Generation of Surfaces: Kinematic Geometry of Surface Machining, Third Edition, addresses the critical challenge of optimizing sculptured part surface machining on multi-axis numerical control (NC) machines while minimizing costs. Drawing from over 40 years of research into part surface generation theory, the new edition includes additional information on part surface geometry, expanded appendices, and updated illustrations and formulas presented in a more complete form. This book presents the modern theory of part surface generation with a focus on the kinematic geometry of part surface machining on a multi-axis (NC) machine, and it introduces key methods for applying the DG/K-based approach to part surface generation. Content progresses systematically through three essential parts: basics, fundamentals, and applications of part surface generation. All analytical results are explained mathematically, using a single set of input parameters – the design parameters of the part surface to be machined. Practical applications encompass part surface machining optimization and cutting-tool design, with extensive examples illustrating the most complex theoretical concepts, while the methodology integrates seamlessly with computer-aided design (CAD) and computer-aided manufacturing (CAM) systems, making it immediately applicable to industrial practice. The rigorous mathematical foundation combined with practical applications makes this an indispensable reference for advancing multi-axis machining capabilities. This book is essential for mechanical engineers, manufacturing engineers, CAD/CAM software developers, and researchers who are actively involved in the development, implementation, and optimization of multi-axis NC machining systems. It is particularly valuable for professionals engaged in creating innovative software packages for sculptured part surface machining, as well as for individuals aiming to enhance surface generation processes while simultaneously minimizing manufacturing costs and improving overall efficiency.