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Reviewer: Nickolas S. Sapidis

Hosaka's aim is to present (1) a review of basic material from analytic and differential geometry, matrix theory, and numerical analysis used in geometric modeling (chapters 1 to 7 and the appendix); (2) a description of standard methods for defining curves and surfaces (chapters 8 to 12 and 15); and (3) a detailed treatment of various design problems appearing in industrial CAD/CAM (Sections 10.5 and 12.7 and chapters 13, 14, 16, and 17). The first task is accomplished successfully, since exactly one-third of the text is devoted to a detailed review of vector and matrix theory, coordinate transformations, analytic geometry, differential geometry, and numerical methods for solving differential equations. This work is the first geometric modeling book that includes such a comprehensive overview of diverse material ranging from differentiation of vectors and reflection of an object by a mirror to a complete treatment of quadrics and the adaptive Runge-Kutta method. Thus, I expect that many CAD/CAM researchers and developers will use it as a reference book to locate standard results and mathematical formulas. Regarding task (2), the book does not quite live up to my expectations. The basic problem is that the author defines and treats the fundamental concept of a Be´zier curve using a “shift operator” E that “is compact but which may be misleading and may even appear to lack precision,” according to Schwartz [1]. More specifically, in Section 9. 3, the author defines E as the operator that increases or decreases the subscript of an object (that is, E p i =p i+1 , and E -1p i+1 =p i ) and concludes with the vague statement that “we can treat E as an algebraic constant.” If this statement were taken literally, then p i =p j should imply that E p i = E p j or p i+1 =p j+1 [1]. The above discussion and the fact that E is used extensively in this book make it obvious that a more precise definition for E should be provided, most probably along the lines of Schwartz [1]. The book's presentation of current methods in curve and surface design is complete but hard to follow, primarily because the shift operator E lacks the direct geometric insight provided by alternative approaches, such as the de Casteljau algorithm and the Bernstein form of the Be´zier curve [2]. The most interesting aspect of this book is that several practical CAD problems are considered (for example, curve and surface smoothing, rounding of corners, and surface/surface intersection) and solutions that the author has implemented in an industrial environment are described. Although no comparison with published techniques is attempted, several interesting ideas are presented together with comments on implementation details.