Farin is well known as the author of another leading text on curves and surfaces [1]. The publishers are gaining a reputation for publishing authoritative books on computer-aided geometric design. Nonuniform rational B-splines (NURBS) have become a de facto standard for curves and surface descriptions. Thus, this book promises much.

The author claims that the book is suitable as a textbook for students and as a sourcebook for professionals in the field. The background required is simple linear algebra and calculus, and a little knowledge of computer graphics.

The book is written from a point of view based in projective geometry, which is introduced at the beginning. Subsequent chapters cover conics and their parametric form in projective space, followed by a switch to affine space for a discussion of rational quadratic conics and conic splines, including their continuity. These topics are then generalized to rational Bézier curves and the important special case of rational cubics. Projective splines and their continuity and rational splines are introduced next. Eventually, NURBS themselves are described. Moving on to surfaces, subsequent chapters cover rectangular patches, rational Bézier triangles, quadrics and their relation to triangular patches, and  Gregory  patches and Gregory triangles. A rather different last chapter describes IGES formats for NURBS, giving some of the limitations of these formats, and provides examples of commonly useful curves and surfaces described in NURBS form. Exercises are given at the end of each chapter; answers would have been helpful for those using the book for self-study.

For a graphics professional who has no experience of NURBS and needs a rapid introduction to the basics in order to solve practical problems in a short time, this book would be of limited use: the chapter on NURBS is just ten pages long, occurring two-thirds of the way through the book, and would offer little to someone who has not worked through the rest of the material. Taking this point further, I find the “practical use” part of the subtitle somewhat misleading. Few algorithms are explicitly detailed in practical form, and only references to the Oslo algorithm are given (Böhm’s knot insertion method is described instead). Few numerical examples are given. There is no discussion of the relative efficiency or numerical stability of different computational approaches. Other topics that are of practical interest but are not addressed include interrogation of NURBS curves and surfaces (for example, the computation of intersections), trimmed NURBS patches, and cyclides (used for blending).

Nevertheless, the book is strongly recommended to its intended audience. For a postgraduate student who can study it from cover to cover, or for someone who already has a fair knowledge of NURBS, this book provides a systematic and insightful development of its subject from the point of view of projective geometry. Projective curves and surfaces are not developed from their affine counterparts, as in most texts, but are treated as the fundamental idea. A minor criticism is that it does not become clear until later in the book how the projective geometry given earlier is going to be fundamentally useful.

Particular topics of interest include knot insertion, degree elevation and reduction, blossoms, base points, duality, and conversion of rational forms to parametric ones, although the treatment of most of these topics is brief. More than 100 references make the book useful as a pointer to the literature. The author’s style is lucid, while concise, and is generally easy to read. This book gives a deeper insight into how NURBS work than do many other texts.