CAD of worms and their machining tools

To develop high-quality products, geometric modeling is needed in gear engineering to calculate the workpiece and tool geometry of worms, threads, or similar mechanical elements that can be described by helical or rotatory surfaces and that are to be generated by milling, grinding, or whirling. A survey will be given on the theoretical background, corresponding procedures, and illustrating examples of application concerning: (a) design of gear or worm profiles by means of curve primitives, their motion and manipulation; (b) calculation of conjugate gear profiles subjected to trochoidal motion; (c) calculation of arbitrary plane intersections of helicoids; and (d) calculation of the rotatory machining tool surface for a given worm and the inverse problem, including the solution of the undercut problem. Basic to the achieved integrated approach to CAD/CAM in this field is a discrete curve representation as sequence of points, tangent lines, and osculating circles providing higher geometrical information about curve and surface shape, which is also of high functional and economical importance for technological decisions.