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Reviewer: Andrew Donald Booth

Computer-aided geometric design is assuming ever-greater importance with the almost universal use of computer-aided manufacturing in industry. A central problem with numerically controlled machine tools is the storage of the data that are to control their operations. When the two- or three-dimensional path to be followed by the cutting tool can be specified by a simple algebraic formula, this may not be difficult but, for an artifact such as the hood of an automobile, the profile of the required press tool is complex and has been designed by an artist rather than by a mathematician. The book is directed to this latter class of object. The original designers used French curves as an aid to two-dimensional curve drawing, and the book points out that, as early as 1959, P. de Casteljau, working for the Citroen Company in Paris, produced an internal report that described the “Be´zier” curve interpolation technique later announced by P. Be´zier of Citroen. Be´zier has provided a historical introduction to this book in which he describes some of the origins of his version of the method. Much of the underlying theory of Be´zier curve fitting derives from that of Bernstein polynomials, and the first few chapters develop all of the relevant theory, particularly as applied to two-dimensional fitting. The text then considers the far more complex problem of surface representation by optimally small sets of generalized polynomials. Generally, apart from questions of stability, no real difficulties occur except in the case of sharp corners and similar discontinuities. The case of the automobile hood and its fairing into the wing structures is a prime example. This matter receives extensive discussion and leads to chapters on Coons patches and their extensions. A readable chapter covers differential geometry as it applies to surfaces and space curves, and the author provides a discussion on smoothing. The main text ends with the author's evaluation of the methods discussed and his recommendations as to selection and implementation. This material is especially valuable because of his previous involvement as a research mathematician with Daimler-Benz. The book ends with a short list of technical terms and definitions. The monumental bibliography lists 480 references, but many of these are to obscure and hard-to-find sources such as theses and internal corporate reports, and the index is excellent. The illustrative diagrams and photographic plates are extensive, and each chapter ends with a set of exercises and a guide to software. The author and publisher have included a floppy disk that contains C source programs for all of the major algorithms in the book. These programs can be used to help solve the example problems or, for the designer, in actual industrial situations. It is worth noting that the C programs produce graphic output as Postscript files, which require a Postscript-compatible printer to view. For those without such a device, it is worth knowing that the program Ghostscript can display Postscript files as well as reproducing them on standard laser printers. It is available as freeware. I strongly recommend the book for classroom use or as a design handbook. The industrial designer should be aware that it requires a good grasp of mathematics.