tried
Preface
very hard to present the mathematics in such a way that the reader will see the moti-
vation for it and understand it. The book is aimed at those individuals who seek such
understanding. Just learning a few formulas is not good enough. I have always appre-
ciated books that tried to provide motivation for the material they were covering and
have been especially frustrated by computer graphics 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. The success
or failure of this book should be judged on how much understanding of the mathe-
matics the reader got, along with whether or not the major topics were covered
adequately.
To accomplish the goal of motivating all of the mathematics needed for geomet-
ric modeling in one book, even if it is large, is not easy and is impossible to do from
scratch. At some places in this book, because of space constraints, few details are pro-
vided and I can only give references. Note that I always have the nonexpert in mind.
The idea is that those readers who are not experts in a particular field should at least
be shown a road map for that field. This road map should organize the material in a
logical manner that is as easy to understand and as motivated as possible. It should
lay out the important results and indicate what one would have to learn if one wanted
to study the field in more detail. For a really in-depth study of most of the major topics
that we cover, the reader will have to consult the references.
Another of my goals was to state everything absolutely correctly and not to make
statements that are only approximately correct. This is one reason why the book is so
long. Occasionally, I had to digress a little or add material to the appendices in order
to define some concepts or state some theorems because, even though they did not
play a major role, they were nevertheless referred to either here or in [AgoM05]. In
those cases involving more advanced material where there is no space to really get
into the subject, I at least try to explain it as simply and intuitively as possible. One
example of this is with respect to the Lebesque integral that is referred to in Chapter
21 of [AgoM05], which forced the inclusion of Section D.4. Actually, the Lebesgue
integral is also the only example of where a concept was not defined.
Not all theorems stated in this book are proved, but at least I try to point out any
potential problems to the reader and give references to where the details can be found
in those cases where proofs are omitted, if so desired. Proofs themselves are not given
for their own sake. Rather, they should be thought of more as examples because they
typically add insight to the subject matter. Although someone making a superficial
pass over the mathematical topics covered in the book might get the impression that
there is mathematics that has little relevance to geometric modeling, that is not the
case. Every bit of mathematics in this book and its appendices is used or referred to
somewhere here or in [AgoM05]. Sometimes defining a concept involved having to
define something else first and so on. I was not trying to teach mathematics for its
own interesting sake, but only in so far as it is relevant to geometric modeling, or at
least potentially relevant. When I say “potentially,” I am thinking of such topics as
algebraic and differential topology that currently appear in only minimal ways in mod-
eling systems but obviously will some day play a more central role.
It is assumed that the reader has had minimally three semesters of calculus and
a course on linear algebra. An additional course on advanced calculus and modern
algebra would be ideal. The role of Appendices B–F is to summarize what is assumed.
They consist mainly of definitions and statements of results with essentially no expla-
vi Preface
nations. The reason for including them is, as stated earlier, to be self-contained.
Readers may have learned the material at some point but forgotten the details, such
as a definition or the precise statement of a theorem. A reader who does not under-
stand some particular material in the appendices may not understand the discussion
at those places in the book where it is used. The biggest of the appendices is Appen-
dix B, which consists of material from modern algebra. This appendix is needed for
Chapters 7, 8, and 10, although not that much of it is needed for Chapters 7 and 8.
Only Chapter 10 on algebraic geometry needs a lot of that background. This is the
one place where using this text in the context of a course would be a big advantage
over reading the material on one’s own because an instructor who knows the mate-
rial would actually be able to explain the important parts of it quite easily and quickly
even to students who have not had a prior course on modern algebra. The actual
applications of Chapter 10 to geometric modeling do not require that much knowl-
edge if one skips over the background and proofs of the theorems that lead up to
them. Hopefully, however, the reader with a minimal mathematics background will
be reduced to simply learning “formulas” in only a few places in this book.
The extensive material on topology, in particular algebraic and differential topol-
ogy, has heretofore not been found in books directed toward geometric modeling.
Although this subject is slowly entering the field, its coming has been slow. Probably
the two main reasons for this are that computers are only now getting to be power-
ful enough to be able to handle the complicated computations, and the material
involves exceptionally advanced mathematics that even mathematics majors would
normally not see until graduate school. It is not very surprising therefore that in cases
like this most of the advancement here will probably come from mathematicians who
either switch their research interest to computer science or who want to use com-
puters to make advances in their field. Having said that though, I also strongly feel
that there is much that can be explained to a nontopologist, and Chapters 6–8 are an
attempt to do this. A similar comment applies to the algebraic geometry in Chapter
10. It is because of my emphasis on explaining things that I suggested earlier that
mathematics students could also use this book to learn about this material, not just
computer scientists.
With regard to the bibliography, it is fairly small because the book is not addressed
to mathematicians per se. This meant that many good but advanced references that
I could have given, but whose intended audience is research mathematicians, are
omitted. This lack of completeness is partially compensated by the fact that additional
references can be found in the references that are given.
The numbering of items in this book uses the following format: x.y.z refers to item
number z in section y of chapter x. For example, Theorem 6.5.7 refers to the seventh
item of type theorem, proposition, lemma, or example in section 5 of Chapter 6. Algo-
rithm 10.11.1 refers to the first algorithm in Section 11 of Chapter 10. Tables are num-
bered like algorithms. Figures are numbered by chapter, so that Figure 9.21 refers to
the twenty-first figure in Chapter 9. Exercises at the end of chapters are numbered by
section.
Cupertino, California Max K. Agoston
Preface vii
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
1 Linear Algebra Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Inner Product Spaces: Orthonormal Bases . . . . . . . . . . . . . . . . . . . . 7
1.5 Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.7 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.8 Principal Axes Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.9 Bilinear and Quadratic Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.10 The Cross Product Reexamined . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
1.11 The Generalized Inverse Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
1.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2 Affine Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.2 Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.2.1 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.2.2 Rotations in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.2.3 Reflections in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.2.4 Motions Preserve the Dot Product . . . . . . . . . . . . . . . . . . . . 76
2.2.5 Some Existence and Uniqueness Results . . . . . . . . . . . . . . . 79
2.2.6 Rigid Motions in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.2.7 Summary for Motions in the Plane . . . . . . . . . . . . . . . . . . . 85
2.2.8 Frames in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.3 Similarities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
2.4 Affine Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
2.4.1 Parallel Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
2.5 Beyond the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
2.5.1 Motions in 3-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
2.5.2 Frames Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Contents
Contents
3 Projective Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.2 Central Projections and Perspectivities . . . . . . . . . . . . . . . . . . . . . . 127
3.3 Homogeneous Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
3.4 The Projective Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
3.4.1 Analytic Properties of the Projective Plane . . . . . . . . . . . . . . 143
3.4.2 Two-Dimensional Projective Transformations . . . . . . . . . . . . 152
3.4.3 Planar Maps and Homogeneous Coordinates . . . . . . . . . . . . 154
3.5 Beyond the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
3.5.1 Homogeneous Coordinates and Maps in 3-Space . . . . . . . . . 161
3.6 Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
3.6.1 Projective Properties of Conics . . . . . . . . . . . . . . . . . . . . . . . 180
3.7 Quadric Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
3.8 Generalized Central Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
3.9 The Theorems of Pascal and Brianchon . . . . . . . . . . . . . . . . . . . . . . 199
3.10 The Stereographic Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
3.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
4 Advanced Calculus Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
4.2 The Topology of Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . 208
4.3 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
4.4 The Inverse and Implicit Function Theorem . . . . . . . . . . . . . . . . . . 232
4.5 Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
4.6 Morse Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
4.7 Zeros of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
4.8 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
4.9 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
4.9.1 Differential Forms and Integration . . . . . . . . . . . . . . . . . . . . 273
4.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
5 Point Set Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
5.2 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
5.3 Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
5.4 Constructing New Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . 298
5.5 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
5.6 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
5.7 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
5.8 Constructing Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . 313
5.9 The Topology of P
n
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
5.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
6 Combinatorial Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
6.2 What Is Topology? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
6.3 Simplicial Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
6.4 Cutting and Pasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
x Contents
6.5 The Classification of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
6.6 Bordered and Noncompact Surfaces . . . . . . . . . . . . . . . . . . . . . . . . 353
6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
7 Algebraic Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
7.2 Homology Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
7.2.1 Homology Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
7.2.2 Induced Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
7.2.3 Applications of Homology Theory . . . . . . . . . . . . . . . . . . . . 384
7.2.4 Cell Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
7.2.5 Incidence Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
7.2.6 The Mod 2 Homology Groups . . . . . . . . . . . . . . . . . . . . . . . 405
7.3 Cohomology Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
7.4 Homotopy Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
7.4.1 The Fundamental Group . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
7.4.2 Covering Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
7.4.3 Higher Homotopy Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 434
7.5 Pseudomanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
7.5.1 The Degree of a Map and Applications . . . . . . . . . . . . . . . . . 443
7.5.2 Manifolds and Poincaré Duality . . . . . . . . . . . . . . . . . . . . . . 446
7.6 Where to Next: What We Left Out . . . . . . . . . . . . . . . . . . . . . . . . . . 449
7.7 The CW Complex P
n
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
7.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
8 Differential Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
8.2 Parameterizing Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460
8.3 Manifolds in R
n
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
8.4 Tangent Vectors and Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
8.5 Oriented Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483
8.6 Handle Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
8.7 Spherical Modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497
8.8 Abstract Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500
8.9 Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
8.10 The Tangent and Normal Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 519
8.11 Transversality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528
8.12 Differential Forms and Integration . . . . . . . . . . . . . . . . . . . . . . . . . 535
8.13 The Manifold P
n
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548
8.14 The Grassmann Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
8.15 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552
9 Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
9.2 Curve Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558
9.3 The Geometry of Plane Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
9.4 The Geometry of Space Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573
9.5 Envelopes of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
Contents xi
9.6 Involutes and Evolutes of Curves . . . . . . . . . . . . . . . . . . . . . . . . . 583
9.7 Parallel Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
9.8 Metric Properties of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589
9.9 The Geometry of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598
9.10 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620
9.11 Envelopes of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
9.12 Canal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
9.13 Involutes and Evolutes of Surfaces . . . . . . . . . . . . . . . . . . . . . . . 640
9.14 Parallel Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643
9.15 Ruled Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
9.16 The Cartan Approach: Moving Frames . . . . . . . . . . . . . . . . . . . . . 649
9.17 Where to Next? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659
9.18 Summary of Curve Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665
9.19 Summary of Surface Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . 667
9.20 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669
10 Algebraic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674
10.2 Plane Curves: There Is More than Meets the Eye . . . . . . . . . . . . . 677
10.3 More on Projective Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684
10.4 Resultants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690
10.5 More Polynomial Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 695
10.6 Singularities and Tangents of Plane Curves . . . . . . . . . . . . . . . . . 702
10.7 Intersections of Plane Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710
10.8 Some Commutative Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715
10.9 Defining Parameterized Curves Implicitly . . . . . . . . . . . . . . . . . . 724
10.10 Gröbner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728
10.11 Elimination Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745
10.12 Places of a Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747
10.13 Rational and Birational Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 764
10.14 Space Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782
10.15 Parametrizing Implicit Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 786
10.16 The Dimension of a Variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 790
10.17 The Grassmann Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796
10.18 N-dimensional Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797
10.19 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805
Appendix A: Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813
Appendix B: Basic Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817
B.1 Number Theoretic Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817
B.2 Set Theoretic Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818
B.3 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821
B.4 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823
B.5 Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831
B.6 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835
B.7 Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 840
B.8 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847
xii Contents