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
