This is the second part of a two-part paper presenting an efficient parametric approach for updating the in-process workpiece represented by the Z-map. With the Z-map representation, the machining process can be simulated by intersecting z-axis aligned vectors with cutter swept envelopes. In this paper the vector-envelope intersections are formulized for the toroidal section of a fillet-end mill which may be oriented arbitrarily in space. For a given tool motion a toroidal surface generates more than one envelope. In NC machining because the torus is considered as one of the constituent parts of a fillet-end mill, only some parts of the torus envelopes, called contact envelopes, can intersect with Z-map vectors. In this paper an analysis is developed for separating the contact-envelopes from the non-contact ones. When a fillet-end mill has an orientation along the vertical z-axis of the Cartesian coordinate system, which happens in 2 1/2 and 3-axis machining, the number of intersections between a Z-map vector and the swept envelope of a toroidal section of the fillet-end mill is maximum one. For finding this single intersection point one of the numerical root finding methods, i.e. bisection, can be applied to the nonlinear function obtained from vector-envelope intersections. On the other hand when a fillet-end mill has an arbitrary orientation, the number of intersections can be more than one and therefore the numerical root finding methods cannot be applied directly. Therefore for addressing those multiple intersections, a system of non-linear equations in several variables, obtained by intersecting a Z-map vector with the envelope surface of the toroidal section of a fillet-end mill, is transformed into a single variable non-linear function. Then developing a nonlinear root finding analysis which guarantees the root(s) in the given interval, those intersections are obtained.

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