In the domain of multi-axis NC machining of sculptured surface parts, the use of orthogonal parameterizations of part and tool surfaces is convenient because it simplifies the transformation of coordinate systems. Using the so-called “differential-geometric method of sculptured surface NC machining,” developed by one of the authors, many parameterizations of part and tool surfaces are easily shown not to be orthogonal. To transform nonorthogonal part and tool surface parameterizations into orthogonal ones, the Jacobian of the transformation may be used. In cases when the Jacobian of the transformation is not known, it is possible to use differential equations for isogonal trajectories on the surfaces (choosing an orthogonal case), or a special kinematic method for obtaining sculptured surface equations. Influences of coordinate system transformations (translations and rotations along and about axes through the origin) on example part and tool surface parameterizations for four types of general helicoidal surfaces are described. The results mentioned above simplify the analytical description of the multi-axis NC machining process, and may be useful for writing NC toolpath generation software.

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