Geometry and tool motion planning for curvature adapted CNC machining

CNC machining is the leading subtractive manufacturing technology. Although it in use since decades, far from fully solved and still a rich source for challenging problems geometric computing. We demonstrate this at hand of 5-axis freeform surfaces, where degrees freedom selecting moving cutting tool allow one to adapt motion optimally surface be produced. aim high-quality finish, thereby reducing need hard-to-control post-machining processes such as grinding polishing. Our work based on careful analysis curvature-adapted via so-called second order line contact between target surface. On side, leads new continuous transition "dual" classical results theory concerning osculating circles curves cones tangentially circumscribed developable surfaces. Practically, serves an effective basis planning. Unlike previous approaches machining, we solve locally optimal positioning planning within single optimization framework achieve curvature adaptation even convex This possible with toroidal cutter that contains negatively curved area. The effectiveness our approach verified digital models, simulations machined parts, including comparison generated commercial software.