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Introduction to the mathematics of subdivision surfaces
Andersson L., Stewart N., Society for Industrial and Applied Mathematics, Philadelphia, PA, 2010. 380 pp.  Type: Book (978-0-898716-97-9)
Date Reviewed: Mar 28 2011

For students in computer science and applied math who deal with computer graphics, solid modeling, and computer-aided design, this book is a must-read. It provides a well-written and comprehensive introduction to the field, explaining the underlying mathematics in a clear and simple manner. The book establishes a hierarchy of subdivision methods, and provides detailed discussions of various methods without being restricted to questions of regularity.

Because parts of the book were used previously as a reference in computer science graduate courses, each chapter ends with some well-crafted exercises. One might expect chapter 1 to be introductory and to provide basic information on topics discussed in subsequent chapters; it does that and much more. The mathematical level of chapter 1 is a bit lower than that of the other chapters. Chapter 1 also contains previews of the topics of later chapters. This unusual approach gives the reader a chance to evaluate his or her own knowledge on the subject, and possibly look for extra information before starting into the book’s detailed, more advanced discussion of methods. I found this approach quite reasonable because many of the topics in the book, such as B-splines, Fourier methods, splitting schema, and subdivision matrices, may not be discussed in other computer science courses; readers have a chance to fill the necessary gaps before reading.

Chapter 2 provides a comprehensive introduction to the central topic of B-spline surfaces, but in terms of subdivision surface methods. As such, the reader should not expect to see classical topics such as nonuniform rational B-splines (NURBS) or the knot insertion algorithm. The book presents B-splines in a slightly unorthodox fashion, since it is based on centered basis functions and the corresponding centered versions of subdivision polynomials introduced in chapter 1.

Chapters 3 and 4 deal with box splines and generalized spline surfaces. Chapter 4 generalizes topics such as the subdivision equation, the nodal-function computation principle, and the polynomial coefficient principle, discussed in previous chapters.

Chapter 5 is one of the most interesting parts of the book. It deals with convergence and smoothness, and discusses the results of convergence for general subdivision polynomial schemes. Chapters 6 and 7 complete the coverage of subdivision surfaces by discussing topics related to the evaluation and estimation of surfaces and shape control.

The notes and appendix at the end of the book are likely to be of great help to the reader. Overall, this is one of the most pleasant, well-organized, and informative books on the subject. Graduate students in computer science will find it useful for their studies, and researchers in fields such as solid modeling and computer graphics will benefit from it, too.

Reviewer:  Alexander Tzanov Review #: CR138933 (1109-0900)
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