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Reviewer: Franz Winkler

Numerical methods in computer-aided geometric design are presented in this easy-to-read book. Affine and projective geometry are introduced in chapter 1, together with translations, rotations, projections, and some problems in visualization, such as removal of hidden vertices or surfaces, shading, and reflection. Chapter 2 deals with parametric curves and surfaces. Curvature and torsion are introduced, and some classical methods of interpolation and approximation are presented. Splines, Be´zier curves and surfaces, and B-splines are the topics of the next chapters. The advantages of the various representations are clearly exhibited. Geometric properties independent of the particular parameterization, such as continuity of curvature or torsion, are also discussed. Chapter 9 reviews the numerics of scattered data interpolation and fitting, leading to bivariate interpolation methods such as Hermite interpolation. Various transformations between different representations of curves and surfaces, such as from a monomial representation to Be´zier form, are described in chapter 10. Chapter 11 generalizes methods from curves and surfaces to higher-dimensional geometric objects. Chapters 12 to 15 cover the intersection of curves and surfaces, as well as topics in smoothing, blending, and offsetting of curves and surfaces. The treatment is purely analytic and numerical, omitting the symbolic algebraic approach to these problems. When symbolic methods are briefly mentioned, such as in implicitization of curves, the treatment is naive. Also missing in these chapters are results in the fast-growing field of research on nonuniform rational B-splines (NURBSs) and their applications in the smoothing of curves. Finally, some practical problems in connection with cutting paths and milling tools are investigated in chapter 16. Proofs of theorems are often omitted and quoted from the literature. This leaves a lot of space for providing valuable intuition. An extensive list of references is provided, but there are no exercises. The weak point of the book is the lack of any in-depth treatment of symbolic algebraic methods in computer-aided geometric design. In particular, such symbolic or combined symbolic-numerical methods would be helpful in parameterization and in the intersection of curves and surfaces. The book is obligatory for anybody who wants an easily accessible and broad overview of numerical methods in computer-aided geometric design.