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Reviewer: Sandu Crivineanu

Based on courses given at the University of Waterloo and the University of California, Berkeley, as well as the Graphics Interface '84 and ACM SIGGRAPH '84, '85, and '86 tutorials, the book relates the experience gained by the authors in developing the mathematical basis for algorithms that can sustain more versatile computer programs for modeling objects and creating realistic images. The spline functions may be considered as a child of approximation theory. However, this book deals with the use of splines for shape approximation from the point of view of the geometric design community. The flavor of this book can be expressed in an outline of its chapters: :9N(1)Introduction (2)Preliminaries (3)Hermite and Cubic Spline Interpolation (4)A Simple Approximation Technique—Uniform Cubic B-splines (5)Splines in a More General Setting (6)The One-Sided Basis (7)Divided Differences (8)General B-splines (9)B-spline Properties (10)Be´zier Curves (11)Knot Insertion (12)The Oslo Algorithm (13)Parametric vs. Geometric Continuity (14)Uniformly-shaped Beta-splines (15)Geometric Continuity, Reparametrization, and the Chain Rule (16)Continuously-Shaped Beta-splines (17)An Explicit Formulation for Cubic Beta-splines (18)Discretely-Shaped Beta-splines (19)B-spline Representations for Beta-splines (20)Rendering and Evaluation (21)Selected Applications The index is extensive and valuable, and there is a copious list of 117 references to related and germinal works. The authors have done an excellent job in presenting the material, using many examples, 10 numerical tables, 268 complex figures and 17 color plates, with the fair and declared purpose of building the reader's intuition and feeling. For example, “we will plot a few low-order discrete B-splines to gain some feeling for them” (p. 269). The subtle difference between parametric and geometric continuity is discussed with clarity and elegance. The treatment of beta-splines is rigorous. The control of shape parameters, called bias and tension, is presented in detail. It provides a good working basis and a stimulus to researchers interested in the advancement of shape control and approximation of curves and surfaces. The beta-spline techniques are now seen as a strong potential basis for an important new generation of algorithms in computer graphics and geometric modeling. Not only the authors' laboratories (the Computer Graphics Laboratory at the University of Waterloo and the Berkeley Computer Graphics Laboratory) but an increasing number of computer graphics research laboratories at universities around the world have already begun to use beta-spline algorithms in design and modeling programs. The last chapter, though I find it too short, has carefully selected applications to animation, picture processing, interpolation, last-squares approximation, and free-form curve input, enabling the new user to become productive quickly. If there is a fault in this book it is that it is designed to be read from cover to cover. I must note that the textbook follows the commendable pattern of lively, friendly writing. The book is highly recommended as essential reading for anyone interested in the subject. It is certainly upon this solid foundation that future work will be built.