Numerically controlled machine tools (lathes and milling machines) are commonly encountered in modern manufacturing processes. They offer the advantages of automation and the production of finished or nearly-finished articles in one operation. The design of these machines, and particularly the software that drives them, is of great practical importance. In the first part of this book, written principally for the software designers, the differential geometry of parametric curves and surfaces is reviewed. The second part shows how this can be applied in various novel ways to the programming of NC machinery. Throughout the book many detailed illustrations help demonstrate theoretical and practical considerations. The book is based on the author's well-known and long-established course on the subject and is fully up-to-date. It will be a valuable resource for graduate students, researchers and and engineers involved in the design of numerically-controlled machinery.