Curves From Motion Motion From Curves

Geometry and kinematics have been intimately connected in their historical evolution and, although it is currently less fashionable, the further development of such connections is crucial to many computer-aided design and manufacturing applications. In this survey, we explore a variety of classical and modern problems that illustrate how simple rules of motion produce interesting curves and, conversely, the computational problems of generating motions with prescribed paths and speeds. These encompass the geometry of trajectories under centripetal forces; the transformation of rotary motion into motion along general curves by mechanisms; real-time curve interpolators for digital motion control; and the description of spatial motions that involve variations of both position and orientation. Such case studies illustrate some of the intellectual appeal, and practical importance, of a sustained dialog between the study of curves and of motions.