On C0 and C1 continuity of envelopes of rotational solids and its application to 5-axis CNC machining

We study the smoothness of envelopes generated by motions of rotational rigid bodies in the context of 5-axis Computer Numerically Controlled (CNC) machining. A moving cutting tool, conceptualized as a rotational solid, forms a surface, called envelope, that delimits a part of 3D space where the tool engages the material block. The smoothness of the resulting envelope depends both on the smoothness of the motion and smoothness of the tool. While the motions of the tool are typically required to be at least $C^2$, the tools are frequently only $C^0$ continuous, which results in discontinuous envelopes. In this work, we classify a family of instantaneous motions that, in spite of only $C^0$ continuous shape of the tool, result in $C^0$ continuous envelopes. We show that such motions are flexible enough to follow a free-form surface, preserving tangential contact between the tool and surface along two points, therefore having applications in shape slot milling or in a semi-finishing stage of 5-axis flank machining. We also show that $C^1$ tools and motions still can generate smooth envelopes.