Can mathematical methods used to design airplanes and rockets and to control them in flight be used to improve the fabrication of parts that form aerospace vehicles? Yes, and this lecture describes one way to do this. Pocket machining is used widely in the aerospace industry to mill metal parts. In this operation, a pocket is excavated a layer at a time by moving the tool along a computed path to remove material. Conventional computer aided manufacturing software packages typically produce a tool path with many rectilinear segments and tight corners whose high curvatures are inherited from a pocket's boundary. This conventional approach is somewhat deficient, especially in today's world of high-speed machining, as high axis drive accelerations can be required to keep tool feed rates high. And if acceleration limits prevent the feed rate limits set by machine spindle horsepower from being achieved, then costly horsepower and feed rate capability are not utilized. A further disadvantage of tight-cornered paths is that if they are used for cutting hard metal like titanium, unnecessary tool wear can result. In this talk, I describe a new approach to pocket machining tool path generation that leads to smooth curvilinear spiral paths. A key component of the approach is the numerical solution of a simple partial differential equation on the pocket region. Once a path is obtained, a constrained trajectory optimization problem is then approximately solved to obtain a feed rate distribution that takes advantage of the path's low curvature. Computational experiments and metal-cutting tests have demonstrated that the approach can save a significant amount of machining time and extend tool life.