This two-part paper presents an efficient parametric approach to updating workpiece surfaces represented by the Z-map vectors. The methodology is developed for up to 3 1/2 1/2-axis machining in which a tool can be arbitrarily oriented. In calculations the Automatically Programmed Tool (APT)-type milling cutters represented by the natural quadrics, planar and the toroidal surfaces are used. The machining process is simulated through calculating the intersections between the Z-map vectors and the tool envelope surface which is modeled by using a tangency function. Part 1 of this two-part paper presents the methodology for the cutters with natural quadrics and planar surfaces. For those surfaces intersection calculations are performed analytically. The geometric complexity of a torus is higher than those of the natural quadric and planar surfaces. Furthermore if the torus has an arbitrary orientation then the intersection calculations for the torus present great difficulties. In NC machining typically a torus is considered as one of the constituent parts of a cutter. In this case only some parts of the torus envelopes, called contact-envelopes, can intersect with Z-map vectors. For this purpose in Part 2 of this two-part paper an analysis is developed for separating the contact-envelopes from the non-contact envelopes. Then a system of non-linear equations in several variables, obtained from intersecting Z-map vectors with contact envelopes, is transformed into a single variable non-linear function. Later using a nonlinear root finding analysis which guarantees the root(s) in the given interval, those intersections are addressed.

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