Vector-valued envelope functions for constructing tool swept surfaces in continuous domains

This paper proposes the vector-valued envelope functions to describe form cutters’ swept profiles at the choice of particular motion instances. The general idea behind such a formulation is to embed the tangency constraint into the tool surface expression and form a single continuous function to describe the segmental curves whose elements lie on the envelopes. This results in skipping unnecessary calculations when no envelope-surface point is associated with the specified input data, reducing the sizeable computational burden due to repetitive constraint implementations, and most importantly, depending on the tool kinematics’ complexity, avoiding the tangency-constraint violations between successive envelope-surface points where the exact fit cannot be obtained. In this study, firstly, the NC tool surface models present in extant literature have been restructured using the four-parameter set of spheres to attain the required one-to-one correspondence between the domain and range sets of these functions, followed by excluding the tangency-constraint evaluations by introducing the systematic parameter-reduction procedures that led to the development of constraint-embedded envelope functions. Next, we introduced the branch-existence test, which allowed us to check whether these functions are continuous over closed domain intervals. Finally, we covered algorithms for implementing the functions.