The main problem with this text is that there’s simply too much in it. The authors attempt to include everything that possibly relates to the mathematics and graphics of implicit curves and surfaces; hence, the book suffers from overly superficial coverage of some topics. It reads like a long literature review, rather than a coherent introduction; as such, the potential readership is limited.

Many mathematicians will be disappointed with the cavalier attitude toward some topics. One example is on page 15, where varieties are said to be “defined by equalities.” If varieties are not to be introduced formally, why have them at all? Another example is in the second chapter, a whistle-stop tour of cell complexes and their topology, without enough depth to be mathematically satisfying. The authors have clearly decided to forgo depth for breadth; while I think that this approach may be valid, I also believe that any topic worth introducing is worth introducing well.

Computer scientists, especially graphics specialists, will look in vain for in-depth programming material. The book does contain many algorithm examples and, later on, some graphics examples, as well as a scattering of data structures, but it would have benefited greatly from a chapter devoted entirely to graphics programming.

That being said, I must admit that the authors took on a huge and difficult task, and I admire their courage. Maybe a second edition will ameliorate the problems mentioned above. In fact, the book might be better as two books--one mostly mathematical and the other mostly computational.

The book consists of four parts: “Mathematics and Data Structures,” “Sampling Methods,” “Reconstruction and Polygonization,” and “Designing Complex Implicit Surface Models.” The first part consists of two chapters and crams in so much material that I was left with a feeling of breathlessness. Topics include geometry; functions and mappings; some elementary topology, including manifolds, immersion, and embedding; cell and simplicial complexes; quadtrees and octrees; binary space partitioning (BSP) trees; and kd-trees. Some data structures are given, but no algorithms. This part would have been better had the authors been more selective with their material, treated the included material in more depth, and mentioned the remaining topics only in passing--maybe in an appendix.

The second part investigates root isolation and root-finding methods, as well as interval arithmetic. The authors point out that these are required for sampling, which is the first step in the graphics process--sampling, polygonization, and rendering. This is the best-paced part of the book; it has careful descriptions of standard methods, such as Newton-Raphson, secant, and false-position, along with Descartes’ rule of signs. However, more modern methods such as Brent’s are not discussed, nor are any methods that are optimized for polynomials, such as the Jenkins-Traub algorithm. This is a missed opportunity.

The third part describes algorithms for actually generating implicit curves and surfaces, with various labeling and continuation methods. Such methods work by moving from one point to the next. Different classes of algorithms are discussed, such as integer labeling, vector labeling, predictor-corrector, and enumeration. Algorithms are provided in pseudocode, with some elegant color diagrams to describe different aspects of the way they work. Again, the constraint of providing so much material in a relatively small space means that although all of the algorithms are described mathematically, there is little computational discussion. Some information about the implementation of such algorithms and their effectiveness would have been a welcome addition.

The final part of the text attempts to pull everything together by showing how various surfaces may be modeled, including a tour de force of a particular seashell. However, a simpler example may have been better suited here; the jump from the previous sections to this complex model is considerable and, once again, no implementation advice is provided, aside from a few lines of code that show blending. Incidentally, chapters 10 and 11 include the odd misspelling, such as “phenomenae” for “phenomena.”

The authors claim in their preface that the book is “directed to graduate students, researchers and developers in computer graphics, geometric modeling and computer games.” However, the mathematical nature of the book makes it unapproachable for most. The book is more of an overview of the field; as such, it would be ideal for beginning researchers--it concludes with 429 references, all of which are cited in the text. A computer graphics specialist who needs to learn how to actually implement the various algorithms would need to look elsewhere. It would be a mistake to use the book as a “core textbook,” as the authors hopefully suggest. For one thing, there are no exercises or worked-out examples, and, apart from the second section, there is too much material crammed into too short a space.

This book would be ideal as an introduction to the field for a beginning researcher, or for a graphics specialist who wants some introductory information on a particular algorithm. I don’t think that it would work as a textbook.